Mahlerʼs Guide to. Financial Economics. Joint Exam MFE/3F. prepared by Howard C. Mahler, FCAS Copyright 2012 by Howard C. Mahler.

Mahlerʼs Guide to Financial Economics Joint Exam MFE/3F prepared by Howard C. Mahler, FCAS Copyright 2012 by Howard C. Mahler. Study Aid 2012-MFE/3F Howard Mahler hmahler@mac.com www.howardmahler.com/teaching

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 1 Mahlerʼs Guide to Financial Economics Copyright 2012 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 Section # Pages Section Name 1 10-20 Introduction 2 21-30 European Options 3 31-54 Properties of Premiums of European Options 4 55-69 Put-Call Parity 5 70-79 Bounds on Premiums of European Options 6 80-91 Options on Currency 7 92-97 Exchange Options 8 98-102 Futures Contracts 9 103-108 Synthetic Positions 10 109-134 American Options 11 135-149 Replicating Portfolios 12 150-159 Risk Neutral Probabilities 13 160-171 Utility Theory and Risk Neutral Pricing 14 172-199 Binomial Trees, Risk Neutral Probabilities 15 200-206 Binomial Trees, Valuing Options on Other Assets 16 207-213 Other Binomial Trees 17 214-226 Binomial Trees, Actual Probabilities 18 227-232 Jensen's Inequality 19 233-242 Normal Distribution 20 243-250 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 2 Section # Pages Section Name 21 251-254 Limited Expected Value 22 255-283 A LogNormal Model of Stock Prices 23 284-313 Black-Scholes Formula 24 314-318 Black-Scholes, Options on Currency 25 319-321 Black-Scholes, Options on Futures Contracts 26 322-326 Black-Scholes, Stocks Paying Discrete Dividends 27 327-339 Using Historical Data to Estimate Parameters of the Stock Price Model 28 340-349 Implied Volatility 29 350-360 Histograms 30 361-375 Normal Probability Plots 31 376-402 Option Greeks 32 403-409 Delta-Gamma Approximation 33 410-415 Option Greeks in the Binomial Model 34 416-429 Profit on Options Prior to Expiration 35 430-438 Elasticity 36 439-441 Volatility of an Option 37 442-445 Risk Premium of an Option 38 446-452 Sharpe Ratio of an Option 39 453-454 Market Makers 40 455-481 Delta Hedging 41 482-491 Gamma Hedging 42 492-493 Relationship to Insurance 43 494 Exotic Options 44 495-511 Asian Options 45 512-532 Barrier Options 46 533-559 Compound Options 47 560-573 Gap Options 48 574-581 Valuing European Exchange Options 49 582-585 Forward Start Options 50 586-592 Chooser Options 51 593-597 Options on the Best of Two Assets 52 598-607 Cash-or-Nothing Options 53 608-619 Asset-or-Nothing Options 54 620-624 Random Walks 55 625-638 Standard Brownian Motion 56 639-651 Arithmetic Brownian Motion 57 652-663 Geometric Brownian Motion 58 664-694 Geometric Brownian Motion Model of Stock Prices 59 695-723 Ito Processes 60 724-739 Ito's Lemma The Table of Contents is continued on the next page.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 3 Section # Pages Section Name 61 740-752 Valuing a Claim on S^a 62 753-765 Black-Scholes Equation 63 766-772 Simulation 64 773-778 Simulating Normal and LogNormal Distributions 65 779-787 Simulating LogNormal Stock Prices 66 788-793 Valuing Asian Options via Simulation 67 794-812 Improving Efficiency of Simulation 68 813-831 Bonds and Interest Rates 69 832-833 The Rendelman-Bartter Model 70 834-855 The Vasicek Model 71 856-882 The Cox-Ingersoll-Ross Model 72 883-889 The Black Model 73 890-901 Interest Rate Caps 74 902-909 Binomial Trees of Interest Rates 75 910-938 The Black-Derman-Toy Model 76 939-981 Important Formulas and Ideas 982-1497 Solutions to Problems My practice exams are sold separately.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 4 Chapter of Derivative Markets Sections of Study Guide 9 3-10 10 2 11, 12, 14, 15. 11.1 11.4 3 10, 13, 16-17, 27, 54 12.1-12.5 4 23-26, 28, 31, 34-38 13 5 32-33, 39-42 14 43-51 18 19-22, 27, 29-30 19.1-19.5 63-67 20.1-20.7 6 55-61 21.1 21.3 7 62 22.1 8 52-53 23.1 23.2 9 27, 28, 58 24.1-24.5 10 68-75 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 334. 3 Including Appendices 11.A and 11.B. 4 Including Appendix 12.A. 5 Including Appendix 13.B. 6 Sections 20.1 20.6 (up to but excluding Multivariate Itôʼs Lemma on pages 665-666) and 20.7 (up to but excluding Valuing a Claim on S a Q b on pages 670-672 and excluding Finding the lease rate on top one-half of page 669). 7 Sections 21.1 21.2 (excluding What If the Underlying Asset Is Not and Investment Asset on pages 688-690) and 21.3 (excluding The Backward Equation on pages 691-692, 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 22.1. 9 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 806-808.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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. http://www.prometric.com/soa/mfe3f_calculator.htm 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) = 0.7764. This should not make a significant difference.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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 21.1 21.2 (excluding What If the Underlying Asset Is Not and Investment Asset on pages 688-690) and 21.3 (excluding The Backward Equation on pages 691-692, 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 23.1 23.2 (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 19.1 19.5 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 3. 16 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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 Prop. of Prems. of Euro. Options 2 4 Put-Call Parity 1 3, 4 1, 4 14, 16 12 5 Bounds on Premiums of Euro. Options 6 Options on Currency 15 9 7 Exchange Options 8 Futures Contracts 9 Synthetic Positions 13 10 American Options 26 12 13 11 Replicating Portfolios 16 17 3 12 Risk Neutral Probabilities 27 14 13 Utility Theory and Risk Neutral Pricing 14 Bin. Trees, Risk Neutral Probs. 4, 49 15, 17 11 18, 19, 23 1 15 Binomial Trees, Options on Other Assets 5, 46 16 Other Binomial Trees 14 17 Binomial Trees, Actual Probabilities 2 7 18 Jensen's Inequality 19 Normal Distribution 20 LogNormal Distribution 21 Limited Expected Value 22 A LogNormal Model of Stock Prices 50 23 Black-Scholes Formula 6 20, 21 3, 8 20 24 Black-Scholes, Options on Currency 7 21 25 Black-Scholes, Options on Futures 55 26 Black-Scholes, Discrete Dividends 15 19 27 Historical Data to Estimate Parameters 17, 51 28 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

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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, 31 17 32 Delta-Gamma Approximation 19 20 33 Option Greeks in the Binomial Model 44, 45, 69 34 Profit on Options Prior to Expiration 40 13 35 Elasticity 20, 41 22 36 Volatility of an Option 5 37 Risk Premium of an Option 38 Sharpe Ratio of an Option 29 39 Market Makers 40 Delta Hedging 9, 47, 65 32 41 Gamma Hedging 33 10 24 42 Relationship to Insurance 43 Exotic Options 44 Asian Options 27 45 Barrier Options 42 34 2 46 Compound Options 28 47 Gap Options 18 17 26 48 Valuing European Exchange Options 49 Forward Start Options 19, 33 50 Chooser Options 25 51 Options on the Best of Two Assets 54 6 52 Cash-or-Nothing Options 28, 53 53 Asset-or-Nothing Options 4 54 Random Walks 55 Standard Brownian Motion 34 56 Arithmetic Brownian Motion 57 Geometric Brownian Motion 58 Geo. Brown. Mot. Model Stock Pr. 10, 11, 32, 37 16 56, 74 59 Ito Processes 12, 23, 48, 61 18 10, 18 63, 66, 67, 70 60 Ito's Lemma 13, 24, 35, 43 35 12 6 64, 68, 73 Continued on the next page

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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, 72 11 62 Black-Scholes Equation 36 8 63 Simulation 64 Simulating Normals & LogNormals 65 Simulating LogNormal Stocks 52 66 Valuing Asian Options via Sim. 67 Improving Efficiency of Simulation 57, 58, 59, 75 68 Bonds and Interest Rates 39 69 The Rendelman-Bartter Model 70 The Vasicek Model 14, 22 36 13 15 71 The Cox-Ingersoll-Ross Model 21, 38, 60 72 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, 76 37 9 14 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 2008. Material on simulation was moved here from exam 4/C in Fall 2009.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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 Sections 1.1-1.4, 2.1-2.6, Appendix 2.A, 3.1-3.5, 4.1-4.4, 5.1-5.4, Appendix 5.B, 8.1-8.2. 19 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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.045 + e.03 + e.015 + e 0 ) = $98.00. 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. 26 24 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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%) = 117.03.] 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%) = 79.67.] 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).

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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 - 80. 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. 33 31 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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 -0.05 = $61.83. 35 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[-0.05-0.06] = $76.15. Adding in the actuarial present value of the payments made in one year, the actuarial present value of the insurerʼs total payments is: $61.83 + $76.15 = $137.98.] 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, 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 230 1.2 (1 point) A stock has a current price of 120. The stock pays dividends at a continuously compounded rate of 1.5%. r = 0.08. What is the 4-year prepaid forward price? A. 113 B. 115 C. 117 D. 119 E. 121 1.3 (1 point) A stock has a two-year forward price of 99.66. 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. 98 1.4 (1 point) A stock has a current price of 90. The stock pays dividends at a continuously compounded rate of 2%. r = 0.06. What is the 5-year forward price? A. 100 B. 105 C. 110 D. 115 E. 120 1.5 (1 point) A stock has a current price of $100. In 3 months the stock will pay a dividend of $2. r = 0.04. What is the 4-month prepaid forward price? 1.6 (1 point) A stock has a four-year forward price of 89.43. 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. 85 1.7 (5B, 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

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 19 1.8 (FM Sample Exam, Q.4) Zero-coupon risk-free bonds are available with the following maturities and yield rates (effective, annual): Maturity (years) Yield 1 0.06 2 0.065 3 0.07 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, 60,184 1.9 (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 105. 1.10 (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

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 20 1.11 (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 1.12 (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, 2007. 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 $200,000 1.13 (8, 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 21 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 $150. 36 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 22 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 < 0. 40 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 140 120 100 80 60 40 20 Stock Price 50 100 150 200 250 300 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 $250. 40 This very useful actuarial notation is not on the syllabus of this exam.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 23 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 250 200 150 100 50 Stock Price 100 200 300 400 500 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 24 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. 45 44 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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 25 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% 100 30% 120 20% 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 13 2.3 (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 13 2.4 (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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 26 2.7 (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. 2.10 (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. 2.11 (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. 2.13 (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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 27 2.14 (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. 2.16 (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. 2.18 (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. 2.19 (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. 2.20 (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.

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 28 2.21(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. 2.22 (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. 2.23 (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) +. 2.24 (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. 2.25 (5B, 11/98, Q.15) (1 point) What combination of stocks, options and borrowing/lending could be represented by the following position diagram? Valueof Position - 20 50 100 150 200 SharePrice - 40-60 - 80-100 1. Sell one share of stock short and borrow the present value of $100. 2. Sell one call with exercise price of $100 and sell one put with exercise price of $100. 3. Sell one share of stock short, sell two puts with exercise price of $100, and lend the present value of $100. A. 1 B. 2 C. 3 D. 2, 3 E. 1, 2, 3

HCMSA-2012-MFE/3F, Financial Economics, 12/14/11, Page 29 2.26 (5B, 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. 2.27 (5B, 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. 2.28 (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 10,000 0.12-strike put options for 84.30 and sell 10,000 0.14-strike call options for 74.80. 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 400