Mahlerʼs Guide to. Financial Economics. Joint Exam MFE/3F. prepared by Howard C. Mahler, FCAS Copyright 2013 by Howard C. Mahler.

Transcription

1 Mahlerʼs Guide to Financial Economics Joint Exam MFE/3F prepared by Howard C. Mahler, FCAS Copyright 2013 by Howard C. Mahler. Study Aid 2013-MFE/3F Howard Mahler

2 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 1 Mahlerʼs Guide to Financial Economics Copyright 2013 by Howard C. Mahler. Concepts in Derivatives Markets by Robert L. McDonald are demonstrated. Information in bold or sections whose title is in bold are more important for passing the exam. Larger bold type indicates it is extremely important. Information presented in italics (and sections whose titles are in italics) should not be needed to directly answer exam questions and should be skipped on first reading. It is provided to aid the readerʼs overall understanding of the subject, and to be useful in practical applications. Highly Recommended problems are double underlined. Recommended problems are underlined. 1 Solutions to the problems in each section are at the end of that section. Section # Pages Section Name Introduction European Options Properties of Premiums of European Options Put-Call Parity Bounds on Premiums of European Options Options on Currency Exchange Options Futures Contracts Synthetic Positions American Options Replicating Portfolios Risk Neutral Probabilities Utility Theory and Risk Neutral Pricing Binomial Trees, Risk Neutral Probabilities Binomial Trees, Valuing Options on Other Assets Other Binomial Trees Binomial Trees, Actual Probabilities Jensen's Inequality Normal Distribution LogNormal Distribution The Table of Contents is continued on the next page. 1 Note that problems include both some written by me and some from past exams. The latter are copyright by the Society of Actuaries and the Casualty Actuarial Society and are reproduced here solely to aid students in studying for exams. The solutions and comments are solely the responsibility of the author; the SOA and CAS bear no responsibility for their accuracy. While some of the comments may seem critical of certain questions, this is intended solely to aid you in studying and in no way is intended as a criticism of the many volunteers who work extremely long and hard to produce quality exams.

3 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 2 Section # Pages Section Name Limited Expected Value A LogNormal Model of Stock Prices Black-Scholes Formula Black-Scholes, Options on Currency Black-Scholes, Options on Futures Contracts Black-Scholes, Stocks Paying Discrete Dividends Using Historical Data to Estimate Parameters of the Stock Price Model Implied Volatility Histograms Normal Probability Plots Option Greeks Delta-Gamma Approximation Option Greeks in the Binomial Model Profit on Options Prior to Expiration Elasticity Volatility of an Option Risk Premium of an Option Sharpe Ratio of an Option Market Makers Delta Hedging Gamma Hedging Relationship to Insurance Exotic Options Asian Options Barrier Options Compound Options Gap Options Valuing European Exchange Options Forward Start Options Chooser Options Options on the Best of Two Assets Cash-or-Nothing Options Asset-or-Nothing Options Random Walks Standard Brownian Motion Arithmetic Brownian Motion Geometric Brownian Motion Geometric Brownian Motion Model of Stock Prices Ito Processes Ito's Lemma The Table of Contents is continued on the next page.

4 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 3 Section # Pages Section Name Valuing a Claim on S^a Black-Scholes Equation Simulation Simulating Normal and LogNormal Distributions Simulating LogNormal Stock Prices Valuing Asian Options via Simulation Improving Efficiency of Simulation Bonds and Interest Rates The Rendelman-Bartter Model The Vasicek Model The Cox-Ingersoll-Ross Model The Black Model Interest Rate Caps Binomial Trees of Interest Rates The Black-Derman-Toy Model Important Formulas and Ideas My practice exams are sold separately.

5 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 4 Chapter of Derivative Markets Sections of Study Guide , 12, 14, , 13, 16-17, 27, , 28, 31, , , 27, , 28, Appendix B.1 1 Appendix C 18 I have included in my early sections, the 9 questions from the 2007 FM Sample Exam for Derivatives Markets, based on earlier chapters of the textbook. Unless otherwise stated chapter appendices are not included in the required readings from this text. 2 Excluding Options on Commodities on page Including Appendices 11.A and 11.B. 4 Including Appendix 12.A. 5 Including Appendix 13.B. 6 Sections (up to but excluding Multivariate Itôʼs Lemma on pages ) and 20.7 (up to but excluding Valuing a Claim on S a Q b on pages and excluding Finding the lease rate on top one-half of page 669). 7 Sections (excluding What If the Underlying Asset Is Not and Investment Asset on pages ) and 21.3 (excluding The Backward Equation on pages , and excluding the paragraph on page 692 that begins If a probability and through the end of the section). 8 But with only those definitions in Tables 22.1 and 22.2 that are relevant to Section Up to but excluding Exponentially Weighted Moving Average on page 746 and through the end of the section. 10 Up to but excluding Forward rate agreements on pages

6 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 5 This study guide covers all of the material on the Joint Exam: CAS 3F and SOA MFE. 11 The syllabus consists of various sections of the 2nd edition of Derivative Markets by Robert L. McDonald, plus a short study note Some Remarks On Derivative Markets by Elias S. W. Shiu. 12 Unless stated otherwise in a question assume: The market is frictionless. There are no taxes, transaction costs, bid/ask spreads, or restrictions on short sales. All securities are perfectly divisible. Trading does not affect prices. Information is available to all investors simultaneously. Every investor acts rationally (i.e., there is no arbitrage.) The risk-free rate is constant The notation is the same as used in Derivative Markets by Robert L. McDonald. The MFE/3F CBT exam will provide a formula document as well as a normal distribution calculator that will be available during the test by clicking buttons on the item screen. Details are available on the Prometric Web Site. Similar to other exam reference buttons, the normal distribution calculator button will be available throughout the exam in the top right corner of every item screen. Click the button to call up the calculator and calculate cumulative normal distribution and inverse cumulative normal distribution values. Use these values to answer the question as needed. When using the normal distribution calculator, values should be entered with five decimal places. Use all five decimal places from the result in subsequent calculations. The normal distribution calculator button replaces the Normal Table. The previous rule on rounding no longer applies. 13 You can try the normal distribution calculator button at the Prometric Web Site. You will benefit from using it at least part of the time when you are studying. The formula sheet contains the same information about the Normal and LogNormal distributions as was provided in the past. 11 In 2007 the CAS and SOA gave separate exams. Starting in 2008 they gave a joint exam. 12 The study note is available on the CAS and SOA webpages. 13 Unfortunately, my solutions were written up using the prior rule: On Joint Exam 3F/MFE, when using the normal distribution, choose the nearest z-value to find the probability, or if the probability is given, choose the nearest z-value. No interpolation should be used. For example, if the given z-value is 0.759, and you need to find Pr(Z < 0.759) from the normal distribution table, then chose the probability for z-value = 0.76: Pr(Z < 0.76) = This should not make a significant difference.

7 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 6 Changes to the Syllabus from 2007: 14 Section 12.6 of Derivative Markets, Perpetual American Options, is no longer on the syllabus. 15 The final portion of Section 20.6, Multivariate Itoʼs Lemma, is no longer on the syllabus. 16 The final portion of Section 24.5, Forward Rate Agreements, is no longer on the syllabus. Added, the first portion of Section 20.7, Valuing a Claim on S a, up to but excluding the last subsection on Valuing a Claim on S a Q b. Changes to the Syllabus for Spring 2009: Chapter 10 of Derivative Markets, exclude Options on Commodities on page 334. Exclude Section 11.5 of Derivative Markets, on Binomial Trees, Discrete Dividends. Add Appendices 11.A and 11.B of Derivative Markets. Add Appendix 13.B of Derivative Markets. Chapter 20 of Derivative Markets: exclude Finding the lease rate on top one-half of page 669. Add parts of Chapter 21: Sections (excluding What If the Underlying Asset Is Not and Investment Asset on pages ) and 21.3 (excluding The Backward Equation on pages , and excluding the paragraph on page 692 that begins If a probability and through the end of the section). Add parts of Chapter 22: Section 22.1 (but with only those definitions in Tables 22.1 and 22.2 that are relevant to Section 22.1.) Add parts of Chapter 23: Sections (up to but excluding Exponentially Weighted Moving Average on page 746 and through the end of the section.) Add Appendix B.1. Add Appendix C. Changes to the Syllabus for Fall 2009: 17 Add Chapter 18 of Derivative Markets, about the LogNormal Stock Price Model. Add Chapter of Derivative Markets, about Monte Carlo Valuation, in other words simulation. Changes for 2011: Computer based testing. 3 hours and approximately 30 questions. 14 Starting in 2008 this is a joint exam, 3F/MFE. 15 In 2007 Section 12.6 was on SOA MFE, but not CAS In 2007 this final portion of Section 20.6 was on CAS 3, but not SOA MFE. 17 Material was moved from Exam 4/C onto Exam 3F/MFE. Exam 3F/MFE was extended from 2 hours to 2.5 hours and will consist of approximately 25 multiple-choice questions.

8 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 7 Exam Questions by Section of This Study Guide: MFE CAS 3 MFE CAS 3 MFE/3F Section Section Name Sample 5/07 5/07 11/07 5/09 1 Introduction 25 2 European Options 3 Properties of Premiums of Euro. Options 2 4 Put-Call Parity 1 3, 4 1, 4 14, Bounds on Premiums of Euro. Options 6 Options on Currency Exchange Options 8 Futures Contracts 9 Synthetic Positions American Options Replicating Portfolios Risk Neutral Probabilities Utility Theory and Risk Neutral Pricing 14 Bin. Trees, Risk Neutral Probs. 4, 49 15, , 19, Binomial Trees, Options on Other Assets 5, Other Binomial Trees Binomial Trees, Actual Probabilities Jensen's Inequality 19 Normal Distribution 20 LogNormal Distribution 21 Limited Expected Value 22 A LogNormal Model of Stock Prices Black-Scholes Formula 6 20, 21 3, Black-Scholes, Options on Currency Black-Scholes, Options on Futures Black-Scholes, Discrete Dividends Historical Data to Estimate Parameters 17, Implied Volatility 29 Histograms 30 Normal Probability Plots The SOA did not release its 11/07 exam MFE. The CAS/SOA did not release the 5/08, 11/08,11/09, and subsequent exams 3F/MFE. Continued on the next page

9 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 8 MFE CAS 3 MFE CAS 3 MFE/3F Section Section Name Sample 5/07 5/07 11/07 5/09 31 Option Greeks 8, Delta-Gamma Approximation Option Greeks in the Binomial Model 44, 45, Profit on Options Prior to Expiration Elasticity 20, Volatility of an Option 5 37 Risk Premium of an Option 38 Sharpe Ratio of an Option Market Makers 40 Delta Hedging 9, 47, Gamma Hedging Relationship to Insurance 43 Exotic Options 44 Asian Options Barrier Options Compound Options Gap Options Valuing European Exchange Options 49 Forward Start Options 19, Chooser Options Options on the Best of Two Assets Cash-or-Nothing Options 28, Asset-or-Nothing Options 4 54 Random Walks 55 Standard Brownian Motion Arithmetic Brownian Motion 57 Geometric Brownian Motion 58 Geo. Brown. Mot. Model Stock Pr. 10, 11, 32, , Ito Processes 12, 23, 48, , 18 63, 66, 67, Ito's Lemma 13, 24, 35, , 68, 73 Continued on the next page

10 2013-MFE/3F, Financial Economics, HCM 12/6/12, Page 9 MFE CAS 3 MFE CAS 3 MFE/3F Section Section Name Sample 5/07 5/07 11/07 5/09 61 Valuing a Claim on S^a 16, 62, 71, Black-Scholes Equation Simulation 64 Simulating Normals & LogNormals 65 Simulating LogNormal Stocks Valuing Asian Options via Sim. 67 Improving Efficiency of Simulation 57, 58, 59, Bonds and Interest Rates The Rendelman-Bartter Model 70 The Vasicek Model 14, The Cox-Ingersoll-Ross Model 21, 38, The Black Model 7 73 Interest Rate Caps 3 74 Binomial Trees of Interest Rates 5 75 The Black-Derman-Toy Model 15, 29, 30, Questions no longer on the syllabus: MFE, 5/07, Q. 16. In August 2010, the SOA/CAS updated the file of MFE Sample Exam questions. There are now a total of 76 sample questions. Check the SOA webpage to see if any additional Sample Exam questions have been added. Valuing a Claim on S a was added to the syllabus in Spring Material on simulation was moved here from exam 4/C in Fall 2009.

11 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 10 Section 1, Introduction Earlier Chapters of Derivative Markets by McDonald are on Exam 2/FM. 18 Some of the ideas covered in those chapters are used in the chapters on your exam. Derivatives: 19 A derivative is an agreement between two people that has a value determined by the price of something else. For example, Alan gives Bob the right to buy from Alan a share of IBM stock one year from now at a price of $120. This is an example of a stock option. The value of this option depends on the price of IBM stock one year from now. Options: A call is an option to buy. For example, Bob purchased a call option on IBM stock from Alan. A put is an option to sell. For example, if Debra purchased a put option on IBM stock from Carol, then Debra will have the option in the future to sell a share of IBM stock to Carol at a specified price. Continuously Compounded Risk Free Rate: 20 If r is the continuously compounded annual risk free rate, then the present value of $1 T years in the future is: e -rt. r as used by McDonald is what an actuary would call the force of interest. Effective Annual Rate: 21 If r is the effective annual risk free rate, then the present value of $1 T years in the future is: 1/(1+r) T. An effective annual rate is what an actuary would call the rate of interest. Effective annual rate will be used in Interest Rate Caps and the Black-Derman-Toy Model, to be discussed in subsequent sections. Otherwise, we will use continuously compounded rates. 18 2nd edition, Sections , , Appendix 2.A, , , , Appendix 5.B, Warren E. Buffett has said, Derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal. 20 See Appendix B.1 of Derivative Markets by McDonald. 21 See Appendix B.1 of Derivative Markets by McDonald.

12 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 11 Selling Short: If we sell a stock short, then we borrow a share of stock and sell it for the current market price. We will give this person a share of stock at the designated time in the future. We also must pay this person any stock dividends they would have gotten on the stock, when they would have gotten them. Forward Contracts: A forward contract is an agreement that sets the terms today, but the buying or selling of the asset takes place in the future. For example, Ed will be moving in a month, and his friend Fred agrees to buy Edʼs TV one month from now for $200. The purchaser of an option has bought the right to do something in the future, but has no obligation to do anything. In contrast, in a forward contract both parties are obligated to fulfill their parts of the contract. Value of a Forward Contract: F 0,T = forward price at time T in the future. For example, if Joe buys a forward contract to buy one share of ABC stock in two years at $120, then F 0,2 = $120. At time 2 years, Joe pays $120 and gets one share of stock. 22 PV[F 0,T ] is the present value at time 0 of a forward contract to be executed at time T. PV[F 0,T ] = F 0,T e -rt. Let us assume the current price of XYZ stock is S 0. Assume XYZ stock pays no dividends. Charlie can buy a forward contract to buy one share of XYZ stock in exchange for paying F 0,T at time T. If Charlie invests F 0,T e -rt at the risk free rate, then at time T he will have F 0,T. 23 He uses that amount to fulfill his forward contract and at time T Charlie has one share of XYZ Stock. Lucy can instead buy one share of XYZ stock now, for the current market price of S 0, and hold onto the share of stock until at least time T. 22 This differs from the prepaid price. Joe might instead be able to pay $110 now and get a share of stock 2 years from now. This is an example of a prepaid futures contract. 23 Charlie could invest in a Treasury Bond.

13 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 12 Both Lucy and Charlie end up in the same situation, with one share of XYZ stock at time T. Therefore, their investments must have equal present value. S 0 = F 0,T e -rt. F 0,T = S 0 e rt, in the absence of dividends. If instead XYZ stock pays dividends, then Lucy would have collected any dividends paid from time 0 to T, while she owned the stock. Charlie would not. Thus Lucyʼs position is equal to Charlieʼs position plus a receipt of dividends. Therefore, S 0 - PV[Div] = F 0,T e -rt. F 0,T = S 0 e rt - PV[Div] e rt. If the dividends are paid at discrete points in time, with amount D t i paid at time t i, then F 0,T = S 0 e rt - e r(t - t i) Dt i. 24 Exercise: The current price of a stock is $100. It will to pay a dividend 3 months from now, a dividend 6 months from now, a dividend 9 months from now, and a dividend 12 months from now. Each dividend is of size $2. r = 6%. Determine the the forward price for a share of stock one year from now [Solution: F 0,1 = (100) e.06 - (2)(e e.03 + e e 0 ) = $ Comment: Both sides of the equation are valued one year from now.] Futures Contracts: A futures contract is similar to a forward contract except: A futures contract is typically traded on an exchange. A futures contract is marked to market periodically. 25 The buyer and the seller post margin See Equation 5.7 in Derivative Markets by McDonald. 25 Marked-to-market means the item is revalued to reflect current market prices. 26 A deposit which compensates the other party to a futures contract in case one of the parties does not fulfill its obligation.

14 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 13 Forward Contracts Versus Futures Contracts: 27 Forward Contract Futures Contract Type of Market Dealer or Broker (Commodities) Exchange Liquidity Low High Contract Form Customized Standard Performance Guarantee Creditworthiness Mark-to-Market Transaction Costs Bid-ask spread Fees or Commissions Continuous Dividends: We often assume that dividends are paid at a continuous rate δ. 28 Over a short period of time dt, stock dividends of: δ S(t) dt are paid, where S(t) is the stock price at time t. So that if one buy a share of stock at time 0, and reinvests the dividends in the stock, at time T one would have e Tδ shares of the stock. 29 Exercise: One buys 1 million shares of a stock that pays dividends at the continuous annual rate of 2%. The dividends are reinvested in that stock. After 3 years how many shares of the stock does one own? [Solution: (1 million)e (3)(.02) = 1,061,837 shares.] If instead of discrete dividends XYZ stock pays continuous dividends, then Lucy would have e Tδ shares of the stock at time T. If Charlieʼs future contract were for e Tδ shares of the stock, then his position would be equal to Lucyʼs. F 0,T e Tδ = S 0 e rt. Therefore, in the case of dividends paid continuously: F 0,T = S 0 e T(r - δ). 30 Prepaid Forward Price: The forward price is the price we would pay in the future for a forward contract. In contrast, the prepaid forward price, F P 0,T, is the price we would pay today for a forward contract. F P 0,T = F0,T e -rt. 27 Taken from Table 2.2 Financial Economics, Harry H. Panjer, editor. 28 This is a good approximation for a stock index fund. 29 δ acts similarly to a force of interest. 30 See Equation 5.8 in Derivative Markets by McDonald. This is like the accumulated value for a force of interest.

15 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 14 For example, let us assume we are paying today in order to own a share stock at time 3. Then the prepaid forward price is F P 0,3 (S). We would pay this price at time 0 in exchange for receiving the stock at time 3. However, we would not receive any dividends the stock would pay between time 0 and 3. Therefore, in the case of discrete dividends, F P 0,T (S) = S0 - PV[Div]. Exercise: The current price of a stock is 120. The stock will pay a dividend of 3 in 2 months. What is the 5 month prepaid forward price of the stock? r = 6%. [Solution: S 0 - PV[Div] = 120-3e -(2/12)(6%) = ] In the case of continuous dividends, F P 0,T (S) = S0 e -δt. Exercise: The current price of a stock is 80. The stock pays dividends at a continuous rate of 1%. What is the 5 month prepaid forward price of the stock? [Solution: S 0 e -δt = 80e -(5/12)(1%) = ] If we pay S 0 e -δt in order to buy e -δt shares of stock today and reinvest the dividends we would have one share of stock at time T. Thus S 0 e -δt is the price we would pay today to own one share of stock at time T. In the case of continuous dividends, somewhat more generally, F P t, T (S) = St e -δ(t-t).

16 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 15 Named Positions: 31 One can buy various combinations of options and stock. The more common such positions have been given names. Bear Spread: The sale of an option together with the purchase of an otherwise identical option with a higher strike price. Can construct a bear spread using either puts or calls. The owner of the Bear Spread hopes that the stock price moves down. Box Spread: Buy a call and sell a put at one strike price, plus at another (higher) strike price sell a call and buy a put. 32 Bull Spread: The purchase of an option together with the sale of an otherwise identical option with a higher strike price. Can construct a bull spread using either puts or calls. The owner of the Bull Spread hopes that the stock price moves up. Butterfly Spread: Buying a K strike option, selling two K + ΔK strike options, and buying a K + 2ΔK strike option. Collar: Purchase a put and sell a call with a higher strike price. Ratio Spread: Buying m of an option and selling n of an otherwise identical option at a different strike. Straddle: Purchase a call and the otherwise identical put. Strangle: The purchase of a put and a higher strike call with the same time until expiration. For example, Gene Green buys a Straddle with K = 80. He buys an 80-strike call and a similar 80-strike put. His payoff at expiration is: Max[0, S T - 80] + + Max[0, 80 - S T ] = S T The further the stock price at expiration is from 80, the larger Geneʼs payoff. Gene is hoping there is a large movement in the stock price See Chapter 3 of Derivative Markets by McDonald, on the syllabus of Exam 2/FM. 32 For European options, the box spread is equivalent ot a zero-coupon bond. 33 In other words, Gene is betting that the stockʼs volatility is high. In contrast, the seller of a straddle is betting that the stockʼs volatility is low.

17 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 16 For example, Vanessa buys a 90 strike call and sells an otherwise identical 100 strike call. This is an example of a Call Bull Spread. Vanessa hopes the stock price increases. If Vanessa bought her 90 strike call from Nathan and sold her 100 strike call to Nathan, than Nathan owns a Call Bear Spread. Nathan hopes the stock price declines. Long and Short Positions: Entering into a long position is buying. Entering into a short position is selling or writing. For example if you long one call option and long the similar put option, then you bought the call and put, and you have purchased a straddle. If instead you short a call option and the similar put option, then you have written (sold) a straddle. If you short a 60-strike 3-month call and long a 80-strike 3-month call, then you have purchased a Bear Spread.

18 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 17 Actuarial Present Values: 34 Let us assume that one year from now an insurer will pay either $50 with probability 70% or $100 with probability 30%. Then the expected payment in one year is: (0.7)(50) + (0.3)(100) = $65. Assume that the continuously compound annual rate of interest is now 5%. Then the actuarial present value of the insurerʼs payment is: 65 e = $ In general in order to calculate an actuarial present value, one takes a sum of the expected payments at each point in time each multiplied by the appropriate discount factor. The discount factor adjusts for the difference between the time value of money at the present and at the time when the payment is made. Exercise: In addition to the payments one year from now, the insurer will pay two years from now either $50 with probability 50%, $100 with probability 40%, or $200 with probability 10%. Assume that one year from now the continuously compound annual rate of interest will be 6%. Determine the actuarial present value of the insurerʼs total payments, including those made one year from now and two years from now. [Solution: The expected payment in two years is: (0.5)(50) + (0.4)(100) + (0.1)(200) = $85. Discounting back to the present: 85 exp[ ] = $ Adding in the actuarial present value of the payments made in one year, the actuarial present value of the insurerʼs total payments is: $ $76.15 = $ ] 34 Covered extensively on CAS Exam 3L and SOA Exam MLC. 35 If instead the 5% were an effective annual rate, then the actuarial present value would be: 65/1.05 = $61.90.

19 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 18 Problems: 1.1 (1 point) A stock price is 160. Assume r = 0.08 and there are no dividends. What is the 4-year forward price? A. less than 200 B. at least 200 but less than 210 C. at least 210 but less than 220 D. at least 220 but less than 230 E. at least (1 point) A stock has a current price of 120. The stock pays dividends at a continuously compounded rate of 1.5%. r = What is the 4-year prepaid forward price? A. 113 B. 115 C. 117 D. 119 E (1 point) A stock has a two-year forward price of The stock pays dividends at a continuously compounded rate of 3%. r = 7%. What is the current price of this stock? A. 90 B. 92 C. 94 D. 96 E (1 point) A stock has a current price of 90. The stock pays dividends at a continuously compounded rate of 2%. r = What is the 5-year forward price? A. 100 B. 105 C. 110 D. 115 E (1 point) A stock has a current price of $100. In 3 months the stock will pay a dividend of $2. r = What is the 4-month prepaid forward price? 1.6 (1 point) A stock has a four-year forward price of The stock pays dividends at a continuously compounded rate of 0.8%. r = 5.2%. What is the current price of this stock? A. 65 B. 70 C. 75 D. 80 E (1 point) Options are extremely risky investments. The variance of returns is great, yet most people are assumed to be risk-averse. Moreover, brokerage commissions on options are high. So why are options and other derivative securities such popular financial instruments?

20 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page (CAS5B, 11/93, Q. 31) (2 points). a. (1.5 points) List and briefly describe three examples of derivative instruments. b. (0.5 points) Why do firms use them? 1.9 (CAS5B, 11/91, Q. 60) (1 point) Which of the following statements concerning corporate securities is FALSE? A. One reason firms utilize derivative instruments is to protect themselves against the effects of adverse changes in various external factors. B. Firms do not issue derivative securities to raise money. C. A futures contract is an order than you place in advance to buy or sell an asset or commodity. D. In a futures contract, the price is fixed when you place the order and paid at the time of the order. E. A forward contract is a tailor-made product that is not traded on an organized exchange (CAS5B, 5/94, Q. 11) (1 point) Which of the following are true? 1. A future is an order that you place in advance to buy or sell an asset or commodity at a price that is agreed upon when the order is placed. 2. A forward contract is traded on an organized exchange. 3. Swaps are agreements that grant bond owners the right to exchange the bond for a predetermined number of common shares by the exercise date. A. 1 B. 2 C. 1, 2 D. 2, 3 E 1, 2, (CAS5B, 11/95, Q. 31) (2 points) a. (1/2 point) Briefly describe warrants and convertible bonds. b. (3/4 points) For each, describe what the rational holder probably will do on the expiration date if the price of stock rises significantly from the date of issuance. c. (3/4 points) For each, describe what the holder probably will do if the price of the company's stock falls significantly (CAS5B, 11/98, Q.10) (1 point) Which of the following are true regarding financial derivatives? 1. Firms typically issue derivatives to raise money on short notice. 2. A forward contract may be traded on an organized exchange. 3. A warrant is a derivative. A. 1 B. 3 C. 1, 3 D. 2, 3 E. 1, 2, 3

21 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page (FM Sample Exam, Q.4) Zero-coupon risk-free bonds are available with the following maturities and yield rates (effective, annual): Maturity (years) Yield You need to buy corn for producing ethanol. You want to purchase 10,000 bushels one year from now, 15,000 bushels two years from now, and 20,000 bushels three years from now. The current forward prices, per bushel, are 3.89, 4.11, and 4.16 for one, two, and three years respectively. You want to enter into a commodity swap to lock in these prices. Which of the following sequences of payments at times one, two, and three will NOT be acceptable to you and to the corn supplier? A. 38,900, 61,650, 83,200 B. 39,083, 61,650, 82,039 C. 40,777, 61,166, 81,554 D. 41,892, 62,340, 78,997 E. 60,184, 60,184, 60, (FM Sample Exam, Q.6) The current price of one share of XYZ stock is 100. The forward price for delivery of one share of XYZ stock in one year is 105. Which of the following statements about the expected price of one share of XYZ stock in one year is TRUE? A. It will be less than 100 B. It will be equal to 100 C. It will be strictly between 100 and 105 D. It will be equal to 105 E. It will be greater than (FM Sample Exam, Q.7) A non-dividend paying stock currently sells for 100. One year from now the stock sells for 110. The risk-free rate, compounded continuously, is 6%. The stock is purchased in the following manner: You pay 100 today You take possession of the security in one year Which of the following describes this arrangement? A. Outright purchase B. Fully leveraged purchase C. Prepaid forward contract D. Forward contract E. This arrangement is not possible due to arbitrage opportunities

22 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page (FM Sample Exam, Q.8) You believe that the volatility of a stock is higher than indicated by market prices for options on that stock. You want to speculate on that belief by buying or selling at-the-money options. What should you do? A. Buy a strangle B. Buy a straddle C. Sell a straddle D. Buy a butterfly spread E. Sell a butterfly spread 1.17 (CAS3, 11/07, Q.25) (2.5 points) On January 1, 2007, the Florida Property Company purchases a one-year property insurance policy with a deductible of $50,000. In the event of a hurricane, the insurance company will pay the Florida Property Company for losses in excess of the deductible. Payment occurs on December 31, For the last three months of 2007, there is a 20% chance that a single hurricane occurs and an 80% chance that no hurricane occurs. If a hurricane occurs, then the Florida Property Company will experience $1,000,000 in losses. The continuously compounded risk-free rate is 5%. On October 1, 2007, what is the risk-neutral expected value of the insurance policy to the Florida Property Company? A. Less than $185,000 B. At least $185,000, but less than $190,000 C. At least $190,000, but less than $195,000 D. At least $195,000, but less than $200,000 E. At least $200, (IOA, CT8, 9/08, Q.6) (6 points) Consider an asset S paying a dividend at a constant instantaneous rate of δ, a forward contract with maturity T written on S and a constant, instantaneous (continuously compounded) risk-free rate of r. Derive the price at time t of the forward contract, using the no-arbitrage principle.

23 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page (CAS8, 5/09, Q.21) (2.25 points) Given the following information about a box spread related to 1,000 shares of Company XYZ stock: The current price of Company XYZ's stock is $100. The strike prices of the European call options underlying the box spread are $110 and $120. The time to maturity of the box spread is 1 year. The continuously compounded risk-free rate is 5% per annum. Investor A is willing to purchase the box spread from you for $9,750. Investor B is willing to sell the box spread to you for $9,750. Assume there are no taxes or transaction costs and you can borrow or lend at the risk-free rate. a. (1.5 point) Explain whether you purchase or sell the box spread. Calculate the profit you earn. Show all work. b. (0.75 point) Assume the options underlying the box spread are American instead of European. The investor with whom you entered into the box spread transaction in part a. above believes the price of Company XYZ will not decrease. Explain the investor's expected actions immediately after entering the box spread transaction with you (CAS8, 5/10, Q.12) (2.5 points) An investor would like to enter into a forward contract whereby in two years the investor exchanges a fixed amount of US Dollars for one million Euros. Assume the current exchange rate is $1.50 per Euro and that the continuously compounded risk-free interest rates are 2% in Europe and 1% in the United States. The investor can borrow and invest at the risk-free rate. a. (1 point) Determine an investment strategy which would give the investor the same cash flows as the forward contract. b. (1.5 points) Assume the two-year forward price for one million Euros is now $1,475,000 and that the investor can either take a long or short position on a forward contract at this price. Determine an investment strategy which would ensure the investor an arbitrage profit. Calculate the present value of this profit.

24 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page 23 Solutions to Problems: 1.1. D. (160)exp[(.08)(4)] = $ A. F P 0,T (S) = S0 e -δt = (120) exp[-(1.5%)(4)] = Comment: This prepaid forward price, is what we pay today for delivery of the stock 4 years from now. We do not receive any of the dividend payments during those 4 years B. The prepaid forward price is the forward price discounted for interest: / e 0.14 = (current price) e -δt = (current price) e = prepaid forward price = current price = e 0.06 = Alternately, forward price = S 0 exp[(r-δ)t] = S 0 exp[(7% - 3%)(2)]. S 0 = C. F 0,T = S 0 e T(r - δ) = (90)exp[( )(5)] = F P 0,T = S0 - PV[Div] = 100-2exp[-.04/4] = $ Comment: This prepaid forward price, is what we pay today for delivery of the stock 4 months from now. We do not receive the dividend payment 3 months from now C. Forward price = S 0 exp[(r-δ)t] = S 0 exp[(5.2% - 0.8%)(4)]. S 0 = Alternately, the prepaid forward price is the forward price discounted for interest: / e = (current price) e -δt = (current price) e = prepaid forward price = current price = e = Options and other derivative securities allow investors to shape their exposure to risks in ways that would otherwise be costly or impossible to attain. Depending on their individual tastes, risk preferences, and circumstances, some investors will use derivatives to increase their exposure to risk while others will decrease their exposure through hedging.

25 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page a. 1. Futures and forward contracts are obligations to buy or sell commodities or securities in the future at prices determined now. Futures are standardized contracts traded on exchanges. Forward contracts are individualized tailor-made contracts that are not traded on exchanges. 2. Options give the purchaser the right to buy (call option) or sell (put option) securities at fixed prices in the future. 3. Swaps are obligations to exchange commodities or values at a number of points in the future. For instance, currency swaps exchange interest payments in different currencies. Interest rate swaps may exchange a fixed coupon payment for a variable coupon payment. b. Firms use these instruments to hedge uncertainties in securities prices (e.g., by call and put options), commodities prices (e.g., by futures contracts), in currency rates (e.g., by currency swaps), and in interest rates (e.g., by interest rate swaps) D. In the case of futures contract, we pay upon delivery; we do not pay cash in advance A. Statement #1 is true. A futures contract is traded on an organized exchange. Thus Statement #2 is false. Statement #3 describes convertible bonds, and thus is false. Swaps are of various types, generally involving the exchange of interest received on fixed-income securities, such as exchanging bond interest paid in U.S. dollars for interest in Euros, or exchanging interest paid at a fixed rate for interest paid at a variable rate a. A warrant is a security that entitles the holder to buy the underlying stock of the issuing company at a fixed strike price until the expiry date. A warrant is similar to an American call option, but it is issued by a firm on its own stock; if the warrant is exercised, new shares of stock are issued. Convertible bond is like conventional debt, but it gives the holder the right to exchange the bond for a fixed number of newly issued shares in the firm. b. If the stock price on the expiration date is significantly higher than the price on the issuance date, then the warrant holder will exercise and the convertible bond holder will convert. c. If stock the price on the expiration date is significantly lower than the price on the issuance date, the warrant holder will not exercise and the convertible bond holder will not convert B. A warrant is an option issued by a firm with its own stock as the underlying asset. A futures contract is traded on an organized exchange.

26 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page B. The present value of the forward prices is: 10,000(3.89)/ ,000(4.11)/ ,000(4.16)/ = 158,968. Any sequence of payments with that present value is acceptable. A: 38,900/ ,650/ ,200/ = 158,968. B: 39,083/ ,650/ ,039/ = 158,193. C: 40,777/ ,166/ ,554/ = 158,969. D: 41,892/ ,340/ ,997/ = 158,968. E: 60,184/ ,184/ ,184/ = 158,967. All but B have the desired present value. Comment: See Page 248 of Derivative Markets by McDonald E. The forward price is a biased downwards predictor of the future price. Therefore, the expected future price of XYZ stock is greater than its forward price of 105. Comment: See Page 140 of Derivative Markets by McDonald. Someone who bought the stock should be compensated for time and risk. The one year forward price is S 0 e r > S 0, where r is the continuously compounded annual risk free rate. If this were the expected future stock price, then someone who bought the stock would only be compensated for time. In order to also be compensated for risk, the expected future stock price must be greater than the forward price C. All four of answers A-D are methods of acquiring the stock. Of these, the prepaid forward has the payment at time 0 and the delivery at time T. Comment: See Table 5.1 in Derivative Markets by McDonald. Since there are no dividends, the prepaid forward price is equal to the current price of 100. In a fully leveraged purchase, you get the stock today and pay 100e.06 one year from now. In a forward contract, you get the stock one year from now and pay the forward price, which since there are no dividends is 100e.06. In an outright purchase you would pay 100 today and also get the stock today.

27 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page B. Only straddles can consist of at-the-money options. Higher volatility means that it is more likely that the future stock price will be either very high or very low. If you buy an at-the-money put and call, then you will benefit on the call if the future stock is high and benefit from the put if the future stock price is low. The higher the future stock price the more you benefit from the call and the lower the future stock price the more you would benefit from the put. Therefore, buying a straddle is correct for this speculation. Comment: See page 78 of Derivative Markets by McDonald. One could speculate on the belief that the volatility of a stock is lower than indicated by market prices for options on that stock by writing (selling) an at-the-money straddle. In a straddle, you would buy a put and a call, both with the same strike price and time until expiration. In a strangle, you would buy a put and a higher strike call, both with the same time until expiration. In a butterfly spread, you buy a call, sell two calls at a higher strike, and buy a fourth call at a still higher strike; the difference between strikes is the same and all of the calls have the same date of expiration. A butterfly spread may also be put together with puts B. We subtract the deductible of $50,000; 1,000,000-50,000 = 950,000. (20%)($950,000)/e.05/4 = $187,640. Comment: This question has nothing to do with derivatives. You are merely being asked to take an actuarial present value using a continuously compound risk-free rate (a force of interest.) Consider two portfolios. Portfolio A: take a long position in the forward contract at time t. Its value at time t is 0. At time T its value is: S T - F t,t [S T ]. Portfolio B: at time t buying a fraction exp[-δ(t - t)] of the underlying asset, and borrowing F t,t [S T ] exp[-r(t - t)]. Its value at time t is: exp[-δ(t - t)] S t - F t,t [S T ] exp[-r(t - t)]. At time T its value is: S T - F t,t [S T ]. Both portfolios have the same value at time T. Thus, using the absence of arbitrage opportunity, both portfolios should have the same value at any intermediate time, in particular at time t. Hence: 0 = exp[-δ(t - t)] S t - F t,t [S T ] exp[-r(t - t)]. F t,t [S T ] = exp[(r-δ)(t - t)] S t. Comment: The forward price is exp[r(t - t)] times the prepaid forward price of: P F t,t [ST ] = exp[-δ(t - t)] S t.

28 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page (a) Box Spread: Buy a call and sell a put at one strike price, plus at another (higher) strike price sell a call and buy a put. For European options, the box spread is equivalent to a zero-coupon bond. Here the buyer of the box spread, would buy a $110 strike call, sell a $110 strike put, sell a $120 strike call, and buy a 120 strike put. The payoff on this box spread is: (S T - 110) + - (110 - S T ) + - (S T - 120) + + (120 - S T ) +. If S T < 110, then the payoff is: 0 - (110 - S T ) + (120 - S T ) = 10. If 120 > S T > 110, then the payoff is: (S T - 110) + (120 - S T ) = 10. If S T > 120, then the payoff is: (S T - 110) - (S T - 120) = 10. The payoff is always 10; there is no stock price risk. Thus the appropriate premium for one box spread is: 10 exp[-0.05] = Therefore, for 1000 box spreads, the appropriate premium is $9512. At $9,750, the box spreads are overpriced. I would sell the box spreads to investor A, earning a profit now of: 9, = $238. Alternately, I sell the box spreads to investor A and invest the money at the risk free rate of 5% for one year, and have: (9750)(e 0.05 ) = 10,250. I then pay investor A (1000)(10) = $10,000. I make a profit of 10,250 - $10,000 = $250 in one year. (b) Investor A owns a $110 strike put as part of the box spread. Since Investor A believes that the price of the stock will not decrease, he expects this put to have a payoff of $110 - $100 = $10 or less. Therefore, Investor A should exercise this $110 strike American put right away. When he does so, I would have to pay him: (1000)($110) = $110,000, for shares of stock that are only worth: (1000)($100) = $100,000.

29 2013-MFE/3F, Financial Economics 1 Introduction, HCM 12/6/12, Page a. The investor will get 1,000,000 Euros two years from now if he now buys Euro-denominated bonds in an amount of: 1 million / exp[(2)(2%)] = 960,789. In order to buy these bonds requires: ($1.50)(960,789) = $1,441,184. Thus the investor borrows $1,441,184 at the U.S. risk-free rate, and lends 960,789 Euros at the Euro risk-free rate. At the end of 2 years, the investor receives 1,000,000 Euros and must pay back: exp[(2)(1%)] $1,441,184 = $1,470,961. b. In part a we saw that the investor can borrow $1,441,184 to set up a position that gives the same cashflows as the forward contract; this involves paying back $1,470,298 two years from now. Thus the forward price of $1,475,000 is too high. The investor can sell the forward contract (take a long position), lend 960,789 Euros, and borrow $1,441,184. (He has bought a synthetic forward contract as well as sold an actual forward contract.) In two years the investor has 1,000,000 Euros, which he delivers to satisfy the forward contract. He receives $1,475,000 from fulfilling the forward contract. After paying back the $1,470,298 he has a net of: $1,475,000 - $1,470,298 = $4702. The present value of this arbitrage profit is: $4702 / exp[(2)(1%)] = $4609.

30 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page 29 Section 2, European Options There are various types of options. The simplest and the most important type for this exam are European Options. The adjective European does not refer to where the option is bought. Call Options: ABC Stock is currently selling for $100. Dick buys from Jane an option to buy one year from today a share of ABC Stock for $ If one year from now ABC Stock has a market price of more than $150 dollars, then Dick should use this option to buy a share of ABC Stock from Jane at $150. Dick could then sell this share of ABC Stock for the market price and make a profit. This is an example of a European Call Option. A European Call Option gives the buyer the right to buy one share of a certain stock at a strike price (exercise price) upon expiration. A European option may only be exercised on one specific day. 37 A call is an option to buy. Dick has purchased a 1 year European Call Option on ABC Stock, with a strike price of $150. Payoff on a Call Option: The eventual value to Dick of this option, depends on the price of ABC Stock one year from now. For example, if ABCʼs market price turned out to be $180 per share one year from today, then Dick could buy a share of ABC from Jane for the $150 strike price, and turn around and sell that share for $180. Dick would make a profit of $30, ignoring what he originally paid Jane to buy the option. 38 If the future price of ABC is $150 or less, then Dick would not exercise his option. 39 In that case, his option turns out to have no value to Dick. Future Price of ABC Payoff on the Option to Dick $120 0 $140 0 $160 $10 $180 $30 36 While for simplicity I have used in the example one share, one could buy an option for 100 shares or 1000 shares. 37 There are other exercise styles. See page 32 of Derivatives Markets by McDonald. Others will be discussed subsequently. 38 And ignoring any transaction costs. 39 Dick has not agreed to buy a share from Jane. Dick does not have an obligation to buy, rather Dick has purchased the right to buy a share if he wishes to.

31 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page 30 When the stock price is greater than the strike price, exercising the option makes money; this call option is in the money. If the stock price and strike price are equal, then the option is at the money. If the stock price is less than the strike price, then the call option is out of the money. Let Y + = Max[0, Y] = Y if Y 0 0 if Y < Then the eventual payoff on Dickʼs call option is (S 1-150) +, where S 1 is the price of ABC Stock one year from now. Here is a graph of the future value of Dickʼs call option: Call Payoff Stock Price In general, the future value of a European call option is: (S T - K) +, where S T is the price of the stock on the expiration date of the call and K is the strike price of the call. Put Options: XYZ Stock is currently selling for $200. Mary buys from Rob an option to sell one year from today a share of XYZ Stock for $250. If one year from now XYZ Stock has a market price of less than $250, then Mary should buy a share of XYZ Stock at the market price and then use her option to sell a share of XYZ Stock to Rob for $250, making a profit. This is an example of a European Put Option. A European Put Option gives the buyer the right to sell one share of a certain stock at a strike price (exercise price) upon expiration. A put is an option to sell. Mary has purchased a 1 year European Put Option on XYZ Stock, with a strike price of $ This very useful actuarial notation is not on the syllabus of this exam.

32 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page 31 Payoff on a Put Option: The eventual payoff to Mary of her option, depends on the price of XYZ Stock one year from now. For example, if XYZ market price turned out to be $220 per share one year from today, then Mary could buy a share of XYZ for $220 and turn around and sell that share for $250 to Rob. Mary would make a profit of $30, ignoring what she originally paid Rob to buy the option. 41 If the future price of XYZ Stock is $250 or more, then Mary would not exercise her option. 42 In this case, her option turns out to have no value to Mary. Future Price of XYZ Payoff on the Option to Mary $220 $30 $240 $10 $260 0 $280 0 The eventual value of Maryʼs put option is (250 - S 1 ) +, where S 1 is the price of XYZ Stock one year from now. 43 Here is a graph of the future value of Maryʼs put option: Put Payoff Stock Price In general, the future value of a European put option is: (K - S T ) +, where S T is the price of the stock on the expiration date and K is the strike price. 41 And ignoring any transaction costs. 42 Mary has not agreed to sell a share to Rob. Mary does not have an obligation to sell, rather Mary has purchased the right to sell a share if she wishes to. 43 Y+ is 0 if Y < 0, and Y is Y 0.

33 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page 32 When the stock price is less than the strike price, exercising the put makes money; this put option is in the money. If the stock price and strike price are equal, then the option is at the money. If the stock price is greater than the strike price, then the put option is out of the money. Expected Future Value of an Option: The future payoff on a European call option is: (S T - K) +, where S T is the price of the stock on the expiration date and K is the strike price. The future payoff on a European put option is: (K - S T ) +. Of course, at the time one could purchase an option, one does not know the future price of the stock. The future price of the stock is a random variable. The expected value of the option can be obtained by averaging using the distribution of future stock prices. The expected future value of a European call option is: E[(S T - K) + ]. The expected future value of a European put option is: E[(K - S T ) + ]. If one knew the distribution of S T, then E[(S T - K) + ] = E[S T - K S T > K] Prob[S T > K] + (0)Prob[S T K] = (E[S T S T > K] - K) Prob[ S T > K]. Similarly, E[(K - S T ) + ] = (0)Prob[S T > K] + E[K - S T S T K] Prob[ S T K] = (K- E[S T S t K]) Prob[ S T K]. Limited Expected Values: 44 Let X K = Min[X, K]. Then the limited expected value is: E[X K]. E[(S T - K) + ] = E[S T ] - E[S T K]. E[(K - S T ) + ] = K - E[S T K]. This manner of writing the expected future value can be useful if for example the distribution of future prices is LogNormal and if one had a formula for the limited expected value of a LogNormal Distribution See Mahlerʼs Guide to Loss Distributions or Loss Models, covering material on the syllabus of Exam 4/C. 45 If the distribution of St were LogNormal, this would lead to the Black-Scholes formula for valuing a put option.

34 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page 33 Problems: Use the following information for the next 3 questions: The price of the stock of the Daily Planet Media Company 1 year from now has the following distribution: Price Probability 60 20% 80 30% % % 2.1 (1 point) Determine the expected stock price of Daily Planet Media Company one year from now. 2.2 (1 point) Determine the expected payoff of a 1 year European call option on one share of Daily Planet Media Company, with a strike price of 85. (A) Less than 7 (B) At least 7, but less than 9 (C) At least 9, but less than 11 (D) At least 11, but less than 13 (E) At least (1 point) Determine the expected payoff of a 1 year European put option on one share of Daily Planet Media Company, with a strike price of 85. A) Less than 7 (B) At least 7, but less than 9 (C) At least 9, but less than 11 (D) At least 11, but less than 13 (E) At least (2 points) Graph the future value of a European call option with a strike price of 100, as a function of the future stock price. 2.5 (2 points) Graph the future value of a European put option with a strike price of 100, as a function of the future stock price. 2.6 (2 points) Graph the payoff on a European call option with a strike price of 100 plus the corresponding put, as a function of the future stock price. This position is called a straddle.

35 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (3 points) Christopher buys a $60 strike European call, sells two $70 strike European calls, and buys an $80 strike European call. The options are on the same stock and have the same expiration date. This position is called a Butterfly Spread. Graph the payoff on this portfolio as a function of the future price of the stock. 2.8 (2 points) Jason buys a $100 strike European put, and sells a $120 strike European put. The puts are on the same stock and have the same expiration date. This position is called a Put Bull Spread. Graph the payoff on this portfolio as a function of the future price of the stock. 2.9 (3 points) Melissa buys a $90 strike European call, sells a $90 strike European put, sells a $130 strike European call, and buys a $130 strike European put. The options are on the same stock and have the same expiration date. This position is called a Box Spread. Graph the payoff on this portfolio as a function of the future price of the stock (3 points) Amanda buys two $100 strike European call, sells three $110 strike European calls, and buys a $130 strike European call. The options are on the same stock and have the same expiration date. This position is called a Asymmetric Butterfly Spread. Graph the payoff on this portfolio as a function of the future price of the stock (3 points) Robert buys 1000 calls on a stock with a strike price of $120. The premium per call is $8. Robert also pays a total commission of $100. Determine the stock price at expiration at which Robert will break even. Graph Robertʼs profit as a percent of his initial investment, as a function of the stock price at expiration of the call. (Ignore the time value of money.) 2.12 (2 points) Tiffany buys a $90 strike European call and sells a $90 strike European put. The options are on the same stock and have the same expiration date. Graph the payoff on this portfolio as a function of the future price of the stock (3 points) Heather buys a $70 strike European put and sells a $90 strike European call. The options are on the same stock and have the same expiration date. This position is called a Collar. Graph the payoff on this portfolio as a function of the future price of the stock.

36 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (1 point) ABC stock costs $78. ABC stock does not pay dividends. Harold enters into a long position on a $80-strike two-year European call on ABC, and enters into a short position on a $80-strike two-year European put on ABC. Harold pays a net of $6.33 for entering these positions. What is the continuously compounded risk free rate? A. 5.0% B. 5.5% C. 6.0% D. 6.5% E. 7.0% 2.15 (3 points) Allen buys a $70 strike European put, sells four $100 strike European puts, and buys three $110 strike European puts. The options are on the same stock and have the same expiration date. This position is called a Asymmetric Butterfly Spread. Graph the payoff on this portfolio as a function of the future price of the stock (3 points) Kimberly buys 100 puts on a stock with a strike price of $80. The premium per put is $5. Kimberly also pays a total commission of $60. Determine the stock price at expiration at which Kimberly will break even. Graph Kimberlyʼs profit as a percent of her initial investment, as a function of the stock price at expiration of the put. (Ignore the time value of money.) 2.17 (2 points) Nicholas buys a share of stock, sells a $110 strike European call on that stock, and buys a $110 strike European put on that stock. The options have the same expiration date. Graph the value of this portfolio when the options expire as a function of the future price of the stock (2 points) Let S(t) be the price of a stock at time t. The stock pays dividends at the continuously compounded rate δ. The continuously compounded risk free rate is r. Assume a contract is purchased at time 0 and pays at time T: Max[S(T), 100 ]. Determine the premium for this contract in terms of the premium of a European option and other known quantities (2 points) Kevin writes (sells) a 60 strike call and a 60 strike put. The options have the same expiration date. Graph the value of this portfolio when the options expire as a function of the future price of the stock. This position is called a written straddle (2 points) Aaron owns a share of stock of the Charming Prints Company. The current price of Charming Prints Company stock is $100. Briefly discuss why might Aaron buy a Collar.

37 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (1 point) Several years ago Warren bought 1000 shares of XYZ stock. Fortunately for Warren the value of XYZ stock has increased substantially since then. However, for tax reasons Warren does not wish to sell his XYZ stock and realize his capital gains. Rather Warren plans to sell his XYZ stock one year from now. Warren is worried that by time he is ready to sell his stock his capital gains may decrease or vanish. Briefly describe how Warren could purchase a European option to hedge this risk (2 points) Lauren buys a 70 strike put and a 90 strike call. The options have the same expiration date. Graph the value of this portfolio when the options expire as a function of the future price of the stock. This position is called a strangle (1 point) You buy a European call with a strike price of 80 and sell a European put with a strike price of 80. You also sell a European call with a strike price of 100 and buy a European put with a strike price of 100. All of these options are on the same stock and have the same expiration date. Which of the following is a graph of the payoff on this portfolio as a function of the future price of the stock? Payoff A Payoff B S S Payoff C. Payoff D S S E. None of A, B, C, or D.

38 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (1 point) The Rich and Fine Stock Index has a current price of 800. An insurer offers a contract that will pay the value of the Rich and Fine Stock Index two years from now; however, the contract will pay a minimum of 750. The insurer buys the index. Briefly describe how the insurer could purchase a European option to hedge its risk (2 points) Options traders often refer to straddles and butterflies. Here is an example of each. Straddle: Buy a call with strike price of $100 and simultaneously buy a put with strike price of $100. Butterfly spread: Simultaneously buy one call with strike price of $100, sell two calls with strike price of $110, and buy one call with strike price of $120. Draw position diagrams for the straddle and butterfly, showing the payoffs from the investor's net position. Each strategy is a bet on variability. Explain briefly the nature of each bet (1 point) You sell a European call with a strike price of 110, and buy a European put with a strike price of 90. The put and call are on the same stock and have the same expiration date. Which of the following is a graph of the payoff on this portfolio as a function of the future price of the stock? Payoff A Payoff B S S Payoff C. 60 Payoff D S S E. None of A, B, C, or D

39 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (1 point) Which of the following statements are true? 1. A European call option can be exercised on or before the expiration date. 2. A put option will be worthless at the expiration date if the share price at that time is less than the strike price. 3. An investor who sells a stock short sells something that he or she does not yet own. A. 1 and 2 only. B. 1 and 3 only. C. 2 and 3 only. D. 1, 2, and 3. E None of A, B, C, or D is correct (2 points) Let Y + equal the maximum of Y and zero. Let S and Q be two random variables. (a) Determine (S - Q) + + (Q - S) +. (b) Determine (S - Q) + - (Q - S) (CAS5B, 11/94, Q.28) (2 points) a. Graph the overall position diagram when an investor simultaneously buys one call with an strike price of $80, sells two calls with strike prices of $90, and buys one call with an strike price of $100. The current price of the stock is $90. Graph the payoff versus the stock price for the total transaction. List both coordinates of all points where the slope of the graph changes. DO NOT include any other graphs or lines in your final answer. b. Assuming that markets are efficient and that the investor is rational with no superior knowledge, what is the investor's prediction on the price movement of the stock described in (a)? Briefly explain the logic underlying your answer (CAS5B, 11/94, Q.30) (2.5 points) Mr. Clean has hired Mr. Slob to run his hog farm. Mr. Clean has given Mr. Slob the following incentive plan: if, in exactly one year, the price of hogs has risen by more than 10% from their current price of $50 each, Mr. Clean will pay Mr. Slob a $10,000 bonus. What is the best estimate of the cost of this incentive scheme for Mr. Clean given the following values of one-year European call options on hogs for the strike prices? Call Option Strike Price Call Option Price $48 $5.38 $50 $4.11 $52 $3.05 $54 $2.20 $56 $1.54 $58 $1.04 Note: I have rewritten this past exam question in order to match the current syllabus.

40 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (CAS5B, 5/95, Q.13) (1 point) You own a share of XYZ stock and are concerned that the price of the stock may fall. Of the following choices, which would allow you to offset (at least partially) potential future losses? 1. Buy a put on the share of stock. 2. Sell the stock short. 3. Sell a call on the share of stock. A. 1 B. 2 C. 1, 2 D. 1, 2, 3 E. None of 1, 2, (CAS5B, 5/95, Q.32) (2 points) The RegLuar Firm, Inc., a publicly held corporation, having current assets of $75 million and no liabilities, borrows $50 million by issuing a zero coupon bond due in two years. Assume no other transactions occur after the bond is issued and before it is redeemed. a. (1/2 point) Briefly describe this transaction in terms of options. b. (3/4 points) If the value of the company's assets falls to $40 million at the end of one year, discuss whether the stock has a nonzero value. c. (3/4 points) At the end of two years, if the value of the company's assets falls to $40 million just before the debt is paid, discuss whether the stock has nonzero value (CAS5B, 5/95, Q.35) (1.5 points) a. (1 point) Explain how a term life insurance policy on the life of an actively employed actuary can function similarly to a put option owned by the actuary's dependents. b. (1/2 point) Under what circumstances does the term life policy fall short of operating like a put?

41 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (CAS5B, 11/98, Q.15) (1 point) What combination of stocks, options and borrowing/lending could be represented by the following position diagram? Valueof Position SharePrice Sell one share of stock short and borrow the present value of $ Sell one call with strike price of $100 and sell one put with strike price of $ Sell one share of stock short, sell two puts with strike price of $100, and lend the present value of $100. A. 1 B. 2 C. 3 D. 2, 3 E. 1, 2, (CAS5B, 11/99, Q.30) (2 points) ABC Insurance Company has purchased a reinsurance contract from Reliable Reinsurer providing coverage for $10 million in excess of $20 million. In other words, Reliable Reinsurer has agreed to pay up to, but no more than, $10 million beyond the initial $20 million in loss dollars retained by ABC. a. (1 point) Draw a position diagram showing the payoff to ABC from the reinsurance as a function of the amount of ABC's total loss. Label both axes. b. (1 point) If we think of ABC's total loss as the "underlying asset," we can model this reinsurance contract as a mixture of simple options. Describe the option position that replicates the payoffs from the reinsurance contract (CAS5B, 11/99, Q.31) (2 points) Norbert Corporation owns a vacant lot with a book value of $50,000. By a stroke of luck, Norbert finds a buyer willing to pay $200,000 for the lot. However, Norbert must also give the buyer a put option to sell the lot back to Norbert for $200,000 at the end of two years. Moreover, Norbert agrees to pay the buyer $40,000 for a call option to repurchase the lot for $200,000 at the end of two years. a. (1 point) What would likely happen if the lot is worth more than $200,000 at the end of two years? What if it is worth less than $200,000? Why? b. (1 point) In effect, Norbert has borrowed money from the buyer. What is the effective annual interest rate per year on the loan? Show all work.

42 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page (FM Sample Exam, Q.3) Happy Jalapenos, LLC has an exclusive contract to supply jalapeno peppers to the organizers of the annual jalapeno eating contest. The contract states that the contest organizers will take delivery of 10,000 jalapenos in one year at the market price. It will cost Happy Jalapenos 1,000 to provide 10,000 jalapenos and todayʼs market price is 0.12 for one jalapeno. The continuously compounded risk-free interest rate is 6%. Happy Jalapenos has decided to hedge as follows (both options are one-year, European): Buy 10, strike put options for and sell 10, strike call options for Happy Jalapenos believes the market price in one year will be somewhere between 0.10 and 0.15 per pepper. Which interval represents the range of possible profit one year from now for Happy Jalapenos? A. 200 to 100 B. 110 to 190 C. 100 to 200 D. 190 to 390 E. 200 to (FM Sample Exam, Q.9) You are given the following information: The current price to buy one share of ABC stock is 100 The stock does not pay dividends The risk-free rate, compounded continuously, is 5% European options on one share of ABC stock expiring in one year have the following prices: Strike Price Call option price Put option price A butterfly spread on this stock has the following profit diagram Which of the following will NOT produce this profit diagram? A. Buy a 90 put, buy a 110 put, sell two 100 puts B. Buy a 90 call, buy a 110 call, sell two 100 calls C. Buy a 90 put, sell a 100 put, sell a 100 call, buy a 110 call D. Buy one share of the stock, buy a 90 call, buy a 110 put, sell two 100 puts E. Buy one share of the stock, buy a 90 put, buy a 110 call, sell two 100 calls.

43 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page 42 Solutions to Problems: 2.1. (20%)(60) + (30%)(80) + (30%)(100) + (20%)(120) = D. (20%)(0) + (30%)(0) + (30%)(100-85) + (20%)(120-85) = A. (20%)(85-60) + (30%)(85-80) + (30%)(0) + (20%)(0) = Graph of the future value of a European call with strike price of 100, E[(S - 100) + ]: Option Value Stock Price Graph of the future value of a European put with strike price of 100, E[(100 - S) + ]: Option Value Stock Price

44 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page Graph of the payoff of a European call plus put each with strike price of 100, E[(S - 100) + ] + E[(100 - S) + ]: Payoff Stock Price Comment: This straddle pays a large amount if the future stock price differs a lot from $100. If $100 is the current price, this is one way to speculate on volatility. Similar to Figure 3.10 in Derivative Markets by McDonald The payoff for the portfolio is: (S T - 60) + - 2(S T - 70) + + (S T - 80) +. If S T 60, then the payoff is nothing. If 70 S T > 60, then the payoff is: S T If 80 S T > 70, then the payoff is: (S T - 60) - 2(S T - 70) = 80 - S T. If S T > 80, then the payoff is: (S T - 60) - 2(S T - 70) + (S T - 80) = 0. A graph of the payoff: Payoff S

45 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page The payoff for the portfolio is: (100 - S T ) + - (120 - S T ) +. If S T 120, then the payoff is nothing. If 120 > S T 100, then the payoff is: -(120 - S T ) = S T If 100 > S T, then the payoff is: (100 - S T ) - (120 - S T ) = -20. A graph of the payoff: Payoff S Comment: The premium for the 100 strike put is less than the premium for the 120 strike put. Joe gained money from setting up this portfolio. Joe is hoping that the future stock price will be at least 120. Similar to Figure 3.7 in Derivative Markets by McDonald The payoff for the portfolio is: (S T - 90) + - (90 - S T ) + - (S T - 130) + + (130 - S T ) +. If S T 90, then the payoff is: -(90 - S T ) + (130 - S T ) = 40. If 130 S T > 90, then the payoff is: (S T - 90) + (130 - S T ) = 40. If S T > 130, then the payoff is: (S T - 90) - (S T - 130) = 40. A graph of the payoff: Payoff Comment: The box-spread has a risk free payoff; buying a box-spread is equivalent to buying a bond. Writing a box-spread is equivalent to borrowing money. S

46 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page The payoff for the portfolio is: 2(S T - 100) + - 3(S T - 110) + + (S T - 130) +. If S T 100, then the payoff is nothing. If 110 S T > 100, then the payoff is: 2(S T - 100). If 130 S T > 110, then the payoff is: 2(S T - 100) - 3(S T - 110) = S T. If S T > 130, then the payoff is: 2(S T - 100) - 3(S T - 110) + (S T - 130) = 0. A graph of the payoff: Payoff Comment: As will be discussed, such a Asymmetric Butterfly Spread may be used to take advantage of certain arbitrage opportunities. λ = ( )/( ) = 2/3. Buy λ of the lowest strike, sell 1 of the middle strike, and buy (1 - λ) of the highest strike. In this case, buy 2/3 of 100 strike, sell 1 of the 110 strike, and buy 1/3 of the 130 strike. Here Amanda has multiplied this position by three. S

47 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page We want: (1000)(S T - 120) + = (1000)(8) = S T = $ If S T < 120, the calls are worthless, and Robertʼs profit is: This is equivalent to -100%. If S T > 120, Robertʼs profit is: (1000)(S T - 120) = 1000S T - 128,100. As a ratio to his initial investment of 8100, this is: S T Here is a graph of Robertʼs profit as a percent of his initial investment as a function of the stock price at expiration: Percent Profit S -100 Comment: Notice the large leverage when one invests in an option. A small change in the stock price at expiration has a large effect on Robertʼs profit The payoff for the portfolio is: (S T - 90) + - (90 - S T ) +. If S T 90, then the payoff is: -(90 - S T ) = S T If S T > 90, then the payoff is: S T A graph of the payoff: Payoff S Comment: I only graphed from a future stock price of 60 to 120. If S T = 60, then the person to whom Tiffany sold the put will require Tiffany to buy the stock for 90 from this person, even though the stock is only worth 60. If S T = 60, then Tiffany has a payoff of = -30.

48 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page The payoff for the portfolio is: (70 - S T ) + - (S T - 90) +. If S T 70, then the payoff is: 70 - S T. If 90 S T > 70, then the payoff is: 0. If S T > 90, then the payoff is: S T A graph of the payoff: Payoff S - 20 Comment: Similar to Figure 3.8 in Derivative Markets by McDonald B. Haroldʼs bought a call and sold the otherwise similar put. The payoff on Haroldʼs position is: (S 2-80) + - (80 - S 2 ) + = S Since ABC pays no dividends, the prepaid forward price for S 2 is just S 0 = 78. The prepaid forward price to receive $80 two years from now is 80e -2r. Therefore, the price for Harryʼs position is: 78-80e -2r. Set 6.33 = 78-80e -2r. r = 5.5%.

49 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page The payoff for the portfolio is: (70 - S T ) + - 4(100 - S T ) + + 3(110 - S T ) +. If S T 70, then the payoff is: (70 - S T ) - 4(100 - S T ) + 3(110 - S T ) = 0. If 100 S T > 70, then the payoff is: -4(100 - S T ) + 3(110 - S T ) = S T If 110 S T > 100, then the payoff is: 3(110 - S T ). If S T > 110, then the payoff is nothing. A graph of the payoff: Payoff S

50 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page We want: (100)(80 - S T ) + = (100)(5) + 60 = 560. S T = $ If S T > 74.40, the puts are worthless, and Kimberlyʼs profit is: This is equivalent to -100%. If S T < 74.40, Kimberlyʼs profit is: (100)(80 - S T ) = S T. As a ratio to her initial investment of 560, this is: S T. Here is a graph of Kimberlyʼs profit as a percent of her initial investment as a function of the stock price at expiration: Percent Profit S Comment: Notice the large leverage when one invests in an option. A small change in the stock price at expiration has a large effect on Kimberlyʼs profit The value of this portfolio when the options expire is: S T - (S T - 110) + + (110 - S T ) +. If S T 110, then the value is: S T + (110 - S T ) = 110. If S T > 110, then the payoff is: S T - (S T - 110) = 110. A graph of the value: Value S

51 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page Max[S(T), 100] = Max[S(T) - 100, 0] = (S(T) - 100) +. (S(T) - 100) + is the payoff on a T-year 100-strike European call on this stock. The present value of a payoff T years from now of 100 is: 100 e -rt. Thus the premium for this contract is: 100 e -rt + C, where C is the premium on a T-year 100-strike European call. Alternately, Max[S(T), 100] = S(T) + Max[100 - S(T), 0] = S(T) + (100 - S(T)) +. (100 - S(T)) + is the payoff on a T-year 100-strike European put on this stock. The prepaid forward price for S(T) is: S(0) e -δt Thus the premium for this contract is: S(0) e -δt + P, where P is the premium on a T-year 100-strike European call. Comment: As will be discussed, the call premium and put premiums are connected via put-call parity. Thus one can show that the two forms of the premium for this contract are equivalent The payoff is: -(S - 60) + - (60 - S) +.

52 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page Collar: Purchase a put and sell a call with a higher strike price. For example, Aaron might buy a 90 strike put and sell a 110 strike call, each of which expire 6 months from now. (There are many other possible Collars.) Then the value of his portfolio 6 months from would be: S.5 + (90 - S.5 ) + - (S.5-110) +. If S.5 90, then Aaronʼs portfolio is worth 90. (Aaron will use his put to sell the stock for 90.) If 90 < S.5 < 110, then Aaronʼs portfolio is S.5. If S.5 110, then Aaronʼs portfolio is worth 110. (The person to whom Aaron sold the call, will use the call to buy the stock for 110 from Aaron.) By buying this collar, Aaron has limited the value of his position in 6 months to be between 90 and 110. Aaron can not make a lot, but also he can not lose a lot. Comment: Using the Black-Scholes formula, to be discussed subsequently, if the stock pays no dividends, the stock has a volatility of 30%, and r = 5%, then the premium for this collar is -2.60; in other words, Aaron will make more money from selling the call than he spends buying the put. If instead for example, Aaron had bought a 120 strike put and sold a 140 strike call, each of which expire 2 years from now, then he would have limited the value of his position in 2 years to be between 120 and Warren could buy one thousand 1-year at-the money European puts on XYZ stock. If one year from now XYZ stock is worth more than its current price, then he can sell his stock and make more in capital gains than he has currently. If one year from now XYZ stock is worth less than its current price, then he could use his puts to sell his stock at its price today and make in capital gains the amount he has currently. Comment: Buying a put protects against the price of a stock you own going down. Buying a call would protect against the price of a stock you shorted going up.

53 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page The payoff is: (70 - S) + + (S - 90) B. (S - 80) + - (80 - S) + = S (S - 100) + - (100 - S) + = S (S - 80) + - (80 - S) + - {(S - 100) + - (100 - S) + } = S (S - 100) = 20. Comment: This position is called a Box Spread The insurer could buy a 2-year 750-strike European put on the Rich and Fine Stock Index. If two years from now the index is worth more than 750, then the insurer can sell the index, pay off the contract, and have some money left over. If two years from now the index is worth less than 750, then the insurer can use its put to sell the index for 750 and pay off the contract.

54 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page Payoff diagram for the straddle: Payoff S The purchaser of the straddle hopes that the stock price moves a lot; the purchaser is betting that there will be high volatility in the stock price. Payoff diagram for the butterfly spread: Payoff S The purchaser of the butterfly spread hopes that the stock price does not move a lot; the purchaser is betting that there will be low volatility in the stock price D. (90 - S) + - (110 - S) + is equal to: S, for S > 100, 0, for 90 < S < S, for S < 90. Comment: This position is called a Collar.

55 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page E. Statement number 1 is false. A European options can be exercised only on the expiration date. The statement is the correct definition for an American option. Statement number 2 is false. In fact, the opposite is true: the lower the market price on the expiration date, the higher the value of the put option. Statement number 3 is correct. Short sellers sell stock which they do not yet own (a) (S - Q) + = S - Q if S Q 0 if S Q. (Q - S) 0 if S < Q + =. Q - S if S < Q S - Q if S Q Therefore, (S - Q) + + (Q - S) + = Q - S if S < Q = S - Q. (b) (S - Q) + = S - Q if S Q 0 if S Q. (Q - S) 0 if S < Q + =. Q - S if S < Q S - Q if S Q Therefore, (S - Q) + - (Q - S) + = = S - Q. S - Q if S < Q Comment: If Q were a constant, then (S - Q) + + (Q - S) + would be the payoff on a Q-strike call and the similar Q-Strike put; in other words the payoff on a straddle is: (S - K) + + (K - S) + = S - K. If Q were a constant, then (S - Q) + - (Q - S) + would be the payoff on a Q-strike call and the sale of a similar Q-Strike put. The fact that (S - K) + - (K - S) + = S - K, is the basis of put-call parity, to be discussed in a subsequent section.

56 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page a. The payoff diagram: Payoff 10 (90, 10) (80, 0) (100, 0) S b. The investor is betting that the volatility of the stock will be less than the market expects. The investor has a large payoff if the stock price moves a small amount from its initial price, and no payoff if the stock price moves a large amount from its initial price. Comment: This is a Butterfly spread; there would have been a net cost to setting up this position If the price of hogs in one year is $55 or more, then Mr. Slob gets his bonus. We can approximate such a payoff by buying a 54-strike call and selling a 56-strike call. 0 if S < 54 The payoff will be: S - 54 if 54 S if S > 56 The cost of one such position is: $ $1.54 = $0.66. If S > 56 the payoff is $2, so Mr. Clean would need to buy $10,000/$2 = 5000 such positions to fund the bonus. Thus the cost of this incentive scheme is: (5000)(0.66) = $3300. Comment: The match between the bonus and the position of calls is approximate. There are other combinations of calls that would also approximate the bonus. An exact match would be provided by 55-strike cash-or-nothing calls, an exotic option to be discussed in a subsequent section.

57 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page D. If the stock price goes down, the put will have a positive payoff, so Statement #1 is true. Owning a share and also selling a share short, you will be unaffected by stock price movements, so Statement #2 is true. If the stock price goes down, the call will have no payoff; you can use the money you got from selling the call to offset some of the losses on the stock. Thus Statement #3 is true. Comment: Buying a put would be the usual way to hedge the risk of the stock price declining a. When a firm borrows, the equity holders exchange their claim on the assets of the firm for a call option on the whole firm with an strike price equal to the maturity value of the debt. In other words, the option holder can either exercise the call option at the exercise date, pay the strike price, and obtain the stock, or the option holder can choose not to exercise the call option and he or she is left with no stock. Similarly, the equity holders can either repay the debt at the maturity date for the par value and retain the assets of the corporation, or they can default on the debt, in which case they are left with nothing since the bondholders take the assets of the corporation. b. There is still a chance that the asset of the company will increase beyond $50 million by the end of the second year, so the stock still has value. This is equivalent to an out-of-the money call with a year until expiration; such a call has a positive if small value. c. The companyʼs assets are less than the money owed to the bondholders, so the stock is now worthless. An out-of-the money call at expiration is worthless a. The actuary's family has a claim on the future wages of the actuary. If the actuary dies, the value of that claim falls to zero. The term life policy is like a put option which pays off if the actuary dies over the term of the policy. b. Events other than death can reduce the actuary's future wages, for example layoff or disability. Under these events, the term life policy will not pay off. Comment: The analogy is a little strained D. This is the diagram for selling (writing) a straddle, position #2. If there are no dividends, then by put call parity, the buying a call is equivalent to buying one share of stock, buying one puts, and borrowing the present value of the strike. Thus, in this case, position #2 would be equivalent to selling a call and selling a put, position #2. The value of position #1 at future time T is: S T - 100e rt, not the given graph.

58 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page a. Payoff is: Min[10 million, (L - 20 million) + ]. Payoff Loss b. Buy a 20 strike call and sell a 30 strike call. 0, S < 20 (S - 20) + - (S - 30) + = S - 20, 20 < S < 30 10, S > 30 Comment: Many insurance and reinsurance arrangements can be thought of in terms of options a. If in two years the lot is worth more than $200,000, then Norbert will use its call to repurchase the lot for $200,000. If in two years the lot is worth less than $200,000, then the put Norbert gave the buyer will be used to sell the lot to Norbert for $200,000. b. In either case, Norbert gets a net of $200,000 - $40,000 = $160,000 today and has $200,000 (without the lot) in two years. Interest rate = (200,000/160,000) 1/2-1 = 11.8% D. The accumulated cost of the hedge is: ( )exp(.06) = Let x be the market price. If x < 0.12, the put is in the money and the payoff is: 10,000(0.12 x) = 1,200-10,000x. The sale of the jalapenos has a payoff of: 10,000x - 1,000. The profit is: 1,200-10,000x + 10,000x - 1, = 190. From 0.12 to 0.14 neither option has a payoff, and the profit is: 10,000x - 1, = 10,000x - 1,010. This ranges from 190 to 390. If x > 0.14, the call is in the money and the payoff is: -10,000(x ) = 1,400-10,000x. The profit is: 1,400-10,000x + 10,000x - 1, = 390. The range of possible profit one year from now is: 190 to 390.

59 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page D. The cost to set up portfolio A is: (2)(1.93) = If S 1 < 90, then the profit is: (90 - S 1 ) + (110 - S 1 ) - 2(100 - S 1 ) e.05 = If 90 < S 1 < 100, then the profit is: (110 - S 1 ) - 2(100 - S 1 ) e.05 = S If S 1 = 100, then the profit is: e.05 = If 100 < S 1 < 110, then the profit is: (110 - S 1 ) e.05 = S 1. If 110 < S 1, then the profit is: -3.19e.05 = The cost to set up portfolio B is: (2)(6.80) = If S 1 < 90, then the profit is: -3.20e.05 = If 90 < S 1 < 100, then the profit is: (S 1-90) e.05 = S If S 1 = 100, then the profit is: e.05 = If 100 < S 1 < 110, then the profit is: (S 1-90) - 2(S 1-100) e.05 = S 1. If 110 < S 1, then the profit is: (S 1-90) - 2(S 1-100) + (S 1-110) e.05 = The cost to set up portfolio C is: = If S 1 < 90, then the profit is: (90 - S 1 ) - (100 - S 1 ) e.05 = If 90 < S 1 < 100, then the profit is: -(100 - S 1 ) e.05 = S If S 1 = 100, then the profit is: 6.32e.05 = If 100 < S 1 < 110, then the profit is: -(S 1-100) e.05 = S 1. If 110 < S 1, then the profit is: -(S 1-100) + (S 1-110) e.05 = The cost to set up portfolio D is: (2)(1.93) = If S 1 < 90, then the profit is: S 1 + (110 - S 1 ) - 2(100 - S 1 ) e.05 = 2S , not matching the given graph. If 90 < S 1 < 100, then the profit is: S 1 + (S 1-90) + (110 - S 1 ) - 2(100 - S 1 ) e.05 = 3S , not matching the given graph. If S 1 = 100, then the profit is: e.05 = -3.71, not matching the given graph. If 100 < S 1 < 110, then the profit is: S 1 + (S 1-90) + (110 - S 1 ) e.05 = S , not matching the given graph. If 110 < S 1, then the profit is: S 1 + (S 1-90) e.05 = 2S , not matching the given graph. The cost to set up portfolio E is: (2)(6.80) = If S 1 < 90, then the profit is: S 1 + (90 - S 1 ) e.05 =

60 2013-MFE/3F, Financial Economics 2 European Options, HCM 12/6/12, Page 59 If 90 < S 1 < 100, then the profit is: S e.05 = S If S 1 = 100, then the profit is: = If 100 < S 1 < 110, then the profit is: S 1-2(S 1-100) e.05 = S 1. If 110 < S 1, then the profit is: S 1-2(S 1-100) + (S 1-110) e.05 = Comment: See Exercise 3.18 on Page 89 of Derivative Markets by McDonald. For each strike price, put-call parity holds. With no dividends, C - P = Ke -.05.

61 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 60 Section 3, Properties of Premiums of European Options The premium of an option is its current price, the actuarial present value of its future payoffs. As will be discussed in subsequent sections, in order to properly price an option we have to use risk-neutral probabilities; we can price options in the risk-neutral environment. Actuarial Present Value of a Call Option: The actuarial present value of a European call option is: E[(S T - K) + ] e -rt. If we let C(S 0, K, T) be the actuarial present value of the call on a stock with current price S 0, strike price K, and time until expiration T: C Eur (S 0, K, T) = E[(S T - K) + ] e -Tr. Exercise: The price of a stock two years from now has the following distribution: 20%, 40%, The continuously compounded annual risk free rate is: r = 4%. Determine the actuarial present value of a European call option on this stock with 2 years to expiration and a strike price of $90. [Solution: e {(20%)(0) + (40%)(10) + (30%)(60) + (10%)(110)} = $30.46.] Exercise: In the previous exercise, change the strike price to $100. [Solution: e {(20%)(0) + (40%)(0) + (30%)(50) + (10%)(100)} = $23.08.] The actuarial present value of this European call option as a function of the strike price: 100 PV K The actuarial present value, in other words premium, of the call decreases as the strike price increases and the curve is concave upwards.

62 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 61 General Properties of European Call Premiums: These are general properties for the behavior of a call as one varies the strike price. 46 The value of an call option with strike price $90 is greater than an otherwise similar call with a strike price of $100. It is less valuable to have the option to buy something at $100 than it is to have the option to buy that same thing at $90. Call premiums decrease as the strike price increases: C For K 1 < K 2, C(K 1 ) C(K 2 ). K 0. For example, assume the future price of a stock at expiration of a call has the following distribution: 20%, 40%, Stock Payoff on Call Payoff on Call Price with K = 90 with K = 100 Difference The difference in payoffs is at most the difference in strike prices, 10. Being able to buy at $90 is worth at most $10 more than being able to buy at $ Therefore the difference in call premiums is at most the difference in strike prices: C For K 1 < K 2, C(K 1 ) - C(K 2 ) K 2 - K 1. K -1. Since European options can only be exercised at expiration, the difference in option premiums cannot be more than the present value of the difference in payoffs. Thus: For K 1 < K 2, C(K 1 ) - C(K 2 ) e -rt C (K 2 - K 1 ). 52 K -e-rt. 46 See pages 292, 299, and 300 in Derivative Markets by McDonald. 47 See equation 9.13 in Derivative Markets by McDonald. 48 This equation holds both for European and American options, to be discussed subsequently. 49 Sometimes being able to buy at $90 and being able to buy at $100 are both worth nothing. 50 See equation 9.15 in Derivative Markets by McDonald. 51 This equation holds both for European and American options, to be discussed subsequently. 52 See Appendix 9.B in McDonald, not on the syllabus.

63 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 62 The third property is referred to as the convexity of the option price with respect to the strike price. 53 For K 1 < K 2 < K 3, C(K 1 ) - C(K 2) K 2 - K 1 C(K 2 ) - C(K 3 ) K 3 - K C K 2 0. For example, let us assume that a call with strike price $90 is worth $7 more than a similar call with a strike price of $100. Then a call with a strike price of $100 exceeds that of a similar call with a strike price of $110, but by less than or equal to $7. Convexity follows from the fact that the second derivative of the call premium with respect to K is positive: 2 C K2 0. C decreases as K increases and the slope is negative; as K increases the slope increases, in other words gets closer to zero. As K decreases the slope decreases; however, the slope can not get less than -1. As mentioned before, the graph of C as a function of K is concave upwards. 53 A convex function has a curve that is concave upwards, shaped like a bowl. 54 See equation 9.17 in Derivative Markets by McDonald. This equation holds both for European and American options, to be discussed subsequently. 55 Increased Limits Factors share this same property for the same underlying mathematical reason. See Sheldon Rosenbergʼs review of On the Theory of Increased Limits and Excess of Loss Pricing, PCAS 1977.

64 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 63 Mathematically the three conditions for call premiums could be summarized as: -1 C K 0, and 2 C K 2 0. This is illustrated in the following graph: 56 C K As K approaches zero, the slope of the above curve approaches -e -rt, which is close to but more than minus one. As K approaches infinity, the slope of the above curve approaches The put premiums are computed via the Black-Scholes formula to be discussed subsequently. This is for a 2-year European call, and σ = 40%, r = 6%, δ = 0.

65 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 64 Explaining the Behavior of the Value of Calls as a Function of K: Let F(S) be the distribution of the future price of the stock. The expected value of the call is: E[(S T - K) + ] = {1 - F(x)} dx. 57 K Therefore, C(K) = e -rt {1 - F(x)} dx. K C K = -e-rt {1 - F(K)} 0. Therefore, C declines as K increases. C(K 1 ) - C(K 2 ) = e -rt K 2 {1 - F(x)} dx e -rt (K 2 - K 1 ) {1 - F(K 1 )} e -rt (K 2 - K 1 ) K 2 - K 1. K 1 2 C K 2 = e-rt f(k) 0. Therefore, C(K) is concave upwards. 57 See Mahlerʼs Guide to Loss Distributions, covering material on Exam 4/C. This is an expression for the expected excess losses.

66 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 65 Lee Diagrams and Call Premiums: 58 One can present the same ideas graphically via Lee Diagrams. 59 Lee Diagrams have the x-axis correspond to probability, while the y-axis corresponds to size of loss. Here we will graph the distribution function of the future price of the stock, with the horizontal axis corresponding to probability and the vertical axis corresponding to the stock price at expiration of the option. The expected payoff of a European Call, is equal to E[(S T - K) + ]. E[(X - K) + ] is the expected losses excess of K, and corresponds to the area on the Lee Diagram above the horizontal line at height K and also below the curve graphing F(x). As K increases, the area above the horizontal line at height K decreases; in other words, the value of the call decreases as K increases. 58 Not on the syllabus of your exam! 59 See The Mathematics of Excess of Loss Coverage and Retrospective Rating --- A Graphical Approach, by Y.S. Lee, PCAS LXXV, Currently on the syllabus of CAS Advanced Ratemaking Exam. See also Mahlerʼs Guide to Loss Distributions, covering material on Exam 4/C.

67 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 66 For an increase in K of ΔK, the value of the call decreases by Area A in the following Lee Diagram: Stock Price K+ΔK K A 1 Prob. The absolute value of the change in the value of the call, Area A, is smaller than a rectangle of height ΔK and width 1 - F(K). Thus Area A is smaller than ΔK {1 - F(K)} ΔK. Thus a change of ΔK in the strike price results in a absolute change in the value of the call option smaller than ΔK. The following Lee Diagram shows the effect of raising the strike price by fixed amounts: The successive absolute changes in the value of the call are represented by Areas A, B, C, and D. We see that the absolute changes in the value of the call get smaller as the strike price increases.

68 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 67 Actuarial Present Value of a Put Option: The actuarial present value of a European put option is: E[(K - S T ) + ] e -rt. If we let P(S 0, K, T) be the actuarial present value of the put on a stock with current price S 0, strike price K, and time until expiration T: P Eur (S 0, K, T) = E[(K - S T ) + ] e -Tr. Exercise: The price of a stock two years from now has the following distribution: 20%, 40%, r = 4%. Determine the actuarial present value of a European put option on this stock with 2 years to expiration and a strike price of $120. [Solution: e {(20%)(70) + (40%)(20) + (30%)(0) + (10%)(0)} = $20.31.] The actuarial present value of the above European put option as a function of the strike price: PV K The actuarial present value of the put increases as the strike price increases and the curve is concave upwards. These are general properties for the behavior of a put as one varies the strike price See pages 292, 299, and 300 in Derivative Markets by McDonald.

69 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 68 General Properties of European Put Premiums: P For K 1 < K 2, P(K 1 ) P(K 2 ). K 0. For example, the value of an put option with strike price $100 is greater than an otherwise similar put with a strike price of $90. It is more valuable to have the option to sell at $100 than to have the option to sell at $ P For K 1 < K 2, P(K 2 ) - P(K 1 ) K 2 - K 1. K 1. For example, the value of a put option with strike price $100 is greater than an otherwise similar put with a strike price of $90 by at most $10. Being able to sell at $100 is worth at most $10 more than being able to sell at $ Since European options can only be exercised at expiration, the difference in option premiums cannot be more than the present value of the difference in payoffs. Thus: For K 1 < K 2, P(K 2 ) - P(K 1 ) e -rt P (K 2 - K 1 ). 66 K e-rt. For K 1 < K 2 < K 3, P(K 2) - P(K 1 ) K 2 - K 1 P(K 3 ) - P(K 2 ) K 3 - K P K 2 0. For example, let us assume that a put with strike price $100 is worth $6 more than a similar put with a strike price of $90. Then a put with a strike price of $110 exceeds that of a similar put with a strike price of $100 by at least $6. This is referred to as the convexity of the option price with respect to the strike price. 68 It follows from the fact that the second derivative of the value of the put with respect to K is positive: 2 P K 2 0. P increases as K increases and the slope is positive; as K increases the slope increases, in other words gets further from zero. However, the slope can not exceed one. 61 See equation 9.14 in Derivative Markets by McDonald. 62 This equation holds both for European and American options, to be discussed subsequently. 63 See equation 9.16 in Derivative Markets by McDonald. 64 This equation holds both for European and American options, to be discussed subsequently. 65 Sometimes being able to sell at $90 and being able to sell at $100 are both worth nothing. 66 See Appendix 9.B in McDonald, not on the syllabus. 67 See equation 9.18 in Derivative Markets by McDonald. This equation holds both for European and American options, to be discussed subsequently. 68 A convex function has a curve that is concave upwards, shaped like a bowl.

70 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 69 As mentioned before, the graph of P as a function of K is concave upwards. 69 Mathematically the three conditions for put premiums could be summarized as: 1 P K 0, and 2 P K 2 0. This is illustrated in the following graph: P K As K approaches infinity, the slope of the above curve approaches e -rt, which is close to but less than one. As K approaches zero, the slope of the above curve approaches The graph of a function is concave upwards if it is shaped like a bowl. If the second derivative is positive, then the graph of a function is concave upwards. A convex function is such that for any 0 t 1, and x y, f(tx + (1-t)y) tf(x) + (1 -t)f(y). A convex function has a graph that is concave upwards. 70 The put premiums are computed via the Black-Scholes formula to be discussed subsequently. This is for a 2-year European put, and σ = 40%, r = 6%, δ = 0.

71 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 70 Explaining the Behavior of the Value of Puts as a Function of K: As before, let F(S) be the distribution of the future price of the stock. The expected value of the put is: 71 E[(K - S T ) + ] = K F(x) dx. 0 Therefore, P(K) = e -rt K F(x) dx. 0 P K = e-rt F(K) 0. Therefore, P increases as K increases. P(K 2 ) - P(K 1 ) = e -rt K 2 F(x) dx e -rt (K 2 - K 1 ) F(K 1 ) e -rt (K 2 - K 1 ) K 2 - K 1. K 1 2 P K 2 = e-rt f(k) 0. Therefore, P(K) is concave upwards. 71 See Mahlerʼs Guide to Loss Distributions, covering material on Exam 4/C. This is an expression for the expected amount by which X is less than K.

72 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 71 Lee Diagrams and Put Premiums: 72 The expected payoff of a European put is E[(K - S T ) + ], which corresponds to Area P below the horizontal line at height K and also above the curve graphing F(x) in the following Lee Diagram: As K increases, the area below the horizontal line at height K increases; in other words, the value of the put increases as K increases. 72 Not on the syllabus of your exam!

73 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 72 For an increase in K of ΔK, the value of the put increases by Area A in the following Lee Diagram: Stock Price K+ΔK K A 1 Prob. The change in the value of the put, Area A, is smaller than a rectangle of height ΔK and width F(K +ΔK). Thus Area A is smaller than ΔK F(K +ΔK) ΔK. Thus a change of ΔK in the strike price results in a change in the value of the put option smaller than ΔK. The following Lee Diagram shows the effect of raising the strike price by fixed amounts: The successive changes in the value of the put are represented by Areas A, B, C, and D. We see that the changes in the value of the put get larger as the strike price increases.

74 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 73 Arbitrage: If there is a possible combination of buying and selling with no net investment that has no risk but generates positive (or at least nonnegative) cashflows, this is an arbitrage opportunity. Taking advantage of such an opportunity is called arbitrage. In other words, arbitrage is free money. If an arbitrage opportunity existed, clever traders would take advantage of it. Relatively quickly, the prices would adjust so as to remove this opportunity for arbitrage. Generally, we assume prices should be such that they do not permit arbitrage. In other words, we assume that there is no free lunch. This is called no-arbitrage pricing. Arbitrage Opportunities with Two Calls: If one of the general properties of option premiums that has been discussed is violated, that creates an opportunity for arbitrage. For example, two otherwise similar calls have the following premiums: 100 strike call costs strike call costs 12. This violates the principal that call premiums should not increase as the strike price increases. The 100 strike call is cheap relative to the 110 strike call; the 110 strike call is expensive relative to the 100 strike call. We can buy a 100 strike call and sell a 110 strike call. We make = 2 from this set of transactions. We invest the 2 at the risk free rate, r. At expiration of the calls at time T, we have: If S 100: 2e rt > 0 If 110 S > 100: (S - 100) + 2e rt > 0. If S > 110: (S - 100) - (S - 110) + 2e rt = e rt > 0. Thus we always end up with a positive (or at least nonnegative) position, having taken no risk. This demonstrates arbitrage.

75 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 74 In another example, two otherwise similar calls have the following premiums: 100 strike call costs strike call costs 13. This violates the principal that call premiums should not decrease by more than the increase of the strike price. The 105 strike call is cheap relative to the 100 strike call. We can sell a 100 strike call and buy a 105 strike call. We make = 7 from this set of transactions. We invest the 7 at the risk free rate, r. At expiration of the calls at time T, we have: If S 100: 7e rt > 0 If 105 S > 100: -(S - 100) + 7e rt 7e rt - 5 > 0. If S > 105: (S - 100) - (S - 105) + 7e rt = 7e rt - 5 > 0. Thus we always end up with a positive (or at least nonnegative) position, having taken no risk. This demonstrates arbitrage. If the principals for put premiums are violated, one can set up similar opportunities for arbitrage to those illustrated for calls. Arbitrage Opportunities When Convexity is Violated by Call Premiums: Three otherwise similar calls have the following premiums: 100 strike call costs strike call costs strike call costs 7. (20-17) / ( ) = 0.3. (17-7) / ( ) = > 0.3. This violates the convexity of the call premium with respect to the strike price. The arbitrage opportunity in such situations involves buying some of the low strike and high strike calls, while selling some of the medium strike calls. While there are many possible positions that demonstrate arbitrage when convexity is violated, McDonald has a technique of coming up with one such a portfolio.

76 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 75 Let λ = K 3 - K 2 K 3 - K λ is the amount of the distance between the high and medium strike prices as a fraction of the distance between the high and low strike prices. In this example, λ = ( ) / ( ) = 3/4. We note that K 2 = λ K 1 + (1 - λ)k 3, a weighted average of K 1 and K 3 with weights λ and 1 - λ. In this example, 110 = (3/4)(100) + (1-3/4)(140). The convexity relationship for call premiums was: For K 1 < K 2 < K 3, {C(K 1 ) - C(K 2 )} / {K 2 - K 1 } {C(K 2 ) - C(K 3 )} / {K 3 - K 2 }. {C(K 1 ) - C(K 2 )}{K 3 - K 2 } {C(K 2 ) - C(K 3 )}{K 2 - K 1 }. (K 3 - K 2 )C(K 1 ) + (K 2 - K 1 )C(K 3 ) (K 3 - K 1 )C(K 2 ). C(K 2 ) λ C(K 1 ) + (1 - λ) C(K 3 ). For convexity to hold, we require C(K 2 ) λ C(K 1 ) + (1 - λ) C(K 3 ). 74 In this example, we require: C(110) (3/4) C(100) + (1-3/4) C(140) = (3/4)(20) + (1/4)(7) = The height of the line between the points (K 1, C(K 1 )) and (K 3, C(K 3 )), at the strike price K is: y = βc(k 1 ) + (1 - β)c(k 3 ), where β = K 3 - K K 3 - K 1. Thus, we require that the point (K 2, C(K 2 )) not be above this line. 73 See equation 9.19 in Derivative Markets by McDonald. It is somewhat arbitrary that McDonald has a numerator of K3 - K2 rather than K2 - K1. 74 See equation 9.20 in Derivative Markets by McDonald.

77 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 76 For this example, here is the line, as well as the premiums for the three calls: C K Convexity is violated because the point (110, 17) is above the line. 75 In other words, the curve of option premium as a function of strike price is not concave upwards The point ( ) is on the line. 76 If concave upwards, the curve should be below any chord drawn between two points on the curve.

78 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 77 In order to demonstrate arbitrage when convexity is violated, in general, one can buy λ of the lowest strike call, buy 1 - λ of the high strike call, and sell 1 of the medium strike call. In this example, we buy 3/4 of the 100 strike calls, buy 1/4 of the 140 strike calls, and sell 1 of the 110 strike calls. Equivalently, we can buy 3 of the 100 strike calls, buy 1 of the 140 strike calls, and sell 4 of the 110 strike calls. When we set up this portfolio, we get: (-3)(20) + (4)(17) + (-1)(7) = 1. We invest this 1 at the risk free rate r. At expiration of the calls at time T, we have: If S 100: e rt > 0 If 110 S > 100: 3(S - 100) + e rt > 0. If 140 S > 110: 3(S - 100) - (4)(S - 110) + e rt = S + e rt > 0. If S > 140: 3(S - 100) - (4)(S - 110) + (S - 140) + e rt = e rt > 0. Thus we always end up with a positive (or at least nonnegative) position, having taken no risk. This demonstrates arbitrage. 79 Arbitrage Opportunities When Convexity is Violated by Put Premiums: If convexity for put premiums are violated, one can set up similar opportunities for arbitrage. For example, three otherwise similar puts have the following premiums: 80 strike put costs strike put costs strike put costs 20. (18-12) / (100-80) = 0.3. (20-18) / ( ) = < 0.3. This violates the convexity of the put premium with respect to the strike price. As with calls, the arbitrage opportunity in such situations involves buying some of the low strike and high strike option, while selling some of the medium strike option. While there are many possible positions that demonstrate arbitrage when convexity is violated, one can use the same technique of coming up with one such a portfolio as was discussed for calls. 77 One could multiply all of the amounts by a constant in order to make them integer. 78 Since convexity is violated, C(K2) > λ C(K1) + (1 -λ) C(K3). Therefore, the medium strike call is overpriced relative to this weighted average of the low strike and high strike calls. Therefore, we sell the medium strike call. 79 Recall, that there are other portfolios that would also demonstrate arbitrage.

79 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 78 In this example, λ = ( ) / (110-80) = 1/3. Thus, we buy 1/3 of the 80 strike puts, buy 2/3 of the 110 strike puts, and sell 1 of the 100 strike puts. Equivalently, we can buy 1 of the 80 strike puts, buy 2 of the 110 strike puts, and sell 3 of the 100 strike puts. When we set up this portfolio, we get: (-1)(12) + (3)(18) + (-2)(20) = 2. We invest this 2 at the risk free rate r. At time T we get back 2e rt. At expiration of the puts at time T, we have: If S 110: 2e rt > 0 If 110 > S 100: (2)(110 - S) + 2e rt > 0. If 100 > S 80: (2)(110 - S) - (3)(100 - S) + 2e rt = S e rt > 0. If S < 80: (2)(110 - S) - (3)(100 - S) + (80 - S) + 2e rt = 2e rt > 0. Thus we always end up with a positive (or at least nonnegative) position, having taken no risk. This demonstrates arbitrage.

80 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 79 Chart of Arbitrage Opportunities: 80 Strike Prices: K 1 < K 2 < K 3. C is the premium for a Call, while P is the premium for a put. Condition C(K 1 ) C(K 2 ). C(K 1 ) - C(K 2 ) K 2 - K 1. Arbitrage if the Condition is Violated Buy the K 1 Call and Sell the K 2 Call (Call Bull Spread) Sell the K 1 Call and Buy the K 2 Call (Call Bear Spread) C(K 1 ) - C(K 2 ) K 2 - K 1 Buy λ = (K 3 - K 2 )/(K 3 - K 1 ) of K 1, Sell 1 of K 2, Buy 1 - λ of K C(K 2) - C(K 3 ) K 3 - K 2. (Asymmetric butterfly spread) P(K 2 ) P(K 1 ). P(K 2 ) - P(K 1 ) K 2 - K 1. Sell the K 1 Put and Buy the K 2 Put (Put Bear Spread) Buy the K 1 Put and Sell the K 2 Put (Put Bull Spread) P(K 2 ) - P(K 1 ) K 2 - K 1 Buy λ = (K 3 - K 2 )/(K 3 - K 1 ) of K 1, Sell 1 of K 2, Buy 1 - λ of K P(K 3) - P(K 2 ) K 3 - K 2. (Asymmetric butterfly spread) 80 See page 300 of Derivative Markets by McDonald. 81 There are other sets of amounts of each call that would also demonstrate arbitrage. 82 There are other sets of amounts of each put that would also demonstrate arbitrage.

81 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 80 Problems: Use the following information for the next 4 questions: The price of the stock of the Willy Wonka Chocolate Company 6 months from now has the following distribution: Price Probability % % % % The continuously compounded annual rate of interest is 5%. 3.1 (1 point) Determine the expected payoff of a 6 month European call option on one share of Willy Wonka Chocolate Company, with a strike price of 180. A. 12 B. 13 C. 14 D. 15 E (1 point) Determine the expected payoff of a 6 month European put option on one share of Willy Wonka Chocolate Company, with a strike price of 160. A. 12 B. 13 C. 14 D. 15 E (1 point) Determine the actuarial present value of a 6 month European call option on one share of Willy Wonka Chocolate Company, with a strike price of 170. A B C D E (1 point) Determine the actuarial present value of a 6 month European put option on one share of Willy Wonka Chocolate Company, with a strike price of 170. A B C D E (1 point) 3 European put options on a stock are otherwise similar except for their strike price. A put with a strike price of 150 has a premium of 30, in other words costs 30. A put with a strike price of 160 has a premium of 34, in other words costs 34. A put with a strike price of 180 has a premium of 40, in other words costs 40. What general property of the value of puts is violated? 3.6 (3 points) Briefly describe an opportunity for arbitrage presented by the situation in the previous question.

82 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page (1 point) Two European put options on a stock are otherwise similar except for their strike price. A put with a strike price of 60 has a premium of 7. A put with a strike price of 80 has a premium of 15. Which of the following is true about the premium of a similar 70 strike put? A. The smallest possible premium is 8 B. The smallest possible premium is 11 C. The largest possible premium is 8 D. The largest possible premium is 11 E. None of A, B, C, or D 3.8 (1 point) Two European call options on a stock are otherwise similar except for their strike price. A call with a strike price of 100 has a premium of 27, in other words costs 27. A call with a strike price of 110 has a premium of 15, in other words costs 15. What general property of the value of calls is violated? 3.9 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the previous question (1 point) Two European call options on a stock are otherwise similar except for their strike price. A call with a strike price of 80 has a premium of 12. A call with a strike price of 85 has a premium of 10. Which of the following is true about the premium of a similar 100 strike call? A. The smallest possible premium is 2 B. The smallest possible premium is 4 C. The largest possible premium is 2 D. The largest possible premium is 4 E. None of A, B, C, or D 3.11 (1 point) 2 European put options on a stock are otherwise similar except for their strike price. A put with a strike price of 150 has a premium of 22, in other words costs 22. A put with a strike price of 170 has a premium of 46, in other words costs 46. What general property of the value of puts is violated? 3.12 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the previous question.

83 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page (3 points) European call and put prices for options on a given stock are available as follows: Strike Price Call Price Put Price $50 $21 $3 $60 $16 $7 $80 $8 $14 All six options have the same expiration date, which is no more than two years from now. After reviewing the information above, Danielle, Eric, and Felicia agree that arbitrage opportunities arise from these prices. Danielle believes that one could use the following portfolio to obtain arbitrage profit: Long two calls with strike price 50; short four calls with strike price 60; long two calls with strike price 80; and either lend or borrow some money. Eric believes that one could use the following portfolio to obtain arbitrage profit: Long two puts with strike price 50; short three puts with strike price 60; long one put with strike price 80; and either lend or borrow some money. Felicia believes that one could use the following portfolio to obtain arbitrage profit: Short one call with strike price 50; long two calls with strike price 60; short one call with strike price 80; long one put with strike price 50; short two puts with strike price 60; long one put with strike price 80; and either lend or borrow some money. Which of the following statements is true? (A) Danielle and Eric are correct, while Felicia is not. (B) Danielle and Felicia are correct, while Eric is not. (C) Eric and Felicia are correct, while Danielle is not. (D) All of them are correct. (E) None of A, B, C, or D (1 point) 3 European call options on a stock are otherwise similar except for their strike price. A call with a strike price of 100 has a premium of 25, in other words costs 25. A call with a strike price of 110 has a premium of 20, in other words costs 20. A call with a strike price of 115 has a premium of 16, in other words costs 16. What general property of the value of calls is violated? 3.15 (3 points) Briefly describe an opportunity for arbitrage presented by the situation in the previous question (1 point) Two European call options on a stock are otherwise similar except for their strike price. A call with a strike price of 120 has a premium of 22, in other words costs 22. A call with a strike price of 125 has a premium of 16, in other words costs 16. What general property of the value of calls is violated? 3.17 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the previous question.

84 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page (1 point) Two European call options on a stock are otherwise similar except for their strike price. A call with a strike price of 80 has a premium of 12, in other words costs 12. A call with a strike price of 90 has a premium of 14, in other words costs 14. What general property of the value of calls is violated? 3.19 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the previous question (2 points) Two European call options are otherwise similar except for their strike prices. A call with a strike price of 85 has a premium of 13. A call with a strike price of 90 has a premium of 7. In order to take advantage of arbitrage, you buy 1000 of the 90 strike calls and sell x of the 85 strike calls. If r = 0%, what is the smallest possible value of x? A. 800 B. 825 C. 850 D. 875 E. 900

85 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page (2 points) A long range forward contract consists of buying a call, and selling a similar put with a lower strike but the same time until expiration, where the strikes are chosen so that the contract has no initial cost. The current exchange rate is 1.58 Canadian Dollars per British Pound. The company Bright, Light, and Powers is based in Canada. It will have to make a payment of 100,000 British Pounds in 4 months. In order to limit its risk, the company buys a 4-month long range forward contract on 100,000 British Pounds. The strike of the call is 1.65 Canadian Dollars. The strike of the put is 1.50 Canadian Dollars. Graph as a function of the future exchange rate the amount of Canadian Dollars that Bright, Light, and Powers has to spend 4 months from now in order to make its payment of 100,000 British Pounds (1 point) The price of a stock one year from now has the following distribution in the risk neutral environment: Price Probability 50 10% % % % % Let X be the expected future value of a 1 year European call option on 100 shares of this stock with a strike price of 170. Let Y be the expected future value of a 1 year European put option on 100 shares of this stock with a strike price of 140. Determine Y - X. A. 150 B. 200 C. 250 D. 300 E. 350

86 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page (MFE Sample Exam, Q.2) Near market closing time on a given day, you lose access to stock prices, but some European call and put prices for a stock are available as follows: Strike Price Call Price Put Price $40 $11 $3 $50 $6 $8 $55 $3 $11 All six options have the same expiration date. After reviewing the information above, John tells Mary and Peter that no arbitrage opportunities can arise from these prices. Mary disagrees with John. She argues that one could use the following portfolio to obtain arbitrage profit: Long one call option with strike price 40; short three call options with strike price 50; lend $1; and long some calls with strike price 55. Peter also disagrees with John. He claims that the following portfolio, which is different from Maryʼs, can produce arbitrage profit: Long 2 calls and short 2 puts with strike price 55; long 1 call and short 1 put with strike price 40; lend $2; and short some calls and long the same number of puts with strike price 50. Which of the following statements is true? (A) Only John is correct. (B) Only Mary is correct. (C) Only Peter is correct. (D) Both Mary and Peter are correct. (E) None of them is correct (IOA, CT8, 4/09, Q.1) (2.25 points) Describe what is meant by an arbitrage opportunity.

87 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 86 Solutions to Problems: 3.1. B. E[(S - 180) + ] = (20%)(0) + (40%)(0) + (30%)(20) + (10%)(70) = 13. Comment: If the future price is low, then the option to buy at 180 is worthless. We have ignored the time value of money E. E[(160 - S) + ] = (20%)(60) + (40%)(10) + (30%)(0) + (10%)(0) = 16. Comment: If the future price is high, then the option to sell at 160 is worthless D. E[(S - 170) + ] e -rt = {(20%)(0) + (40%)(0) + (30%)(30) + (10%)(80)} e -(.05)(1/2) = E. E[(170 - S) + ] e -rt = {(20%)(70) + (40%)(20) + (30%)(0) + (10%)(0)} e -(.05)(1/2) = & 3.6. The absolute value of the changes in the value of the option over the absolute value of the changes in the strike price are: (34-30)/( ) = 0.4, and (40-34)/( ) = > 0.3, which violates the proposition that the rate of change of the put option premium must increase as the strike price rises. In other words, convexity is violated. λ = ( )/( ) = 2/3. We can buy 2/3 puts with strike price 150, buy 1/3 puts with a strike price of 180, and sell 1 put with strike price 160. Equivalently, we can buy 2 puts with strike price 150, buy 1 put with a strike price of 180, and sell 3 puts with strike price 160. Then you collect a net of: (3)(34) - (2)(30) - (1)(40) = 2. You could invest this 2 at the risk free rate and have 2e rt at the expiration of the options. If S T 180, then all the puts turn out to worthless. You end up with 2e rt 2. If 180 > S T 160, then the put that you have bought with strike price of 180 will be exercised. You will make S T by exercising this put. You end up with: 2e rt + (180 - S T ) 2. If 160 > S T 150, then the puts that you have sold will be exercised. You will pay (3)(160) = 480 for the 3 shares you buy from the person holding the puts. Then you can sell 1 share at 180 and 2 shares at S T. You end up with: 2e rt S T = 2e rt + 2S T (2)(150) = 2. If S T < 150, then you will pay (3)(160) = 480 for the 3 shares you buy from the person holding the puts. Then you can sell 1 share at 180 and 2 shares at 150. You end up with: 2e rt = 2e rt 2. In all possible situations you end up with a positive amount at time T, not having invested a positive amount of money. Comment: See Example 9.9 in Derivative Markets by McDonald.

88 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page D. If the premiums were on a straight line, then the premium at a 70 strike would be 11. For convexity to hold, the line between (60, 15) and (80, 7) must not be below the point (70, premium). Therefore, the premium for a 70 strike must be 11 or less & 3.9. The absolute value of the change in the value of the option is: = 12, which is greater than the absolute value of the change in the strike price: = 10. This should not occur. You could buy a call with strike price 110 and sell a call with strike price 100. Then you collect a net of: = 12. You could invest this 12 at the risk free rate and have 12e rt, at the expiration of the call options. If S T 100, then both calls turn out to worthless. You end up with 12e rt > 12. If 110 S T > 100, then the call that you have sold will be exercised. You can buy a share at S T and then will get 100 for the share when you sell it to the person holding the call you sold. You end up with: 12e rt - (S T - 100) > = 2. If S T 110, then the call that you have sold will be exercised, buying a share at 110. Then will get 100 for the share when you sell it to the person holding the call. You end up with: 12e rt - 10 > = 2. In all possible situations you end up with a positive amount at time T, not having invested a positive amount of money. Comment: See Example 9.7 in Derivative Markets by McDonald. Arbitrage is a transaction where you always end up with a positive position, with no net investment and no risk; in other words, arbitrage represents a free lunch. Arbitrage involves the simultaneous buying and selling of related assets B. If the premiums were on a straight line, then the premium for a 100 strike would be 4. For convexity to hold, the line between (80, 12) and (100, premium) must not be below the point (85, 10). Therefore, the premium for a 100 strike must be 4 or more.

89 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page & The absolute value of the change in the value of the option is: = 24, which is greater than the absolute value of the change in the strike price: = 20. This should not occur. You could buy a put with strike price 150 and sell a put with strike price 170. Then you collect a net of: = 24. You could invest this 24 at the risk free rate and have 24e rt, at the expiration of the options. If S T 170, then both puts turn out to worthless. You end up with 24e rt > 24. If 170 > S T 150, then the put that you have sold will be exercised. You buy a share at 170 from the person who owns the put you sold, and then you sell the share for S T. You end up with: 24e rt - (170 - S T ) > = 4. If S T < 150, then the put that you have sold will be exercised. You buy a share at 170 from the person who owns the put you sold, and then you sell the share for 150, using the put you own. You end up with: 24e rt - ( ) > = 4. In all possible situations you end up with a positive amount at time T, not having invested a positive amount of money.

90 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page C. The call premiums satisfy all three properties, and thus there is no arbitrage opportunity. If Danielle buys her portfolio of calls, then she would get: (4)(16) - (2)(21) - (2)(8) = 6. She would lend out 6. If for example, S T = 80, then the payoff from the calls is: (2)(30) + (-4)(20) + (2)(0) = -20. Thus she would have 6e rt - 20, which is almost surely not positive. Danielleʼs portfolio does not demonstrate arbitrage. The put premiums do not satisfy convexity, and thus there are opportunities for arbitrage. If Eric buys his portfolio of puts, then he would get: (3)(7) - (2)(3) - (1)(14) = 1. He would lend out 1. If S T 50, then the puts payoff: (2)(50 - S) - (3)(60 - S) + (1)(80 - S) = 0. He ends up with e rt > 0. If 50 < S T 60, then the puts payoff: (-3)(60 - S) + (1)(80 - S) = 2S > 0. He ends up with more than e rt > 0. If 60 < S T 80, then the puts payoff: (1)(80 - S) > 0. He ends up with more than e rt > 0. If 80 < S T, then all of the puts expire worthless. He ends up with e rt > 0. Ericʼs portfolio does demonstrate arbitrage. If Felicia buys her portfolio of puts and calls, then she would spend: (-1)(21) + (2)(16) + (-1)(8) + (1)(3) + (-2)(7) + (1)(14) = 6. She would borrow 6. If S T 50, then the payoff is: (1)(50 - S) + (-2)(60 - S) + (1)(80 - S) = 10. If 50 < S T 60, then the payoff is: (-1)(S - 50) + (-2)(60 - S) + (1)(80 - S) = 10. If 60 < S T 80, then the payoff is: (-1)(S - 50) + (2)(S - 60) + (1)(80 - S) = 10. If 80 < S T, then the payoff is: (-1)(S - 50) + (2)(S - 60) + (-1)(S - 80) = 10. So she always ends up with 10-6e rt 10-6e r2 > 0. Feliciaʼs portfolio does demonstrate arbitrage. Comment: Similar to MFE Sample Exam, Q.2. I have assumed that 10 > 6e r2. r < 25.5%.

91 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page & The absolute value of the changes in the value of the option over the absolute value of the changes in the strike price are: (25-20)/( ) = 0.5, and (20-16)/( ) = < 0.8, which violates the proposition that the rate of absolute change of the call option premium must decrease as the strike price rises. In other words, convexity is violated. λ = ( )/( ) = 1/3. We can buy 1/3 calls with strike price 100, buy 2/3 calls with a strike price of 115, and sell 1 call with strike price 110. Equivalently, we can buy 1 call with strike price 100, buy 2 calls with a strike price of 115, and sell 3 calls with strike price 110. Then you collect a net of: (3)(20) - (1)(25) - (2)(16) = 3. You loan out this 3 and collect interest at the risk free rate. At expiration of the calls at time T, we have: If S 100: 3e rt > 0 If 110 S > 100: (S - 100) + 3e rt > 0. If 115 S > 110: (S - 100) - (3)(S - 110) + 3e rt = 230-2S + 3e rt > 0. If S > 115: (S - 100) - (3)(S - 110) + (2)(S - 115) + 3e rt = 3e rt > 0. Thus we always end up with a positive (or at least nonnegative) position, having taken no risk. This demonstrates arbitrage. Comment: See Example 9.8 in Derivative Markets by McDonald. The given call premiums as a function of strike price: 25 C K Since (110, 20) is above the line between (100, 25) and (115, 16), convexity is violated. The line between (100, 25) and (115, 16) can be written as a weighted average of 25 and 16: y = 25 β + 16 (1 - β), where β = (115 - K)/( ). At K = K 2 = 110, β = 1/3 = (K 3 - K 2 )/(K 3 - K 1 ) = λ, and the height of the line is: (25)(1/3) + (16)(2/3) = 19 < 20. For convexity to hold, we require that: λ C(K1) + (1 - λ) C(K3) C(K2). There are other possible portfolios that would demonstrate arbitrage, but the one given in my solution is based on how in the textbook arbitrage is demonstrated for this situation.

92 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page For K 1 K 2, C(K 1 ) - C(K 2 ) K 2 - K 1. However, here = 6 > 5 = The absolute change in call premium, should not be greater than the absolute change in the strike prices Sell one 120 strike call and buy one 125 strike call. You get a net of: = 6. If the future stock price is less than 120, then both calls are worthless. If the future stock price is between 120 and 125, then you lose money from the first call, while the second call is worthless. Your position is worth: 6e rt - (S - 120) > 6-5 = 1. If the future stock price is more than 125, then you lose money from the 120 strike call that you sold: S - 120, while making money on the second call: S Your position is worth: 6e rt - (S - 120) + (S - 125) > 6e rt - 5 > 1. For an initial gain, you make money or break even, demonstrating arbitrage The premium for the call should not increase as the strike price increases Buy one 80 strike call and sell one 90 strike call. You get a net of: = 2. If the future stock price is less than 80, then both calls are worthless. If the future stock price is between 80 and 90, then you make money from the first call, while the second call is worthless. If the future stock price is more than 90, then you make money from the first call: S - 80, while losing money on the second call: S Overall, you make: (S - 80) - (S - 90) = 10. For an initial gain, you make money or break even, demonstrating arbitrage.

93 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page D. If the future stock price is S T < 85, then both calls are worthless. We want 13x - (1000)(7) 0. x > 538. If the future stock price is 90 > S T > 85, then the 90 strike calls are worthless, but we lose money on the 85 strike calls we sold. The worst case is when S T = 90 and we lose 5 on each 85 strike call. We want 13x - (1000)(7) 5x. x 875. If the future stock price is S T > 90, then we make money on the 90 strike calls, and we lose money on the 85 strike calls we sold. We want 13x - (1000)(7) x(s T - 85) - (1000)(S T - 90). x(98 - S T ) (1000)(97 - S T ). If 98 - S T > 0, then we require that x (1000)(97 - S T )/(98 - S T ) = 1000{1-1/(98 - S T )}. For 90 < S T < 98. The left hand side is largest for S T = 90; (1000)(97-90)/(98-90) = 875. We need x 875. If 98 - S T < 0, then we require that x (1000)(97 - S T )/(98 - S T ) = 1000{1 + 1/(S T - 98)}. Thus we need x We conclude that we can take 875 x The smallest possible value of x is 875. Comment: For example, if x = 900, then we get for setting up the portfolio: (900)(13) - (1000)(7) = If S T < 85 both calls are worthless. If S T = 90, we lose (5)(900) = 4500 on the 85 strike calls we sold, but still come out ahead. If for example, S T = 100, we lose (15)(900) = 13,500 on the 85 strike calls we sold, and make (1000)(10) = 10,000 on the 90 strike calls we bought, for a net loss of 3500; but we still come out ahead due to the 4700 we got for setting up the portfolio.

94 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page If the exchange rate is greater than 1.65, it uses its call to buy 100,000 British Pounds for 165,000 Canadian Dollars. If the exchange rate is less than 1.50, the person who bought the put uses it to sell 100,000 British Pounds to Bright, Light, and Powers for 150,000 Canadian Dollars. For exchange rates in the middle, both options expire worthless, and Bright, Light, and Powers pays the current exchange rate to buy 100,000 British Pounds. Amount rate D. E[(S - 170) + ] = (10%)(0) + (20%)(0) + (40%)(0) + (20%)(30) + (10%)(80) = 14. E[(140 - S) + ] = (10%)(90) + (20%)(40) + (40%)(0) + (20%)(0) + (10%)(0) = 17. Y - X = 100 E[(140 - S) + ] E[(S - 170) + ] = = 300.

95 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page D. Both Mary and Peter are correct; the prices are not arbitrage-free. The call option prices do not satisfy their convexity condition: = 0.5 > = 0.6. Mary buys one 40-strike calls; sells three 50-strike calls; lends $1; and buys some 55-strike calls λ = = 1/3. Buy λ of K 1. Sell 1 of K 2. Buy 1 - λ of K 3. So Mary can buy two of the 55-strike calls. To buy and sell the calls brings in $1: (3) (6) (2) (3) = 1. Thus Mary does indeed lend out $1. The prices are not arbitrage-free. To show that Maryʼs portfolio yields arbitrage profit: Time 0 Time T Time T Time T Time T S T < S T < S T < 55 S T 55 Buy 1 call S T 40 S T 40 S T 40 Strike 40 Sell 3 calls (S T 50) -3(S T 50) Strike 50 Lend $1-1 e rt e rt e rt e rt Buy 2 calls (S T 55) Strike 55 Total 0 e rt > 0 e rt + S T - 40 e rt + 2(55 - S T ) e rt > 0 The total at time 0 is zero, while in all cases, the total at time T is positive, proving arbitrage.

96 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 95 Strike Price Call Price Put Price $40 $11 $3 $50 $6 $8 $55 $3 $11 Peter buys 1 40-strike call & sells 1 40-strike put: 11-3 = 8 money out the door. Buys 2 55-strike calls & sells 2 55-strike puts: (2)(11) - (2)(3) = 16 money in the door. Sells x 50-strike calls & buys x 50-strike puts: x8 - x6 = 2x money out the door. We are told that Peter lends $2. He must get $2 in the door net. 2 = x. 2x = 6. x = 3. Let a position be the purchase of a K-strike call and sale of a K-strike put. Peter buys one 40-strike position, sells three 50-strike positions, buys two 55-strike positions, and lends $2. The payoff on the purchase of a K-strike call and sale of a K-strike put is: S (S T - K)+ - (K - S T ) + = T - K if S T > K -(K - S T ) if S T < K At expiration, the payoff on his positions is: S T (3) (S T - 50) + (2) (S T - 55) = 0. Thus at time T he has: 2 e rt > 0. = S T - K. The total at time 0 is zero, while the total at time T is positive, proving arbitrage. Alternately, Peterʼs portfolio makes arbitrage profit, because: Time-0 cash flow Time-T cash flow Buy 1 call & sell 1 put Strike = -8 S T - 40 Sell 3 calls & buy 3 puts Strike 50 3(6-8) = -6 3(50 - S T ) Lend $2-2 2e rt Buy 2 calls & sells 2 puts Strike 55 2( ) = 16 2(S T - 55) Total 0 2e rt The total at time 0 is zero, while the total at time T is positive, proving arbitrage. Comment: See Table 9.7 in Derivative Markets by McDonald. The call option prices do not satisfy their convexity condition: (11-6)/(50-40) = 0.5, while (6-3)/(55-50) = 0.6 which is larger.

97 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 96 These call premiums as a function of strike price: C K Since the point (50, 6) is above the line between (40, 11) and (55, 3), convexity is violated. In other words, the curve of option premium versus strike price is not concave upwards. The line between (40, 11) and (55, 3) can be written as a weighted average of 11 and 3: y = 11 β + 3 (1 - β), where β = 55 - K At K = K 2 = 50, β = 1/3 = K 3 - K 2 K 3 - K 1 = λ, and the height of the line is: (11)(1/3) + (3)(2/3) = 5.67 < 6. For convexity to hold, we require that: λ C(K 1 ) + (1 - λ) C(K 3 ) C(K 2 ).

98 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page 97 In contrast, the put option prices do satisfy their convexity condition: = 0.5 > = 0.6 which is larger. These put premiums as a function of strike price: 11 P K Since the point (50, 8) is below the line between (40, 3) and (55, 11), convexity is satisfied. In other words, the curve of option premium versus strike price is concave upwards. Long Buy. Short Sell. The payoff on Peterʼs combination of positions is always zero; thus its correct price is 0. Since its price is not 0, there is an opportunity for arbitrage. Depending on the sign of this price, either we would buy and sell Peterʼs combination of positions, or we would do the exact opposite of what Peter did.

99 2013-MFE/3F, Financial Economics 3 Properties of Premiums, HCM 12/6/12, Page Put in simple terms, an arbitrage opportunity is a situation where we can make a sure profit with no risk. This is sometimes described as a free lunch. Put more precisely an arbitrage opportunity means that: (a) We can start at time 0 with a portfolio which has a net value of zero (implying that we are long in some assets and short in others). (b) At some future time T: the probability of a loss is 0. the probability that we make a strictly positive profit is greater than 0.

100 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 99 Section 4, Put-Call Parity The value of an otherwise similar call and put option on the same asset are related. Adam and Eve Example: XYZ stock pays no dividends. Adam buys a 2 year call on XYZ stock with strike price $100. Adam also loans out $100 e -2r at the risk free rate. In two years, Adam will have $100. If XYZ stock has a price > $100, Adam will use $100 and his call to buy a share of XYZ. Otherwise, Adam keeps his $100. Two years from now, Adam ends up with a share of XYZ or $100, whichever is worth more. Eve buys a 2 year put on XYZ stock with strike price $100 from Seth. Eve also buys a share of XYZ stock. In two years, if XYZ stock has a price < $100, Eve will use her put to sell a share of XYZ stock to Seth for $100. Otherwise, Eve does not exercise her put. Two years from now, Eve ends up with a share of XYZ or $100, whichever is worth more. Future Stock Price Adam Eve < $100 $100 $100 > $100 Stock Stock Two years from now Adam and Eve have the same position. Therefore, Adam and Eveʼs initial positions must have the same price. Call + K e -rt = Put + Stock. If the stock had paid dividends, then Eve would have collected them, while Adam will not. If we subtract the present value of these dividends, then their two positions would still be equal. PV[F 0,T ] = S 0 - PV[Div]. Setting equal the values of Adamʼs position and Eveʼs position minus any dividends we have: Call + K e -rt = Put + PV[F 0,T ].

101 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 100 Put-Call Parity formula: Thus we have the following put-call parity formula: 83 C Eur (K, T) = P Eur (K, T) + PV[F 0,T ] - PV[K]. Exercise: In the absence of stock dividends, determine the future value of the sum of: (a) A share of stock. (b) A European put option on that stock with a strike price of K. (c) Having sold a call option on that stock with a strike price of K and the same expiration as the put. [Solution: Future value is: S T + (K - S T ) + - (S T - K) +. If S K, then this future value is: S T + K - S T - 0 = K. If S K, then this future value is: S T (S T - K) = K. Comment: (X-d) + - (d-x) + = X - d. X + (d-x) + - (X-d) + = d.] Thus we again have put-call parity. As in the Adam and Eve example, in the absence of dividends: Share of Stock at Time T + Put - Call = K. Value of Call at Time T - Value of Put at Time T = Value of Stock at Time T - Strike Price. Stocks With Discrete Dividends: If dividends are paid at discrete times, then PV[F 0,T ] = S 0 - PV[Div]. 84 C Eur (K, T) = P Eur (K, T) + S 0 - PV[Div] - K e -rt. 85 Stocks With Continuous Dividends: If dividends are paid continuously, then PV[F 0,T ] = S 0 e -δt. C Eur (K, T) = P Eur (K, T) + S 0 e -δt - K e -rt See Equation 3.1 in Derivatives Markets by McDonald. 84 F0,T, the future value of the share of stock, is the expected value of owning a share at time T, excluding any dividends that may have been paid from time 0 to time T. 85 See equation 9.2 in Derivatives Markets by McDonald. 86 See below equation 9.2 in Derivatives Markets by McDonald. In the case of no dividends, δ = 0.

102 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 101 Exercise: S 0 = $100, r = 6%, δ = 2%, T = 1/2. The present value of a European put option with a strike price of $120 is $30. Determine the present value of a European call option with the same strike price of $120. [Solution: $30 + $100 e $120 e = $12.55.] Thus given r, δ, S 0, K and T, if we know the value of either the put or call, then we know the value of the other one. Exercise: S 0 = $100, r = 6%, δ = 2%, T = 1/2. The present value of a European call option with a strike price of $110 is $15. Determine the present value of a European put option with the same strike price of $110. [Solution: $15 - $100 e $110 e = $22.74.] Exercise: On a stock that pays continuous dividends, at what strike price are a European put and call worth the same? [Solution: 0 = C Eur (K, T) - P Eur (K, T) = S 0 e -δt - K e -rt. K = S 0 e (r-δ)t. Comment: In a risk-neutral environment, the expected stock price at time T is: S 0 e (r-δ)t.] Strike Price Equal to Current Stock Price: 87 If the stock pays no dividends, then C Eur (K, T) - P Eur (K, T) = S 0 - K e -rt. Let us assume S 0 = K = $100. Then C - P = $100(1 - e -rt ). Let us assume we have $100 today available to spend. If we buy the stock now for $100, we can hold it until time T. If we instead buy the call and sell the put, then this will cost $100(1 - e -rt ). Subtracting this from our $100, we would have $100 e -rt left. Assuming we invest $100 e -rt at the risk free rate, when time T comes we would have $100. At time T if S T 100, we can buy a share for 100 using our call. At time T if S T < 100, then we buy a share for 100 from the person who bought our put. In both situations, we end up with the stock and have invested $100 at time 0 for it. By buying the call and selling the put, we have deferred the payment of $100 until T. This deferral costs interest on the $100 of: $100(1 - e -rt ). Thus the option premiums differ by interest on the deferral of the payment for the stock. 87 See Example 9.1 in Derivative Markets by McDonald. When S = K, the option is said to be at the money.

103 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 102 Derivation of Put-Call Parity: The payoff on a call is (S - K) +. The payoff on a put is (K - S) +. If one buys a call and sells a similar put, the payoff is: (S - K) + - (K - S) + = S - K. Therefore, taking the prepaid forward prices, in other words the actuarial present values: C - P = P F 0,T [S T ] - P F 0,T [K] = PV[F 0,T ] - PV[K]. Bonds: A bond pays coupons to its owner. These coupon payments act mathematically like stock dividends paid at discrete times. Therefore, if B 0 is the current price of the bond: 88 C Eur (K, T) = P Eur (K, T) + B 0 - PV[Coupons] - K e -rt. Summary: 89 As will be discussed in subsequent sections, there are similar relationships for other assets. Asset Stock, No Dividends Parity Relationship S 0 = C - P + e -rt K Stock, Discrete Dividends S 0 - PV[Div] = C - P + e -rt K Stock, Continuous Dividends e -δt S 0 = C - P + e -rt K Bond B 0 - PV[Coupons] = C - P + e -rt K. Futures Contract e -rt F 0,T = C - P + e -rt K Currency exp[-r f T] x 0 = C - P + e -rt K Exchange Options F P 0,T (S0 ) = C - P + F P 0,T (Q0 ) exp[-δ S T] S 0 = C - P + exp[-δ Q T] Q 0 88 While listed by McDonald, he does not discuss further this formula involving bonds. In this formula, e -rt is the price of a zero-coupon bond that pays 1 at time T. 89 See Table 9.9 at page 305 of Derivatives Markets by McDonald.

104 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 103 Problems: 4.1 (2 points) Jay Corp. common stock is priced at $80 per share. The company just paid its $1 quarterly dividend. Interest rates are 6.0%. A $75 strike European call, maturing in 8 months, sells for $10. What is the price of a 8-month, $75 strike European put option? A. 3 B. 4 C. 5 D. 6 E (2 points) A bond currently costs $830. The bond has $10 quarterly coupons. A coupon has just been paid. r = 4%. What is the difference in price between a 2-year $800 strike European call option and the corresponding put? A. 15 B. 20 C. 25 D. 30 E. 35 Use the following information for the next two questions: The Rich and Fine stock index is priced at Dividends are paid at the rate of 2%. The continuously compounded risk free rate is 5%. 4.3 (1 point) A 1800 strike European call, maturing in 4 years, sells for 196. What is the price of a 4-year, 1800 strike European put option? A. Less than 400 B. At least 400, but less than 450 C. At least 450, but less than 500 D. At least 500, but less than 550 E. At least (1 point) A 1500 strike European put, expiring in 3 years, sells for 293. What is the price of a 3-year, 1500 strike European call option? A. 150 B. 175 C. 200 D. 225 E (2 points) For a stock that does not pay dividends, the difference in price between otherwise similar European call and put options is The current stock price is 100. r = 6%. The time until expiration is 2 years. Determine the strike price. A. 80 B. 85 C. 90 D. 95 E (2 points) 160 = current market price of the stock 130 = strike price of the option 5% = annual risk free force of interest 18 months = time until the exercise date of the option 3% = (continuously compounded) annual rate at which dividends are paid on this stock If a European put has an option premium of 13, determine the premium for a similar European call. A. 35 B. 40 C. 45 D. 50 E. 55

105 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 104 Use the following information for the next four questions: You have the following bid and ask prices on 3.25 month European options on HAL stock with a strike price of $160. Call Bid Call Ask Put Bid Put Ask $13.00 $13.40 $2.90 $3.20 You buy at the ask and sell at the bid. Your continuously compounded lending rate is 1.9% and your borrowing rate is 2%. The current price of HAL stock is $ Ignore transaction costs on the stock. HAL will pay a $0.36 dividend one month from today. 4.7 (2 points) You sell the call, buy the put, buy the stock, and borrow the present value of the strike price plus dividend. What is the cost? 4.8 (2 points) Briefly discuss whether parity is violated by the situation in the previous question. 4.9 (2 points) What is the cost if you buy the call, sell the put, short the stock, and lend the present value of the strike price plus dividend? 4.10 (2 points) Briefly discuss whether parity is violated by the situation in the previous question (2 points) The price of a non-dividend paying stock is $85 per share. A 18 month, at the money European put option is trading for $5. If the interest rate is 6.5%, what is the price of a European call at the same strike and expiration? A. 12 B. 13 C. 14 D. 15 E (2 points) As a function of the future stock price, graph the future value of the sum of a share of the stock and a European put option on that same stock with a strike price of (2 points) A stock that pays continuous dividends at 1.5%, has a price of 130. r = 4%. At what strike price are a 2 year European put and call worth the same? A. Less than 126 B. At least 126, but less than 128 C. At least 128, but less than 132 D. At least 132, but less than 136 E. At least 136

106 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (2 points) The prices for otherwise similar European puts and calls are: Strike Call premium Put premium Describe a spread position involving puts and calls that you can use to effect arbitrage, and demonstrate that arbitrage occurs (2 points) Kay Corp. common stock is priced at $120 per share. Yesterday, the company paid its $2 quarterly dividend. The continuously compounded interest rate is 5.0%. The company just paid its $2 quarterly dividend. Interest rates are 5.0%. A $130 strike European put, maturing in 4 months, sells for $15. What is the price of a 4-month, $130 strike European call option? A. 3 B. 4 C. 5 D. 6 E (3 points) You are given the following premiums for European options: Strike Price Call Price Put Price $60 $3.45 $80 $23.85 $9.88 $90 $19.56 $14.38 $110 $13.09 All of the options are on the same stock and have the same expiration date. Determine the sum of the price of the 60 strike call plus the price of the 110 strike put. A B C D E (2 points) Joe buys a share of a stock, buys a European put option on that stock with a strike price of 100, and sells a European call option on that stock with a strike price of 100. Graph the future value of Joeʼs investments as a function of the stock price at expiration of the options (2 points) 150 = current market price of the stock 140 = strike price of the option 6% = annual risk free force of interest 6 months = time until the exercise date of the option 3% = (continuously compounded) annual rate at which dividends are paid on this stock If a European call has an option premium of 23, determine the premium for a similar European put. A. 7 B. 8 C. 9 D. 10 E (2 points) A European put and call on the same stock have the same time until maturity and the same strike price. S 0 = 100. K = 115. r = 6%. δ = 2%. If the put and the call have the same premium, what is the time until maturity? A. 1.5 years B. 2.0 years C. 2.5 years D. 3.0 years E. 3.5 years

107 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (2 points) A stock is priced at $100 per share. The stock is forecasted to pay dividends of $0.80, $1.20, and $1.50 in 3, 6, and 9 months, respectively. r = 5.5%. A $125 strike European call, maturing in 9 months, sells for $8. What is the price of a 9-month, $125 strike European put option? A. Less than 25 B. At least 25, but less than 30 C. At least 30, but less than 35 D. At least 35, but less than 40 E. At least (2 points) Consider 6 month 80 strike European options on a nondividend paying stock. Two months after being purchased, the call and the put would have the same value for a stock price of Determine r. A. 4.0% B. 4.5% C. 5.0% D. 5.5% E. 6.0% 4.22 (2 points) Near market closing time on a given day, you lose access to stock prices, but some European call and put prices for a stock are available as follows: Strike Price Call Price Put Price $70 $11 $3 $80 $6 $7 $90 $3 All of the options have the same expiration date. Determine the price of the 90 strike put. A B C D E (2 points) A one year European put and call on the same stock with the same strike price have the same premium. The current stock price is 84. The stock will pay dividends of 2, one month from now, four months from now, seven months from now, and ten months from now. If r = 5%, what is the strike price? A. 75 B. 80 C. 85 D. 90 E (2 points) A two-year Call Bull Spread has strike prices that differ by 10. The premium for this Call Bull Spread is r = 5.6%. Determine the premium for a similar Put Bear Spread. A B C D E. 4.75

108 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (2 points) You are given the following information on European options on a given stock index that does not pay dividends. All the options have the same strike price and all expire on July 1, As of January 1, 2008 As of March 1, 2008 Stock Index Price Call Premium Put Premium Determine the continuously compounded risk rate of interest, r. A. 4.5% B. 5.0% C. 5.5% D. 6.0% E. 6.5% 4.26 (2 points) A stock is priced at $93 per share. The stock pays dividends at a continuously compounded rate δ. r = 5.2%. The premium for a $90-strike 2-year European call is $ The premium for a $90-strike 2-year European put is $9.36. Determine δ. A. 0.5% B. 1.0% C. 1.5% D. 2.0% E. 2.5% 4.27 (3 points) You observe the prices of various European call and put options all on the same stock and all with the same expiration date: Strike Price Call Price Put Price $60 $21 $6 $70 $17 $11 $90 $12 $25 After reviewing the information above, Andrew tells Brittany and Christopher that no arbitrage opportunities can arise from these prices. Brittany disagrees with Andrew. She argues that one could use a portfolio of various calls to obtain arbitrage profit. Christopher also disagrees with Andrew. He claims that one could use a portfolio of various puts to obtain arbitrage profit. Which of the following statements is true? (A) Only Andrew is correct. (B) Only Brittany is correct. (C) Only Christopher is correct. (D) Both Brittany and Christopher are correct. (E) None of them is correct.

109 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (3 points) A nondividend paying stock has a current price of 76. You observe the prices of various 2-year European options on this stock: 70-strike call has a premium of strike put has a premium of strike call has a premium of Determine the possible values of r (3 points) Consider European options all on the same stock and all with the same time until expiration. Dustin buys a 90-strike call and sells a 100-strike put. His net cost is Edith buys a 100-strike call and sells a 90-strike put. Her net cost is Frank buys a 120-strike call and sells a 120-strike put. His net cost is Gabrielle buys a 110-strike call and sells a 110-strike put. What is Gabrielleʼs net cost? A. 8.4 B. 8.6 C. 8.8 D. 9.0 E (2 points) Consider European options all on the same stock and all with the same time until expiration. Ashley buys a 60-strike call and sells an 80-strike put. Her net cost is Chelsea buys an 80-strike call and sells a 60-strike put. Her net cost is Benjamin buys an 70-strike call and sells a 70-strike put. What is Benjaminʼs net cost? A. 2.8 B. 2.9 C. 3.0 D. 3.1 E (2 points) A stock pays dividends at a continuously compounded rate of 2%. The current price of the stock is 82. The effective annual risk-free interest rate is 10%. You buy a three-year 80-strike European call option on the stock and sell a similar put. Determine the premium for this position. (A) 16 (B) 17 (C) 18 (D) 19 (E) (3 points) Consider European options all on the same stock and all with the same time until expiration. Let C(K) be the premium for a call with strike K. Let P(K) be the premium for a put with strike K. Let A(K) = C(K) - C(K + 10). Let B(K) = P(K + 10) - P(K). A(95) = B(95) = B(80) = Determine A(80). A. 4.7 B. 4.8 C. 4.9 D. 5.0 E. 5.1

110 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (CAS5B, 11/92, Q.64) (1 point) For a certain non-dividend paying stock, the price of a put option with a strike price of $100 exercisable in one year is $25. The current stock price is $120. The cost to borrow money is 10% for one year. What is the price of a call with an strike price of $100 exercisable in one year? A. Less than $50 B. At least $50, but less than $55 C. At least $55, but less than $60 D. At least $60, but less than $65 E. $65 or greater 4.34 (CAS5B, 11/94, Q.29) (4.5 points) An investor would like to sell short ABC company's stock, currently valued at $100, and will close out this position in one year. a. (1 point) What does it mean "to sell short ABCʼs stock"? b. (1 point) In a graph, depict the cumulative payoff in one year to this investor, relative to ABC's stock price. Ignore interest and transaction costs. Assume ABC pays no dividends. For (c) and (d) below, assume the investor does not want to lose more than $100 on this transaction when she closes out her position at the end of the year. Further assume a European options market exists for ABC stock. Assume ABC pays no dividends. c. (1.5 points) Construct an option that would achieve this goal, ignoring the cost of the option itself, interest charges, and any related transaction costs. Draw two graphs: the first to show the payoff of this option, and the second to show the investor's net payoff after implementing this strategy. d. (1 point) Given the following table, what is the expected cost of this "insurance"? Assume a continuously compounded risk-free interest rate of 10% per year, and ignore transaction costs and potential broker margin calls. Term: One Year Strike Price Value of Put

111 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (CAS5B, 5/99, Q.1) (1 point) You sell a 1-year European call option on Greystokes Inc. with an strike price of 110 and buy a 1-year European put option with the same strike price and term. The current continuously compounded risk-free rate is 12% and the value of your combined position is zero. Greystokes Inc. is a non-dividend-paying stock. What is the price of a share of Greystokes Inc.? A. Less than B. At least 95.00, but less than C. At least 97.50, but less than D. At least , but less than E. At least (IOA 109, 9/00, Q.1) (8.25 points) (i) (1.5 points) State what is meant by put-call parity. (ii) (4.5 points) Derive an expression for the put-call parity of a European option that has a dividend payable prior to the exercise date. (iii) (2.25 points) If the equality in (ii) does not hold, explain how an arbitrageur can make a riskless profit (MFE Sample Exam, Q.1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more than the put option. (iii) Both the call option and put option will expire in 4 years. (iv) Both the call option and put option have a strike price of $70. Calculate the continuously compounded risk-free interest rate. (A) (B) (C) (D) (E) (FM Sample Exam, Q.1) Which statement about zero-cost purchased collars is FALSE? A. A zero-width, zero-cost collar can be created by setting both the put and call strike prices at the forward price. B. There are an infinite number of zero-cost collars. C. The put option can be at-the-money. D. The call option can be at-the-money. E. The strike price on the put option must be at or below the forward price.

112 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (FM Sample Exam, Q.2) You are given the following information: The current price to buy one share of XYZ stock is 500. The stock does not pay dividends. The risk-free interest rate, compounded continuously, is 6%. A European call option on one share of XYZ stock with a strike price of K that expires in one year costs A European put option on one share of XYZ stock with a strike price of K that expires in one year costs Using put-call parity, determine the strike price, K. A. 449 B. 452 C. 480 D. 559 E (FM Sample Exam, Q.5) You are given the following information: One share of the PS index currently sells for 1,000. The PS index does not pay dividends. The effective annual risk-free interest rate is 5%. You want to lock in the ability to buy this index in one year for a price of 1,025. You can do this by buying or selling European put and call options with a strike price of 1,025. Which of the following will achieve your objective and also gives the cost today of establishing this position? A. Buy the put and sell the call, receive B. Buy the put and sell the call, spend C. Buy the put and sell the call, no cost D. Buy the call and sell the put, receive E. Buy the call and sell the put, spend 23.81

113 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (CAS3, 5/07, Q.3) (2.5 points) For a dividend paying stock and European options on this stock, you are given the following information: The current stock price is $ The strike price of options is $ The time to expiration is 6 months. The continuous risk-free rate is 3% annually. The continuous dividend yield is 2% annually. The call price is $2.00. The put price is $2.35. Using put-call parity, calculate the present value arbitrage profit per share that could be generated, given these conditions. A. Less than $0.20 B. At least $0.20 but less than $0.40 C. At least $0.40 but less than $0.60 D. At least $0.60 but less than $0.80 E. At least $ (CAS3, 5/07, Q.4) (2.5 points) The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and in five months. The term structure is flat, with all continuously compounded, risk-free rates being 10%. Calculate the price of a European put option on the same stock with expiration in six months and a strike price of $30. A B C D E (MFE, 5/07, Q.1) (2.6 points) On April 30, 2007, a common stock is priced at $ You are given the following: (i) Dividends of equal amounts will be paid on June 30, 2007 and September 30, (ii) A European call option on the stock with strike price of $50.00 expiring in six months sells for $4.50. (iii) A European put option on the stock with strike price of $50.00 expiring in six months sells for $2.45. (iv) The continuously compounded risk-free interest rate is 6%. Calculate the amount of each dividend. (A) $0.51 (B) $0.73 (C) $1.01 (D) $1.23 (E) $1.45

114 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (MFE, 5/07, Q.4) (2.6 points) For a stock, you are given: (i) The current stock price is $ (ii) δ = 0.08 (iii) The continuously compounded risk-free interest rate is r = (iv) The prices for one-year European calls (C) under various strike prices (K) are shown below: K C $40 $ 9.12 $50 $ 4.91 $60 $ 0.71 $70 $ 0.00 You own four special put options each with one of the strike prices listed in (iv). Each of these put options can only be exercised immediately or one year from now. Determine the lowest strike price for which it is optimal to exercise these special put option(s) immediately. (A) $40 (B) $50 (C) $60 (D) $70 (E) It is not optimal to exercise any of these put options (CAS3, 11/07, Q.14) (2.5 points) Given the following information about a European call option on Stock Z: The call price is The call has a strike price of 47. The call expires in two years. The current stock price is 45. The continuously compounded risk-free rate is 5%. Stock Z will pay a dividend of 1.50 in one year. Calculate the price of a European put option on Stock Z with a strike price of 47 that expires in two years. A. Less than 3.00 B. At least 3.00, but less than 3.50 C. At least 3.50, but less than 4.00 D. At least 4.00, but less than 4.50 E. At least 4.50

115 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (CAS3, 11/07, Q.16) (2.5 points) An investor has been quoted a price on European options on the same non-dividend paying stock. The stock is currently valued at 80 and the continuously compounded risk-free interest rate is 3%. The details of the options are: Option 1 Option 2 Type Put Call Strike Time to expiration 180 days 180 days Based on his analysis, the investor has decided that the prices of the two options do not present any arbitrage opportunities. He decides to buy 100 calls and sell 100 puts. Calculate the net cost of this transaction. (Hint: A positive net cost means the investor pays money from the transaction. A negative cost means the investor receives money.) A. Less than -60 B. At least -60, but less than -20 C. At least -20, but less than 20 D. At least 20, but less than 60 E. At least (MFE/3F, 5/09, Q.12) (2.5 points) You are given: (i) C(K, T) denotes the current price of a K-strike T-year European call option on a nondividend-paying stock. (ii) P(K, T) denotes the current price of a K-strike T-year European put option on the same stock. (iii) S denotes the current price of the stock. (iv) The continuously compounded risk-free interest rate is r. Which of the following is (are) correct? (I) 0 C(50, T) - C(55, T) 5e -rt (II) 50e -rt P(45, T) - C(50, T) + S 55e -rt (III) 45e -rt P(45, T) - C(50, T) + S 50e -rt (A) (I) only (B) (II) only (C) (III) only (D) (I) and (II) only (E) (I) and (III) only

116 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (CAS8, 5/10, Q.16) (4 points) Given the following stock option information: The current stock price is $60. The strike price is $57. The time to expiration is six months. The risk-free rate is 3.0% compounded continuously. European call option price is $3. Dividend amount to be paid in three months is $2. European put option price is $2. Assume there are no transaction costs, it is possible to borrow or lend at the risk-free rate, and there are no taxes to consider. a. (1 point) Use put-call parity to demonstrate why an arbitrage opportunity exists with respect to this stock option. b. (3 points) Assume an investor takes a position in one option. Fully discuss how an investor would take advantage of the arbitrage opportunity, including the investments that an investor would buy and sell, the decision points during the arbitrage period, and the risk-free profit that would be made (IOA, CT8, 4/10, Q.2) (6 points) Consider a stock paying a dividend at a rate δ and denote its price at any time t by S t. The dividend earned between t and T, T t, is: S t {e δ(t-t) - 1}. Let C t and P t be the price at time t of a European call option and European put option respectively, written on the stock S, with strike price K and maturity T t. The instantaneous risk-free rate is denoted by r. Prove put-call parity in this context by adapting the proof of standard put-call parity that applies to put and call options on a non-dividend paying stock (IOA, CT8, 9/10, Q.8) (5.25 points) Consider a particular stock and denote its price at any time t by S t. This stock pays a dividend D at time Tʼ. Let C t and P t be the price at time t of a European call option and European put option respectively, written on S, with strike price K and maturity T Tʼ t. The instantaneous risk-free rate is denoted by r. Prove the put-call parity in this context by adapting the proof of standard put-call parity. Hint: assume that when the dividend is paid it is used to pay off any borrowed positions required as part of the proof.

117 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 116 Solutions to Problems: 4.1. B. C Eur (K, T) = P Eur (K, T) + S 0 - PV[Div] - K e -rt. 10 = P {(1)e (1)e -.03 } - (75)e P = $ A. C Eur (K, T) = P Eur (K, T) + B 0 - PV[Coupons] - K e -rt. C - P = (10)(e e e e e e e e -.08 ) - (800)e -.08 = $ C. C Eur (K, T) = P Eur (K, T) + S 0 e -δt - K e -rt. 196 = P e e -.2. P = D. C Eur (K, T) = P Eur (K, T) + S 0 e -δt - K e -rt. C = e e -.15 = A. With no dividends, δ = 0 and: C Eur (K, T) - P Eur (K, T) = S 0 - K e -rt = K e K = C. C Eur (K, T) = P Eur (K, T) + S 0 e -δt - K e -rt = 13 + (160)e -(.03)(1.5) - (130)e -(.05)(1.5) =

118 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page & 4.8. Sell the call for $ Buy the put at $3.20. We buy a share of stock for $ Borrowing the present value of the $160 strike price we receive: 160 exp[-(2%)(3.25/12)] = $ Borrowing the present value of the $0.36 dividend we receive:.36 exp[-(2%)(1/12)] = $ The cost of our position is: = $ So we need to come up with a positive amount of money in order to set up this position. We get the dividend of $0.36 in one month and repay the lender. At expiration if S T > K, then the person to whom we sold the call will exercise it and buy our share of HAL for K. We will then pay K to the lender. If instead S T K, then we sell our share of HAL for K to the person from whom we bought the put. We will then pay K to the lender. In either case, we end up with nothing at time T. We needed to come up with a positive amount of money in order to set up a position which turns out to be worth nothing. Thus this is not a case of arbitrage. Put-call parity would have been violated if there was arbitrage available. Thus parity was not violated. Comment: Similar to question 9.17 in Derivative Markets by McDonald. Beyond what I expect you to be asked on your exam.

119 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page & Buy the call for $ Sell the put at $2.90. We borrow a share of stock, sell it for $169.70, and will give this person a share of stock in 3.25 months when the option expires. We also must pay this person the stock dividend they would have gotten on the stock, when they would have gotten it. Lending the present value of the $160 strike price we receive: 160 exp[-(1.9%)(3.25/12)] = $ Lending the present value of the $0.36 dividend we receive:.36 exp[-(1.9%)(1/12)] = $ The cost of our position is: = $ So we need to come up with a positive amount of money in order to set up this position. We will receive from the person to whom we made the loan the money to pay the stock dividend to the person from whom we borrowed the stock. We will receive from the person to whom we made the loan the strike price at expiration of the options 3.25 months from now. If S T K, then we exercise our call and buy a share of HAL for K, and give the share to the person from whom we borrowed the stock. If instead S T < K, then we buy a share of HAL for K from the person who exercises the put option we sold, and give the share to the person from whom we borrowed the stock. In either case, we end up with nothing at time T. We needed to come up with a positive amount of money in order to set up a position which turns out to be worth nothing. Thus this is not a case of arbitrage. Put-call parity would have been violated if there was arbitrage available. Thus parity was not violated B. With no dividends, δ = 0. At the money means: K = S 0 = 85. C Eur (K, T) = P Eur (K, T) + S 0 e -δt - K e -rt = e -(1.5)(.065) = Comment: For a non-dividend paying stock the difference between similar at the money options is: C Eur (K, T) - P Eur (K, T) = S 0 (1 - e -RT ).

120 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page Graph of the future value of a share of the stock plus a European put with a strike price of S +100 S =100, for S < , S + E[(100 - S) + ] = = Max[S, 100]: S + 0 = S, for S 100 Option Value Stock Price Comment: If today one buys a share of the stock plus a put option to sell the stock at 100, then at the expiration of the option, you can sell the stock for 100 (by exercising the put option) or the market price of the stock, whichever is higher E. 0 = C Eur (K, T) - P Eur (K, T) = S 0 e -δt - K e -rt. K = S 0 e (r-δ)t = (130)exp[( )(2)] =

121 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page Buy a call with strike 100: cost 27. Sell a put with strike 100: get 25. Sell a call with strike 120: get 23. Buy a put with strike 120: cost 26. Cost is: = 5. We borrow 5 at the risk free rate. The payoff on the options is: (S T - 100) + - (100 - S T ) + - {(S T - 120) + - (120 - S T ) + } = S T (S T - 120) = 20. For any reasonable values of r and T, 5e rt < 20. After repaying the loan, we have: 20-5e rt > 0. Thus for no investment, we end up with a position whose value is always positive. Comment: Similar to Q. 9.8 in Derivative Markets by McDonald. The given call premiums are OK by themselves. The given put premiums are OK by themselves. The given 100 strike premiums are OK by themselves. The given 120 strike premiums are OK by themselves. The arbitrage results from the relationship between all of the given premiums. By put-call parity, C - P = PV[F 0,T ] - PV[K]. For the 100 strike options: 2 = PV[F 0,T ] - 100e -rt. For the 120 strike options: -3 = PV[F 0,T ] - 120e -rt. e -rt = 5/20. rt = ln(4) = If in fact rt = 1.386, then there is no arbitrage. (For example, T = 10 years and r = 13.86%.) We assume that for any application to a real world situation, rt < Therefore, put-call parity must be violated. Therefore, there must be arbitrage.

122 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page C. C Eur (K, T) = P Eur (K, T) + S 0 - PV[Div] - K e -rt. C = (2)e -.05/4 - (130)e -.05/3. C = $ D. By Put-Call Parity, C - P = Se -δt - Ke -rt. Therefore, = = Se -δt - 80e -rt, and = 5.18 = Se -δt - 90e -rt. Therefore subtracting these two equations, 10e -rt = e -rt = Therefore, Se -δt = (80)(.879) = Therefore, for K = 60, C = P - Ke -rt + Se -δt = (60)(.879) = Therefore, for K = 110, P = C + Ke -rt - Se -δt = (110)(.879) = Sum of the prices of the 60 strike call and the 110 strike put is: = $ S +100 S - 0 =100, for S < S + E[(100 - S) + ] - E[(100 - S) + ] = = 100: S (S - 100) = 100, for S 100 Option Value Stock Price Comment: If today you buy a share of the stock plus a put option to sell the stock at 100, and also sell to Fred a call option with a strike price of 100, then at the expiration of the options: (a) If the future stock price is below 100, you can sell the stock for 100 by exercising the put option. Fred has no interest in exercising his call option and buying your share of the stock for 100, since 100 is higher than the market price of the stock. (b) If the future stock price is above 100, Fred will exercise his call option to buy your share of the stock for 100, which is lower than the market price of the stock. In either case (a) or (b), you end up with 100. By buying a share, buying a put, and selling a call, you have removed the uncertainty in the future value of your investment. An example of put-call parity.

123 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page E. C Eur (K, T) = P Eur (K, T) + S 0 e -δt - K e -rt. 23 = P + (150)e -.03/2 - (140)e -.06/2. P = E. 0 = C - P = S 0 e -δt - K e -rt. lns 0 - δt = lnk - rt. T = ln[k/s 0 ]/(r - δ) = ln[1.15]/.04 = 3.49 years. Comment: S 0 e -δt = 100 e -(.02)(3.49) = 93.3 = 115 e -(.06)(3.49) = K e -rt C. C Eur (K, T) = P Eur (K, T) + S 0 - PV[Div] - K e -rt. 8 = P {(.8)e -.055/4 + (1.2)e -.055/2 + (1.5)e -(.055)(.75) } - (125)e -(.055)(.75). P = $ C. In 2 months, the options will have 4 months until expiration. By put call parity, C - P = S e -δt - K e -rt = exp[-r/3]. Setting C = P: = 80 exp[-r/3]. r = 5.0% C. By Put-Call Parity, C - P = Se -δt - Ke -rt. Therefore, 11-3 = 8 = Se -δt - 70e -rt, and 6-7 = -1 = Se -δt - 80e -rt. Therefore subtracting these two equations, 10e -rt = 9. e -rt = 0.9. Therefore, Se -δt = 8 + (70)(.9) = 71. Therefore, for K = 90, P = C + Ke -rt - Se -δt = 3 + (90)(.9) - 71 = 13. Alternately, if there were discrete dividends, then by Put-Call Parity, C - P = S - PV[Div] - Ke -rt. Therefore, 11-3 = 8 = = S - PV[Div] - 70e -rt, and 6-7 = -1 = S - PV[Div] - 80e -rt. Proceed as before B. By Put-Call parity, C - P = S 0 - PV[Div] - K e -rt. 0 = 84-2e -.05/12-2e -.05(4/12) - 2e -.05(7/12) - 2e -.05(10/12) - K e -.05(1). K =

124 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page E. Let us assume the strikes for the Call Bull Spread are K and K Call Bull Spread: Buy a K strike call and sell a K + 10 strike call. Premium for Call Bull Spread is: C(K) - C(K+10) = Put Bear Spread: Sell a K strike put and buy a K + 10 strike put. By put-call parity: P(K) = C(K) + K e -rt - S 0 e -δt. By put-call parity: P(K+10) = C(K+10) + (K+10) e -rt - S 0 e -δt. Premium for the Put Bear Spread is: P(K+10) - P(K) = C(K+10) - C(K) + 10e -rt = 10e -(0.056)(2) = Comment: Premium for the Put Bull Spread is Bull Spread: The purchase of an option together with the sale of an otherwise identical option with a higher strike price. One can construct a bull spread using either puts or calls. Bear Spread: The sale of an option together with the purchase of an otherwise identical option with a higher strike price. One can construct a bear spread using either puts or calls A. Applying put call parity on January 1: = Ke -r/2. Ke -r/2 = Applying put call parity on March 1: = Ke -r/3. Ke -r/3 = Therefore dividing the two equations, e r/6 = / = r = K = e.045/2 = C. e -δt S 0 = C - P + e -rt K. e -δ2 93 = e -(0.052)(2) 90. δ = 1.5%.

125 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page E. The call premiums satisfy all three properties. The put premiums satisfy all three properties. However, applying put-call parity: F P 0,T [S] = Ke -rt + C - P = 60 e -rt F P 0,T [S] = 70 e -rt F P 0,T [S] = 90 e -rt Combining the first two equations: 60 e -rt = 70 e -rt e -rt = Combining the last two equations: 70 e -rt = 90 e -rt e -rt = Thus the sets of prices for the puts and calls are not consistent. Thus one could use a portfolio of puts and calls to obtain arbitrage profit. Comment: Similar to MFE Sample Exam, Q.2. A graph of the call premiums: 21 C S A graph of the put premiums: P S

126 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 125 One can demonstrate arbitrage by: buying two 60-strike calls, selling two 60-strike puts, selling three 70-strike calls, buying three 70-strike puts, buying one 90-strike call, and selling one 90-strike put. You get in the door initially for setting up this portfolio: (2)(6) - (2)(21) + (3)(17) - (3)(11) + (1)(25) - (1)(12) = 1. We invest this 1 at the risk free rate, and on the expiration date of the options we have e rt. If S T < 60, all of the calls turn out to be worthless, and the payoffs from the puts are: -(2)(60 - S) + (3)(70 - S) - (1)(90 - S) = 0. If 60 S T < 70, then the payoffs from the options are: (2)(S - 60) + (3)(70 - S) - (1)(90 - S) = 0. If 70 S T < 90, then the payoffs from the options are: (2)(S - 60) - (3)(S - 70) - (1)(90 - S) = 0. If 90 S T, then the puts are all worthless and the payoffs from the calls are: (2)(S - 60) - (3)(S - 70) + (1)(S - 90) = 0. So regardless, the payoffs are zero, and we end up with e rt, demonstrating arbitrage. (S - K) + - (K - S) + = S - K. Thus the payoff from buying a call and selling the similar put is: S - K. Thus the payoff for this portfolio of puts and calls is: 2 (S T - 60) - 3 (S T - 70) + (S T - 90) = Let C(80) be the premium of the 80-strike call. By put-call parity, C(80) = e -2r = e -2r. This is positive. The various properties of call premiums must hold < C(80) < < e -2r < < 80 e -2r < < e -2r < % > r > 2.1% The difference between the premiums must be less than the discounted difference in strikes C(80) < 10 e -2r ( e -2r ) < 10 e -2r. 70 e -2r < r > 3.8%. C(80) < 20 e -2r e -2r < 20 e -2r < 100 e -2r. r < 13.3%. For convexity to hold, we must have C(80) < (2/3)(19.77) + (1/3)(7.93) = e -2r < < 80 e -2r. r < 7.5%. Combining the conditions: 3.8% < r < 7.5%. Comment: For example, if r = 6%, then C(80) = e = Then C(80) < 15.82, so convexity holds > > C(80) = 6.11 < 8.87 = 10 e -2r. C(80) = 5.73 < = 20 e -2r. Thus all three properties of call premiums would hold.

127 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page E. By put-call parity, P(100) = C(100) + 100e -rt - PV[S]. Therefore, Dustinʼs premium is: C(90) - C(100) - 100e -rt + PV[S] = C(90) - C(100) - 100e -rt + PV[S]. By put-call parity, P(90) = C(90) + 90e -rt - PV[S]. Therefore, Edithʼs premium is: C(100) - C(90) - 90e -rt + PV[S] = C(100) - C(90) - 90e -rt + PV[S]. Adding the equations from Dustin and Edith: = 2 PV[S] - 190e -rt. By put-call parity, Frankʼs premium is: C(120) - P(120) = PV[S] - 120e -rt = Therefore, (2)(1.61) = 240e -rt - 190e -rt. e -rt = PV[S] = By put-call parity, Gabrielleʼs premium is: C(110) - P(110) = PV[S] - 110e -rt = (110)(0.762) = Alternately, Gabrielleʼs premium is equal to Frankʼs premium plus 10e -rt : (10)(0.762) = B. By put-call parity, P(80) = C(80) + 80e -rt - PV[S]. Therefore, Ashleyʼs premium is: C(60) - C(80) - 80e -rt + PV[S] = C(60) - C(80) - 80e -rt + PV[S]. By put-call parity, P(60) = C(60) + 60e -rt - PV[S]. Therefore, Chelseaʼs premium is: C(80) - C(60) - 60e -rt + PV[S] = C(80) - C(60) - 60e -rt + PV[S]. Adding the equations from Ashley and Chelsea: 5.74 = 2 PV[S] - 140e -rt. By put-call parity, Benjaminʼs premium is: C(70) - P(70) = PV[S] - 70e -rt = 5.74/2 = Comment: Benjaminʼs premium is an average of Ashleyʼs and Chelseaʼs.

128 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page B. By put-call parity: Call Premium - Put Premium = 82 e -(3)(0.02) = Comment: Assume Adam bought a three year call and loaned out Then at time 3, Adam can use 80 to exercise his call if it makes sense to do so. Assume Eve bought a three year put and a three-year prepaid forward contract on the stock. Then at time 3, Eve could use her put to sell the stock for 80 if it makes sense to do so. Then if S 2 > 80, both Adam and Eve end up with the stock. Then if S 2 < 80, both Adam and Eve end up with 80. Therefore, Call = Put + S 0e -3δ. A form of put-call parity D. A(K) + B(K) = C(K) - C(K + 10) + P(K + 10) - P(K) = {C(K) - P(K)} - {C(K + 10) - P(K + 10)} = S 0 e -δt - K e -rt - {S 0 e -δt - (K + 10) e -rt } = 10e -rt. Thus A(K) + B(K) is independent of K. A(95) + B(95) = = A(80) = B(80) = = B. C = P + S - K e -rt = e -0.1 = $ Comment: If instead you assume the 10% is an effective annual rate, then: C = P + S - K e -rt = /1.1 = $54.09.

129 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page a. One borrows shares of ABC stock from someone who owns them, and then sell the shares. You promise to return the shares of stock in the future, in this case in one year, along with the present value of any dividends issued on the stock. b. Assuming ABC stock pays no dividends, then one year from now the investor will have to buy a share of ABC stock with price S 1 ; ignoring interest the payoff is: S 1. Payoff S c. Buying a one-year 200-strike European call would limit her lose to $100. The payoff on the call: Payoff S

130 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page 129 The investor's net payoff: Payoff S d. Assuming no dividends, C = P + S - Ke -rt = e -0.1 = C. C = P + Se -δt - Ke -rt. Since C = P and δ = 0, S = Ke -rt = 100e -.12 =

131 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (i) Put-call parity expresses a relationship between the price of a put option and the price of a call option on a stock where the options have the same exercise dates and strike prices. (ii) Consider a portfolio A which contains one European call and an amount of cash D + Xe -r(t-t), where X = exercise price, r = risk-free rate, T - t = time to exercise of the option, D = present value of dividends payable. At the exercise date if the share price S T X then the call will be exercised and portfolio A will have a value of D e r(t-t) + S T. If at T we have S T < X then the call will not be exercised and portfolio A will be worth D e r(t-t) + X. Now consider portfolio B consisting of one European put and a share. At the exercise date if S T X then the put will not be exercised and portfolio B will have value of S T + D e r(t-t) If at the exercise date T, we have S T < X then the put will be exercised and portfolio B will have a value of X + D e r(t-t). Portfolios A and B have the same value in all circumstances at the exercise date T. Hence they must be equivalent at all earlier times. the portfolios are of equal value. Therefore c + D + Xe -r(t-t) = p + S t. c = value of European call with strike X and exercise date T. p = value of European put with strike X and exercise date T. S t = value of stock at time t. (iii) Let D be the present value of dividends payable and consider: c + D + X e -r(t-t) < p + S t. Then for some amount A: A + c + D + X e -r(t-t) = p + S t. Hence we can short one share and sell a put and receive p + S t At the exercise date we know the value of this portfolio is: max[s T + D e r(t-t), X + D e r(t-t) ]. However we know that the value of a portfolio invested in a European call and D + Xe -r(t-t) at time t will be worth max[s T + D e r(t-t), X + D e r(t-t) ] at time T. This is the same as the amount we must repay at time T. Hence we are left with a profit of Ae r(t-t). Therefore, the strategy in order to make a riskless profit is short 1 share and sell a put, buy 1 call and put on deposit A + D + X e -r(t-t). The net investment is zero at time t, and we end up with Ae r(t-t) at time T. If the inequality is reversed also reverse investment (i.e. swap long positions for short positions and vice versa) A. The put-call parity formula for a European call and a European put on a nondividend-paying stock with the same strike price and maturity date is: C - P = S 0 - Ke -rt = 60-70e -r4. r =

132 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page D. By put call parity, C - P = S - PV[Div] - Ke -rt. For at the money options, S = K, and C - P = K - PV[Div] - Ke -rt. Assuming the dividend rate is less than r, C - P < 0. Therefore, the premium of the at-the-money put is less than the premium of the at-the-money call. Therefore, if the call is at-the-money, the put option with the same cost will have a higher strike price. However, a purchased collar requires that the put have a lower strike price. Thus statement D is false. Comment: See pages of Derivative Markets by McDonald. On the syllabus of an earlier exam; beyond the level of detail you are likely need for your exam. A collar is the purchase of a put option and the sale of a similar call with a higher strike; both options are on the same stock and have the same expiration date. If the collar is written rather than purchased, then the put is sold and the call is bought. In the case of a zero-cost collar, the two option premiums are the same, and the net cost of the collar is zero C. For no dividends, by put call parity: C - P = S - Ke -rt = 500 Ke K = E. You want to buy the call and sell the put. Let x be the future value of the index one year from now. If x > 1,025, then you will receive x - 1,025 from the call. After buying the index for x you will have spent a total of 1,025. If x < 1,025, then you will pay 1,025 x to the person who bought the put from you. After buying the index for x you will have spent a total of 1,025. One way to get the cost is to note that the forward price is: 1,000(1.05) = 1,050. You want to pay 1,025, which is 25 less. Therefore, you must spend 25/1.05 = today, in order to guarantee that outcome. Alternately, by put-call parity with no dividends, using effective annual rates rather than continuously compounded rates: C - P = S - K/(1 + r) T = /1.05 =

133 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page B. Based on put-call parity, we would expect the call to have a price of: P - Ke -rt + Se -δt = e -.03/ e -.02/2 = $2.30. Therefore, the call is underpriced at $2.00. The present value arbitrage profit is: $ $2.00 = $0.30. Alternately, based on put-call parity, we would expect the put to have a price of: C + Ke -rt - Se -δt = e -.03/ e -.02/2 = $2.05. Therefore, the put is overpriced at $2.35. The present value arbitrage profit is: $ $2.05 = $0.30. Comment: We can take advantage of the arbitrage by: buying a call, selling a put, investing Ke -rt, and shorting e -δt = e -.02/2 =.990 shares of stock. (We borrow.990 shares of stock from someone at time 0 and return 1 share of stock at time 1/2. If one owned.990 shares and reinvested the dividends, one would have.990e.02/2 = 1 share at time 1/2.) When we set up the position we receive: (49.70)e -.02/2 = We invest this at the risk free rate, and end up with e.03/2 = , at time 1/2. At time 1/2 we need 1 share of stock to return to the person we borrowed it from. If S 1/2 > K = 50, then we use our call to buy a share of stock at 50. If S 1/2 < K = 50, then the person to whom we sold the put uses it to sell us a share of stock at 50. In either case, we use $50 to buy a share of stock, and are left with $ $50 = $ The present value is: $0.305 e -.03/2 = $0.30. Put another way, one can buy a synthetic put by: buying a call, lending at the risk free rate K e -rt, and shorting e -δt = e -.02/2 shares of stock. The cost of this synthetic put is: e -.03/ e -.02/2 = $2.05. We can sell an actual put for $2.35 and buy the synthetic put for $2.05, making $ E. Based on put-call parity, P = C + Ke -rt - (S - PV[Dividends]) = e -.10/2 - { e -.10/6 -.50e -.10(5/12) } = $ B. By put-call parity, C = P + S - PV[Div] - K e -rt. Let x be the size of each dividend = x{e -(.06)(2/12) + e -(.06)(5/12) } - (50)e -(.06)(6/12). x =

134 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page E. By put-call parity, P = C - Se -δt + Ke -rt = C - 50e Ke K C Immediate Payoff P Exercise Immediately $40 $9.12 $0 $1.40 No $50 $4.91 $0 $6.79 No $60 $0.71 $10 $12.20 No $70 $0.00 $20 $21.10 No We exercise immediately if the payoff for the put, (K - 50) +, is more than the premium for a European Put. That is not the case for any of the four examples here. Comment: The present value of an option if one does not exercise it, is called the continuation value. American options can be exercised any time up through expiration. Thus unlike here, one would need to compare the payoff and the continuation value at more times than just initially, in order to decide whether it is optimal to exercise early D. P = C + PV[K] + PV[Div] - S = e -(2)(.05) e = Alternately, the prepaid forward price for the stock is: e -.05 = P = e -(2)(.05) = A. C - P = S - K exp[-rt] = e -(.03)(1/2) = (C - P) = Comment: Since the call is worth less than the put, the investor receives money E. A 55 strike call is worth less than a 50 strike call, so 0 C(50, T) - C(55, T). The difference in payoffs between a 50 strike and 55 strike call is at most 5. Since these are European options, these payoffs takes place T years in the future. Therefore, C(50, T) - C(55, T) 5e -rt. Statement I is true. By put-call parity, P(45, T) = C(45, T) + 45e -rt - S. Therefore, P(45, T) - C(50, T) + S = C(45, T) - C(50, T) + 45e -rt. However, 0 C(45, T) - C(50, T) 5e -rt. 45e -rt C(45, T) - C(50, T) + 45e -rt 50e -rt. 45e -rt P(45, T) - C(50, T) + S 50e -rt. Statement III is true. Since Statement III is true, Statement II is not true.

135 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page (a) Using put-call parity, the price of the call should be: e -0.03/4-57 e -0.03/2 = $3.86. Thus the call is underpriced at $3. Alternately, using put-call parity, the price of the put should be: e -0.03/2 - (60-2e -0.03/4 ) = $1.14. Thus the put is overpriced at $2. (b) We can buy the call and sell a synthetic call. To sell the synthetic call, we sell a put, short sell a share of stock, lend 2e -0.03/4 the present value of dividends, and lend the strike price for 6 months. We net $0.86 from setting up this position. In six months, if the stock price is greater than the strike price of $57, we use the call to buy the stock for $57, repay the stock and accumulated value of dividends to the person from whom we borrowed the stock. We are left with $0.86 accumulated for 6 months of interest. In six months, if the stock price is less than the strike price of $57, then the person to whom we sold the put uses it to sell us a share of stock for $57; we repay the stock and accumulated value of dividends to the person from whom we borrowed the stock. We are left with $0.86 accumulated for 6 months of interest Consider two portfolios. Portfolio A: buying the call and selling the put at time t. Its value at time t is C t - P t and at time T, it is S T - K in all states of the universe. Portfolio B: buying a fraction exp[-δ(t - t)] of the underlying asset for S t exp[-δ(t - t)] and borrowing K exp[-r(t - t)] at time t. Its value at time t is then S t exp[-δ(t - t)] - K exp[-r(t - t)]. Its value at maturity is then S T - K by taking into account the dividends which are paid continuously at rate δ. Using the absence of arbitrage opportunity, both portfolios should have the same value at any intermediate time, in particular at time t. Hence: C t - P t = S t exp[-δ(t - t)] - K exp[-r(t - t)]. Comment: There are other sets of two portfolios of equal value one could set up in order to prove put-call parity. For example: Portfolio A: At time t, buying a call option and lending K exp[-r(t - t)]. Portfolio B: At time t, buying the put option and buying exp[-δ(t - t)] shares of stock.

136 2013-MFE/3F, Financial Economics 4 Put-Call Parity, HCM 12/6/12, Page Adam buys the call and loans out Ke -r(t-t). At time T, Adam has the call and K. Eve buys the stock, the put, and borrows D exp[-r(tʼ-t)]. At time Tʼ, Eve gets the dividend of size D, and pays off her loan. At time T, Eve has the stock and the put. If S T > K, Adam uses his call to buy the stock for K, and he ends up with the stock. If S T > K, Eveʼs put expires worthless, and she ends up with the stock. If S T < K, Adamʼs call expires worthless, and he ends up with K. If S T < K, Eve uses her put to sell the stock for K, and she ends up with K. Since they end up in the same position, the initial costs of their portfolios must have been the same. C t + Ke -r(t-t) P = P t + S t - D exp[-r(tʼ-t)] = P t + S t - PV[Dividends] = P t + F t,t [S].

137 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 136 Section 5, Bounds on Premiums of European Options Option premiums must stay within certain bounds. Maximum and Minimum Prices of Calls: 90 Since an option does not require its owner to do anything, it never has a negative value. E[(S T - K) + ] 0. E[(K - S T ) + ] 0. If Joe buys a call option, then he must pay K > 0 if he wants to own a share of stock at time T in the future. For S 0 Mary can buy a share of the stock now and if she holds on to it, guarantee she owns a share at time T in the future. After Mary buys her share of stock and after Joe buys his option, Mary is in a better position than Joe, even ignoring any dividends Mary may get on the stock. Therefore, the value of the call option is less than (or equal to) S 0, the current market share of the stock. Owning a share of stock now is worth at least as much as the option to buy a share of stock in the future. A call is never worth more than the current stock price. S 0 C(S 0, K, T) 0. In fact, one can pay the prepaid forward price, F P 0,T (S0 ), and will own a share of stock at time T. This is more valuable than the option to buy a share of stock at time T in the future. For example, Tim pays Howard the prepaid forward price for a share of ABC stock and in exchange Howard agrees to give Tim a share of ABC stock two years from now. By paying this amount of money today, Tim will own a share of ABC stock two years from now. Instead, Tim could buy a two year $100 strike call on ABC stock from John. Two years from now Tim will have the option to pay $100 to buy a share of ABC stock from John. Owning a share of stock two years from now is more valuable than having the option to buy a share of stock two years from now for $100. The future value of the former is S 2. The future value of the latter is: (S 2-100) + < S 2. Therefore, a somewhat more stringent inequality holds: F P 0,T (S0 ) C Eur (S 0, K, T) See page 292 and 293 of Derivative Markets by McDonald. 91 McDonald does not mention this in Derivative Markets. In the absence of dividends, the prepaid forward price is just S0.

138 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 137 Phyllis can buy a forward contract to buy a share of stock at T in the future. At time T in the future Phyllis will pay F 0,T for a share of stock. Harry can instead buy a call option for this same stock at time T in the future, for a price of C(S 0, K, T), and can in addition invest PV 0,T [K] in a risk-free investment. At time T Harry will then have the money to buy a share of stock if he chooses to exercise his option. Phyllis will own a share of stock at time T. Harry can choose to own a share of stock at time T if it is advantageous to him to exercise his option. Therefore, Harryʼs position is worth more than Phyllisʼs. C(S 0, K, T) + PV 0,T [K] PV 0,T [F 0,T ]. C(S 0, K, T) PV 0,T [F 0,T ] - PV 0,T [K]. 92 Therefore, recalling that an option can never have a negative value: S 0 C(S 0, K, T) (PV 0,T [F 0,T ] - PV 0,T [K]) However, PV 0,T [F 0,T ] = S 0 - PV[Div]. Therefore, C(S 0, K, T) (S 0 - PV[Div] - PV 0,T [K]) +. Exercise: K = 100, T = 2, r = 6%, and δ = 2%. What are the bounds on the call premium as a function of S 0? [Solution: PV 0,T [F 0,T ] - PV 0,T [K] = S 0 e -δt - K e -rt = S 0 e e = S S 0 C(S 0, K, T) (0.961 S ) +. Comment: S = 0, for S 0 = 92.3 = Ke -(r-δ)t.] For these inputs, here is a graph of the possible call premiums as a function of S 0 : C S0 92 From put-call parity: C = P + PV[F0,T] - PV[K]. Also P 0. C PV[F0,T] - PV[K]. 93 See Equation 9.9 in Derivative Markets by McDonald.

139 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 138 Exercise: S 0 = 90, T = 2, r = 6%, and δ = 2%. What are the bounds on the call premium as a function of K? [Solution: PV 0,T [F 0,T ] - PV 0,T [K] = S 0 e -δt - K e -rt = 90 e K e = K. 90 C(S 0, K, T) ( K) +. Comment: K = 0, for K = 97.5 = S 0 e (r-δ)t.] For these inputs, here is a graph of the possible call premiums as a function of K: C K Maximum and Minimum Prices of Puts: If one owns a put option, then the most one can gain is K, if the stock price goes to zero. Therefore, the value of a put is at most its strike price K. 94 K P(S 0, K, T). In fact, since the possible gain from the put occurs at time T, the value of the put is at most the present value of K, e -rt K: PV[K] P(S 0, K, T). Susan buys a put option on a share of stock with expiration T, and also enters a forward contract to buy the stock at T, at price F 0,T. Then if Susan invests PV 0,T [F 0,T ], she will have the money at time T to buy the stock as per her forward contract. If the price of the stock is greater than K she will not exercise her put option and instead she keeps the stock. If the price of the stock at time T is less than K she will exercise her option, and sell the share of stock for K. In either case, her position at time T is worth at least K. 94 PEur(S0, K, T) = E[(K - ST)+] e -Tr E[(K - ST)+] K.

140 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 139 Therefore, P(S 0, K, T) + PV 0,T [F 0,T ] PV 0,T [K]. P(S 0, K, T) PV 0,T [K] - PV 0,T [F 0,T ]. 95 Therefore, recalling that an option can never have a negative value: K P(S 0, K, T) (PV 0,T [K] - PV 0,T [F 0,T ]) However, PV 0,T [F 0,T ] = S 0 - PV[Div]. Therefore, P(S 0, K, T) (PV 0,T [K] - S 0 + PV[Div]) +. Exercise: K = 80, T = 3, r = 5%, and δ = 1%. What are the bounds on the put premium as a function of S 0? [Solution: PV 0,T [K] - PV 0,T [F 0,T ] = K e -rt - S 0 e -δt = 80 e S 0 e = S P(S 0, K, T) ( S 0 ) +. Comment: S 0 = 0, for S 0 = 71.0 = Ke -(r-δ)t.] For these inputs, here is a graph of the possible put premiums as a function of S 0 : P S Exercise: S = 70, T = 3, r = 5%, and δ = 1%. What are the bounds on the put premium as a function of K? [Solution: PV 0,T [K] - PV 0,T [F 0,T ] = K e -rt - S 0 e -δt = K e e = K K P(S 0, K, T) (0.861 K ) +. Comment: K = 0, for K = 78.9 = S 0 e (r-δ)t.] 95 From put-call parity: P = C + PV[K] - PV[F0,T]. Also C 0. P PV[K] - PV[F0,T]. 96 See Equation 9.10 in McDonald.

141 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 140 For these inputs, here is a graph of the possible put premiums as a function of K: P K For a European put, the most you can be paid is K at time T, which has present value Ke -rt. Therefore, Ke -rt P(S 0, K, T). 97 Relationship to Put-Call Parity: From Put-Call Parity, we have: C Eur (K, T) = P Eur (K, T) + PV[F 0,T ] - PV[K]. Since, P Eur (K, T) 0, C Eur (K, T) PV[F 0,T ] - PV[K], as discussed previously. Since, C Eur (K, T) 0, P Eur (K, T) + PV[F 0,T ] - PV[K] 0. Therefore, P Eur (K, T) PV[K] - PV[F 0,T ], as discussed previously. 97 McDonald does not mention this in Derivative Markets. See for example, Options, Futures, and Other Derivatives by John C. Hull, not on the syllabus.

142 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 141 An Increasing Strike Price: 98 Rather than being fixed as is usual, the strike price could increase with the time until expiration. Let the strike price for an option with time until expiration T be: K T = Ke rt. In other words, we keep the present value of the strike price constant. 99 On a stock that does not pay dividends, when the strike price grows at the risk free rate, then the premiums on European puts increase with time to expiration. 100 Exercise: Demonstrate that an arbitrage opportunity exists in the following situation. The risk free rate is 6%. The premium for a one-year 100-strike European put on a non-dividend paying stock is 5. The premium for the two-year strike European put on the same stock is 4. [Solution: Buy the two-year put and sell the one-year put. You get 1, which you invest at the risk-free rate. At time 1 you have e 0.06 = If S 1 > 100 then you do not pay off on the one-year put you sold. You have plus the two year put; your position is positive. If S 1 < 100 then you pay off on the one-year put: S 1. However, the premium of the strike put that you still own is greater than or equal to: (PV[K] - PV[F 0,T ]) + = ( e S 1 ) + = S 1. Thus the put you still own is worth at least as much as the amount you pay off on the put you sold. Thus you have at least ; your position is positive. Demonstrating arbitrage. Comment: Note that the strike for the two year put is: (100) e 0.06 = Since the stock pays no dividends, its prepaid forward price is just its current price. The result did not depend on the particular values other than the fact that 4 < 5.] If the options start at the money, then the increasing strike price is: S 0 e rt. From put-call parity, C - P = Ke -rt - Se -δt. Thus, in this situation with no dividends and K = S 0 e rt : C - P = 0. The call premium is equal to the put premium. They both increase with T. In general, for a European call on a non-dividend paying stock, when the strike price grows at the risk free rate, then the premiums increase with time to expiration See page 298 of Derivatives Markets by McDonald. 99 This is analogous to keeping a insurance policy deductible or limit up with inflation. 100 Since the put premium increases with the strike price, if the strike price increases with T at a rate greater than r, then the put premium increases even faster with T. 101 Since the call premium decreases with the strike price, if the strike price increases with T at a rate less than r, then the call premium increases even faster with T.

143 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 142 Exercise: For a stock that does not pay dividends, demonstrate that when the strike price grows at the risk free rate, the premium of a European call increases with time to expiration. [Solution: K T = Ke rt. Take two similar calls, with different times to expiration T < U. Assume the premium of the first call with less time to expiration is more than that of the second call. Buy the U-year call and sell the T-year call Since we have assumed that the premium of the T-year call is greater than the premium of the U-year call, we get money in the door, which we invest at the risk-free rate. If S T Ke rt then we do not pay off on the T-year call we sold. We have money plus the T-year call; our position is positive. If S T > Ke rt then we pay off on the T-year call: S T - Ke rt. However, the premium of the call we own with strike Ke ru, that has U - T until expiration, is greater than or equal to: (PV[Forward contract on the stock] - PV[K]) + = (S T - e -r(u-t) Ke rt ) + = S T - e -r(u-t) Ke ru = S T - Ke rt. Thus the call we still own is worth at least as much as the amount we pay off on the call we sold. Thus we have at least the money we invested at time 0, plus the interest we earned on it; your position is positive. Demonstrating arbitrage.] Portfolio Insurance for the Long Run: 102 If you owned a share of a stock that pays no dividends and purchased a put with strike price S 0 e rt, then if at time T: S T < S 0 e rt, you could use your put to sell the stock for S 0 e rt. You would guarantee that over the period time from 0 to T you would earn at least the risk free rate on the stock plus put, not including the cost to buy the put. However, as we have seen the cost of such puts increases with T. 103 The longer the period of time we wish to insure, the more the premium. One can apply the same idea to a portfolio of different stocks. 102 See page 299 of Derivatives Markets by McDonald. See also page 603, to be discussed subsequently. 103 When there are no dividends. If there are dividends, we would instead have the strike increase at the rate r - δ; after taking into account the dividends we receive, we would earn at least the risk free rate.

144 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 143 Problems: 5.1 (1 point) A European 140 strike 2 year call is an option on a stock with current price 100 and forward price of 108. r = 5%. Which of the following intervals represents the range of possible prices of this option? A. [0, 100] B. [0, 108] C. [9.05, 100] D. [9.05, 108] E. [8, 108] 5.2 (1 point) A European 140 strike 2 year put is an option on a stock with current price 100 and forward price of 108. r = 5%. Determine the range of possible prices of this option. 5.3 (3 points) One has a European option on a stock with current market price of 100. The time until expiration is 2 years. r = 5%. The two year forward price of the stock is 110. Which the following statements are true? A. The premium of a call option with a strike price of 80 could not be 80. B. The premium of a put option with a strike price of 140 could not be 105. C. The premium of a call option with a strike price of 80 could not be 25. D. The premium of a put option with a strike price of 140 could not be 30. E. The premium of an at-the-money put must be at least the premium of an at-the-money call. 5.4 (1 point) A European 110 strike 2 year put is an option on a stock with current price 100 and forward price of 108. r = 5%. Which of the following intervals represents the range of possible prices of this option? A. [0, 100] B. [0, 110] C. [1.81, 108] D. [1.81, 110] E. [2, 108] 5.5 (1 point) A European 105 strike 2 year call is an option on a stock with current price 100 and forward price of 108. r = 5%. Which of the following intervals represents the range of possible prices of this option? A. [0, 100] B. [0, 108] C. [2.72, 100] D. [2.72, 108] E. [8, 108]

145 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page (2 points) ABC stock pays no dividends. r = 8%. For a 90 strike 6 month European put on ABC Stock, which of the following graphs represents the bounds on the put premium as a function of S 0? P P A S B S0 P P C S D S0 P E S0

146 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page (2 points) XYZ stock pays dividends at a continuously compounded rate of 2%. XYZ stock has a current price of 80. r = 10%. For a 3 year European call on XYZ Stock, which of the following graphs represents the bounds on the call premium as a function of K? C 80 C A K B K C C C K D K C E K 5.8 (CAS5B, 11/94, Q.29) (1 point): Explain why the price of a European call option approaches the prepaid forward price of the underlying stock less the present value of the strike price as the stock rises further and further beyond the strike price.

147 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page 146 Solutions to Problems: 5.1. A. S 0 C(S 0, K, T) (PV 0,T [F 0,T ] - PV 0,T [K]) C (108 e e -.1 ) + = 0. Comment: While McDonald does not mention this in Derivative Markets, for a European call, C F P 0,T (S0 ) = PV 0,T [F 0,T ] = 108 e -.1 = K P(S 0, K, T) (PV 0,T [K] - PV 0,T [F 0,T ]) P (140 e e -.1 ) + = Comment: While McDonald does not mention this in Derivative Markets, for a European put, P Ke -rt = 140 e -.1 = C. S 0 C(S 0, K, T) (PV 0,T [F 0,T ] - PV 0,T [K]) +. Therefore, 100 the premium of the call. More precisely, C F P 0,T (S0 ) = PV 0,T [F 0,T ] = 110 e -.1 = Statement A is not true. The premium of the call (110-80)/e.1 = Statement C is true. K P(S 0, K, T) (PV 0,T [K] - PV 0,T [F 0,T ]) +. Therefore, 140 the premium of the put. More precisely, P PV 0,T [K] = 140 e -.1 = Statement B is not true. The premium of put ( )/e.1 = Statement D is not true. By put-call parity: P - C = PV 0,T [K] - PV 0,T [F 0,T ] = 100e e -.1 = Statement E is not true D. K P(S 0, K, T) (PV 0,T [K] - PV 0,T [F 0,T ]) P (110 e e -.1 ) + = Comment: While McDonald does not mention this in Derivative Markets, for a European put, P Ke -rt = 110 e -.1 =

148 2013-MFE/3F, Financial Economics 5 Bounds on Premiums Euro., HCM 12/6/12, Page C. S 0 C(S 0, K, T) (PV 0,T [F 0,T ] - PV 0,T [K]) C (108 e e -.1 ) + = Comment: While McDonald does not mention this in Derivative Markets, for a European call, C F P 0,T (S0 ) = PV 0,T [F 0,T ] = 108 e -.1 = C. PV 0,T [K] - PV 0,T [F 0,T ] = K e -rt - S 0 e -δt = 90 e S 0 = S P ( S 0 ) A. PV 0,T [F 0,T ] - PV 0,T [K] = S 0 e -δt - K e -rt = 80 e K e -.3 = K. 80 C(S 0, K, T) ( K) +. C K