Introduction. Financial Economics Slides Howard C. Mahler, FCAS, MAAA These are slides that I have presented at a seminar or weekly class. The whole syllabus of Exam MFE is covered. At the end is my section of important ideas and formulas. Use the bookmarks / table of contents in the Navigation Panel in order to help you find what you want. This provides another way to study the material. Some of you will find it helpful to go through one or two sections at a time, either alone or with a someone else, pausing to do each of the problems included. All the material, problems, and solutions are in my study guide, sold separately. 1 These slides are a useful supplement to my study guide, but are self-contained. There are references to page and problem numbers in the latest edition of my study guide, which you can ignore if you do not have my study guide. The slides are in the same order as the sections of my study guide. At the end, there are some additional questions for study. Section # Section Name 1 Introduction 2 European Options 3 Properties of Premiums of European Options 4 Put-Call Parity 5 Bounds on Premiums of European Options 6 Options on Currency 7 Exchange Options 8 Futures Contracts 9 Synthetic Positions 10 American Options 11 Replicating Portfolios 12 Risk Neutral Probabilities 13 Utility Theory and Risk Neutral Pricing 14 Binomial Trees, Risk Neutral Probabilities 15 Binomial Trees, Valuing Options on Other Assets 16 Other Binomial Trees 17 Binomial Trees, Actual Probabilities 18 Jensen's Inequality 19 Normal Distribution 20 LogNormal Distribution 1 My practice exams are also sold separately.
Section # Section Name 21 Limited Expected Value 22 A LogNormal Model of Stock Prices 23 Black-Scholes Formula 24 Black-Scholes, Options on Currency 25 Black-Scholes, Options on Futures Contracts 26 Black-Scholes, Stocks Paying Discrete Dividends 27 Using Historical Data to Estimate Parameters of the Stock Price Model 28 Implied Volatility 29 Histograms 30 Normal Probability Plots 31 Option Greeks 32 Delta-Gamma Approximation 33 Option Greeks in the Binomial Model 34 Profit on Options Prior to Expiration 35 Elasticity 36 Volatility of an Option 37 Risk Premium of an Option 38 Sharpe Ratio of an Option 39 Market Makers 40 Delta Hedging 41 Gamma Hedging 42 Relationship to Insurance 43 Exotic Options 44 Asian Options 45 Barrier Options 46 Compound Options 47 Gap Options 48 Valuing European Exchange Options 49 Forward Start Options 50 Chooser Options 51 Options on the Best of Two Assets 52 Cash-or-Nothing Options 53 Asset-or-Nothing Options 54 Random Walks 55 Standard Brownian Motion 56 Arithmetic Brownian Motion 57 Geometric Brownian Motion 58 Geometric Brownian Motion Model of Stock Prices 59 Ito Processes 60 Ito's Lemma
Section # Section Name 61 Valuing a Claim on S^a 62 Black-Scholes Equation 63 Simulation 64 Simulating Normal and LogNormal Distributions 65 Simulating LogNormal Stock Prices 66 Valuing Asian Options via Simulation 67 Improving Efficiency of Simulation 68 Bonds and Interest Rates 69 The Rendleman-Bartter Model 70 The Vasicek Model 71 The Cox-Ingersoll-Ross Model 72 The Black Model 73 Interest Rate Caps 74 Binomial Trees of Interest Rates 75 The Black-Derman-Toy Model 76 Important Formulas and Ideas
Chapter of Third Edition Derivatives Markets Sections of Study Guide 9 3-10 10 2 11, 12, 14, 15, 27 11.1 11.3 3 10, 13, 16-17, 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 20.1-20.6 6 55-61 21.1 21.3 7 62 23.1 8 52-53 24.1 24.2 9 27, 28, 58 25.1-25.5 10 68-75 Appendix B.1 1 Appendix C 18 2 Excluding Options on Commodities on pages 315 and 316 3 Including Appendices 11.A and 11.B. 4 Including Appendix 12.A. 5 Including Appendix 13.B. 6 Sections 20.1 20.3 (up to but excluding Modeling Correlated Asset Prices on pages 612-613), 20.4 (excluding Multivariate Itôʼs Lemma on pages 616-617), 20.5 20.6 (up to but excluding Valuing a Claim on S a Q b on pages 621-622). 7 Sections 21.1 21.2 (excluding What If the Underlying Asset Is Not an Investment Asset on pages 635 637) and 21.3 (excluding The Backward Equation on pages 637 638, and excluding the last two paragraphs of the section on page 639). 8 But with only those definitions in Tables 23.1 and 23.2 that are relevant to Section 23.1. 9 Up to the second paragraph on page 721, but including footnote 4 on page 721 and the top panel in Figure 24.3 on page 723. 10 Sections 25.1 25.4 (up to the first paragraph on page 773), 25.5 (excluding LIBOR Market Model on pages 781-783), Appendix 25.A (this appendix contains only a reference to the following site for download, http://wps.aw.com/wps/media/objects/14728/15081864/appendices/mcdonald-web-25-a.pdf )
Author Biography: Howard C. Mahler is a Fellow of the Casualty Actuarial Society, and a Member of the American Academy of Actuaries. He has taught actuarial exam seminars and published study guides since 1994. He spent over 20 years in the insurance industry, the last 15 as Vice President and Actuary at the Workers' Compensation Rating and Inspection Bureau of Massachusetts. He has published many major research papers and won the 1987 CAS Dorweiler prize. He served 12 years on the CAS Examination Committee including three years as head of the whole committee (1990-1993). Mr. Mahler has taught live seminars and/or classes for Exam C, Exam MFE, CAS Exam ST, CAS Exam 5, and CAS Exam 8. He has written study guides for all of the above. hmahler@mac.com www.howardmahler.com/teaching
Exam MFE Financial Economics Seminar Style Slides prepared by Howard C. Mahler, FCAS Copyright 2016 by Howard C. Mahler. Howard Mahler hmahler@mac.com www.howardmahler.com/teaching
Introduction A derivative is an agreement between two people that has a value determined by the price of something else. A call is an option to buy. A put is an option to sell.
Continuously Compounded Risk Free Rate: r as used by McDonald is what an actuary would call the force of interest. The discount factor is: exp[-t r]. In contrast, an effective annual rate is what an actuary would call the rate of interest. The discount factor is: 1 / (1 + r)t.
A forward contract is an agreement that sets the terms today, but the buying or selling of the asset takes place in the future. The forward price, F 0,T, is the price we would pay at time T for a forward contract. Cash on delivery. In contrast, the prepaid forward price, F P 0,T, is the price we would pay today for a forward contract. Cash in advance. F P 0,T = PV[F0,T ] = F 0,T e-rt.
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. If discrete dividends: F P 0,T (S) = S0 - PV[Div]. If continuous dividends at rate δ: F P 0,T (S) = S0 e-δt.
Page 18. There are named positions involving combinations of puts and calls. For example, a Straddle is the purchase of a call and the otherwise identical put. If they use the name of one of these positions in a question, the exam committee expects you to know what is meant.
European Options 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. Future Price Payoff on a of Stock $150 Strike Call Option $120 0 $140 0 $160 $10 $180 $30
Future Price Payoff on a of Stock $150 Strike Call Option $120 0 $140 0 $160 $10 $180 $30 Payoff 140 120 100 80 60 40 20 Stock Price 50 100 150 200 250 300
The 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. Y + = Y if Y > 0, Y + = 0 if Y 0.
If the stock price is greater than the strike price, S > K, then the call is in the money. If the stock price and strike price are equal, S = K, then the option is at the money. If the stock price is less than the strike price, S < K, then the call is out of the money.
A European option may only be exercised on one specific day.
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. The payoff on a European put option is: (K - S T ) +. Future Price Payoff on a of Stock $250 Strike Put Option $220 $30 $240 $10 $260 0 $280 0
Future Price Payoff on a of Stock $250 Strike Put Option $220 $30 $240 $10 $260 0 $280 0 Payoff 250 200 150 100 50 Stock Price 100 200 300 400 500
Properties of Premiums of European Options r = 4%. The price of a stock two years from now has the following distribution: $50 @ 20%, $100 @ 40%, $150 @30%, $200 @10%. Determine the present value of a European call option on this stock with 2 years to expiration and a strike price of $90. (20%)(0) + (40%)(10) + (30%)(60) + (10%)(110) exp[0.08] = $30.46. The actuarial present value of a European call option is: E[(S T - K) + ] e-tr.
As will be discussed subsequently, in order to price the option these should be risk-neutral probabilities.
The call premium if K = $90: $30.46. The call premium if K = $100: (20%)(0) + (40%)(0) + (30%)(50) + (10%)(100) exp[0.08] = $23.08 < $30.46. The call premium decreases as the strike price increases. For K 1 < K 2, C(K 1 ) C(K 2 ). The option to buy at 100 is less valuable than the option to buy at 90.
Page 63. Stock Payoff on Call Payoff on Call Price with K = 90 with K = 100 Difference 50 0 0 0 100 10 0 10 150 60 50 10 200 110 100 10 The difference in payoffs is at most the difference in strike prices, 10. Therefore the difference in call premiums is at most the difference in strike prices: For K 1 < K 2, C(K 1 ) - C(K 2 ) K 2 - K 1. In fact, since a European Option can only be exercised at expiration, the difference in the call premiums cannot be more than the discounted difference in strike prices: C(K 1 ) - C(K 2 ) (K 2 - K 1 ) e-rt.
For K 1 < K 2 < K 3, C(K 1 ) - C(K 2 ) K2 - K1 C(K 2) - C(K 3 ) K 3 - K 2. This is called the convexity of the option price with respect to the strike price.
Convexity is equivalent to the graph of the option price as a function of the strike price being concave upwards, in other words having a positive second derivative: 2 C K 2 0. 100 80 60 40 20 C 50 100 150 200 K The call premium decreases; the slope is negative.
Page 75. Arbitrage: If there is a possible combination of buying and selling with no net investment that has no risk but generates nonnegative cashflows, this is an arbitrage opportunity. Taking advantage of such an opportunity is called arbitrage. In other words, arbitrage is free money. When one of the properties is violated, there is an opportunity for arbitrage.
Page 76 For three otherwise similar calls: 100 strike call costs 20 110 strike call costs 17. 140 strike call costs 7. 20-17 110-100 = 0.3. 17-7 140-110 = 0.333. 0.333 > 0.3. Violating convexity.
For three otherwise similar calls: 100 strike call costs 20 110 strike call costs 17. 140 strike call costs 7. These call premiums as a function of strike price: C 20 17 7 100 110 140 K
Since the point (110, 17) is above the line between (100, 20) and (140, 7), convexity is violated. C 17 110 K In other words, the curve of option premium versus strike price is not concave upwards.
Let λ = K 3 - K2 K 3 - K1. In this example, λ = 140-110 140-100 = 3/4. One way to demonstrate arbitrage: 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, 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, buy 3 of the 100 strike calls, buy 1 of the 140 strike calls, and sell 4 of the 110 strike calls.
3.14. Three European call options on a stock are otherwise similar except for their strike price. K Premium 100 25 110 20 115 16 What general property is violated? 3.15. Briefly describe an opportunity for arbitrage presented by this situation.
The absolute value of the changes in the value of the option over the absolute value of the changes in the strike price are: 25-20 110-100 = 0.5, and 20-16 115-110 = 0.8. 0.5 < 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. In other words, the curve of option premium versus strike price is not concave upwards.
These call premiums as a function of strike price: C 25 20 16 100 110 115 K Since (110, 20) is above the line between (100, 25) and (115, 16), convexity is violated.
λ = 115-110 115-100 = 1/3. 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, 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. Loan out this 3 and collect interest at the risk free rate, r.
Buy 1 call with strike price 100, buy 2 calls with a strike price of 115, and sell 3 calls with strike price 110. At expiration of the calls at time T, we have: If S 100: 3 ert > 0. If 110 S > 100: (S - 100) + 3 ert > 0. If 115 S > 110: (S - 100) - (3) (S - 110) + 3 ert = 230-2S + 3 ert > 0. If S > 115: (S - 100) - (3) (S - 110) + (2) (S - 115) + 3 ert = 3 ert > 0. Thus we always end up with a positive position, demonstrating arbitrage.
C 25 20 16 100 110 115 K 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 115-100. 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(K 1 ) + (1 - λ) C(K 3 ) C(K 2 ). See my page 77.
Page 69. Actuarial present value (premium) of a European put option is: E[(K - S T ) + ] e-tr. The price of a stock two years from now has the following distribution: $50 @ 20%, $100 @ 40%, $150 @30%, $200 @10%. r = 4%. Determine the present value of a European put option on this stock with 2 years to expiration and a strike price of $120. (20%)(70) + (40%)(20) + (30%)(0) + (10%)(0) exp[0.08] = $20.31.
The premium of the put increases as the strike price increases and the curve is concave upwards. For K 1 < K 2, P(K 1 ) P(K 2 ). For K 1 < K 2, P(K 2 ) - P(K 1 ) K 2 - K 1. In fact since a European Option can only be exercised at expiration: P(K 2 ) - P(K 1 ) (K 2 - K 1 ) 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 2. This is called the convexity of the option price with respect to the strike price.
Convexity is equivalent to the graph of the option price as a function of the strike price being concave upwards, in other words having a positive second derivative: 15 P 10 5 20 40 60 80 100 K The put premium increases; the slope is positive.
Page 81. Strike Prices: K 1 < K 2 < K 3 Arbitrage if Condition Condition is Violated C(K 1 ) C(K 2 ). C(K 1 ) - C(K 2 ) K 2 - K 1. 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 of K 1 {C(K 2 ) - C(K 3 )} / {K 3 - K 2 }. Sell 1 of K 2 Buy 1 - λ of K 3 (Asymmetric butterfly spread) λ = (K 3 - K 2 ) / (K 3 - K 1 )
{C(K 1 ) - C(K 2 )} / {K 2 - K 1 } Buy λ of K 1 {C(K 2 ) - C(K 3 )} / {K 3 - K 2 }. Sell 1 of K 2 Buy 1 - λ of K 3 (Asymmetric butterfly spread) λ = (K 3 - K 2 ) / (K 3 - K 1 ) Butterfly spread: buy the wings and sell the middle.
Page 81. Strike Prices: K 1 < K 2 < K 3 Arbitrage if Condition Condition is Violated 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 of K 1 {P(K 3 ) - P(K 2 )} / {K 3 - K 2 }. Sell 1 of K 2 Buy 1 - λ of K 3 (Asymmetric butterfly spread) λ = (K 3 - K 2 ) / (K 3 - K 1 )
Put-Call Parity
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. 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]. put-call parity: C Eur (K, T) = P Eur (K, T) + PV[F 0,T ] - PV[K]. With dividends paid continuously: C Eur (K, T) = P Eur (K, T) + S 0 e-δt - K e-rt.
Note the information above Q. 4.3 4.4. The Rich and Fine stock index is priced at 1300. Dividends are paid at the rate of 2%. The continuously compounded risk free rate is 5%. 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? C = 293 + 1300 e-0.06-1500 e-0.15 = 226.23.
4.15. 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%. 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? C = 15 + 120 - (2) e-0.05/4 - (130) e-0.05/3 = $5.17.
4.35 (2 points) For a dividend paying stock and European options on this stock, you are given the following information: The current stock price is $57. The strike price of options is $60. The time to expiration is 9 months. The continuous risk-free rate is 4% annually. The continuous dividend yield is 1% annually. The call price is $3.80. The put price is $5.00. Using put-call parity, calculate the present value arbitrage profit per share that could be generated, given these conditions. A. Less than $0.10 B. At least $0.10 but less than $0.20 C. At least $0.20 but less than $0.30 D. At least $0.30 but less than $0.40 E. At least $0.40
4.35. E. Based on put-call parity, we would expect the call to have a price of: P - K e-rt + S e-δt = 5.00-60 e-(0.04)(3/4) + 57 e-(0.01)(3/4) = $3.35. Therefore, the call is overpriced at $3.80. The present value arbitrage profit is: $3.80 - $3.35 = $0.45. Alternately, based on put-call parity, we would expect the put to have a price of: C + K e-rt - S e-δt = 3.80 + 60 e-(0.04)(3/4) - 57 e-(0.01)(3/4) = $5.45. Therefore, the put is underpriced at $5.00. The present value arbitrage profit is: $5.45 - $5.00 = $0.45.
Comment: Similar to CAS3, 5/07, Q.3. We can take advantage of the arbitrage by: buying a put, selling a call, and buying e-δt = e-(0.01)(3/4) = 0.9925 shares of stock. When we set up the position we receive: 3.80-5.00-57 e-(0.01)(3/4) = -57.774. We borrow this at the risk free rate, and will need to repay 57.774 e(0.04)(3/4) = 59.533, at time 3/4. At time 3/4 we have 1 share of stock. If S 3/4 < K = 60, then we use our put to sell our share of stock for 60. If S 3/4 > K = 60, then the person to whom we sold the call uses it to buy our share of stock for 60. In either case, after repaying our loan are left with: 60-59.533 = $0.467. The present value is: 0.467 e-(0.04)(3/4) = $0.45.
4.43. CAS3, 5/07, Q.3 For a dividend paying stock and European options on this stock, you are given the following information: The current stock price is $49.70. The strike price of options is $50.00. 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 $0.80
CAS3, 5/07, Q.3. B. Based on put-call parity, we would expect the call to have a price of: P - K e-rt + S e-δt = 2.35-50 e-0.03/2 + 49.70 e-0.02/2 = $2.30. Therefore, the call is underpriced at $2.00. The present value arbitrage profit is: $2.30 - $2.00 = $0.30. Alternately, based on put-call parity, we expect the put to have a price of: C + K e-rt - S e-δt = 2.00 + 50 e-0.03/2-49.70 e-0.02/2 = $2.05. Therefore, the put is overpriced at $2.35. The present value arbitrage profit is: $2.35 - $2.05 = $0.30.
Comment: We can take advantage of the arbitrage by: buying a call, selling a put, investing K e-rt, and shorting e-δt = e-0.02/2 = 0.990 shares of stock. (We borrow 0.990 shares of stock from someone at time 0 and return 1 share of stock at time 1/2. If one owned 0.990 shares and reinvested the dividends, then one would have 0.990 e0.02/2 = 1 share at time 1/2.)
Bounds on Premiums of European Options Since an option does not require its owner to do anything, it never has a negative value. Owning a share of stock now is worth at least as much as the option to buy a share of stock in the future. S 0 C(S 0, K, T). A call is never worth more than the current stock price. From put-call parity: C = P + PV[F 0,T ] - PV[K]. C PV[F 0,T ] - PV[K]. F 0,T is the price of a forward contract to buy the stock at time T. S 0 C(S 0, K, T) (PV 0,T [F 0,T ] - PV 0,T [K]) + 0.
The payoff on a put is: (K - S T ) + K. A put is never worth more than the strike price. From put-call parity: C = P + PV[F 0,T ] - PV[K]. P = C + PV[K] - PV[F 0,T ]. P PV[K] - PV[F 0,T ]. K P(S 0, K, T) (PV 0,T [K] - PV 0,T [F 0,T ])+ 0.
Options on Currency
One may be able today to buy one Euro for 1.25 U.S. Dollars. 1 = $1.25. 0.8 = $1. Then one could buy one U.S. Dollar for 0.8 Euros. A dollar-denominated call on Euros would give one the option to obtain Euros at some time in the future for a specified number of dollars. For example, if T = 2 years and K = $1.20/, then the owner of the call would have the option in 2 years to use $1.20 to obtain one Euro. A dollar-denominated put on Euros would give one the option to sell Euros at some time in the future for a specified number of dollars. For example, if T = 2 years and K = $1.20/, then the owner of the put would have the option in 2 years to sell one Euro for $1.20.
For a dollar-denominated call on Euros: C $ (x 0, K, T) = P $ (x 0, K, T) + x 0 exp[-t r ] - K e-tr. Similar to: C(K, T) = P(K, T) + S 0 e-tδ - K e-tr. x 0 S 0. x 0 is the price of the asset euros. S 0 is the price of the asset stock. Dollars act as money: r $ r. Euros act as the asset: r δ.
x 0 = $1.25/. T = 2 years. K = $1.20/. The dollar-denominated interest rate is 6% and the euro-denominated interest rate is 4%. The option premium for a dollar-denominated put on one Euro is $0.10. Determine the premium for a dollar-denominated call on one Euro. $0.10 + ($1.25)e-(2)(.04) - ($1.20)e-(2)(.06) = $0.190. Note that each term is in dollars.
A Euro-denominated call on dollars would give one the option to obtain dollars at some time in the future for a specified number of Euros. A Euro-denominated put on dollars would give one the option to sell dollars at some time in the future for a specified number of Euros. For a Euro-denominated call on dollars: C (x 0, K, T) = P (x 0, K, T) + x 0 exp[-t r $ ] - K exp[-t r ], x 0 is the exchange rate for dollars in terms of Euros, 0.8 Euro = $1, and K is in Euros. Similar to: C(K, T) = P(K, T) + S 0 e-tδ - K e-tr. x 0 S 0. x 0 is the price of the asset dollars. S 0 is the price of the asset stock. Euros act as money: r r. Dollars act as the asset: r $ δ.
6.8. Currently one can buy 1 U.S. Dollar for 11 Mexican Pesos. Dollar and Peso interest rates are 4.0% and 6.0%, respectively. The price of a 12 Peso strike 1-year put option on dollars is 1.36 Pesos. What is the price in Pesos of a similar call?
6.8. B. C Peso (x 0, K, T) = P Peso (x 0, K, T) + x 0 exp[-t r $ ] - K exp[-t r Peso ]. = 1.36 + (11) e-0.04 - (12) e-0.06 = 0.63 Pesos. Comment: We have a peso denominated option, and therefore pesos act as money, while dollars act as the asset. r Peso r = return on money r $ δ = return on asset (stock) This is the reverse of what we would do for a dollar denominated option. x 0 is the price of the asset dollars. S 0 is the price of the asset stock. Note that all terms in the equation are in pesos.
Page 157. Different Points of View Currently, 1 Euro = $1.25. 0.8 Euro = $1. Sam is in the United States. Sam has 1000 dollar denominated calls with strike price $1.30, giving him the option one year from now to buy 1000 Euros for 1300 dollars.
Johann is in Germany. Johann has 1300 Euro denominated puts with strike price: 1/1.30 = 0.7692, giving him the option one year from now to sell 1300 dollars for: (1300) (0.7692) = 1000 Euros.
Both Samʼs and Johannʼs positions give them the option one year from now to use 1300 dollars to obtain 1000 Euros. Their positions must be worth the same. If each of Samʼs 1000 calls is currently worth $0.085, then his position is worth $85. Johannʼs position is currently worth: (0.8) ($85) = 68 Euros. Each of Johannʼs 1300 puts is worth: 68 / 1300 = 0.0523 Euros.
A dollar denominated call to buy euros is equal to a euro denominated put to sell dollars. A call from one point of view is a put from an opposite point of view. In the above example: C $ (1.25, 1.30, 1) = 0.085 = (1.25) (1.30) (0.0523) = (1.25) (1.30) P (1/1.25, 1/1.30, 1). C $ (x 0, K, T) = x 0 K P f (1/x 0, 1/K, T).
6.9. The spot exchange rate of dollars per South Korean Won is 0.00097. Dollar and Won interest rates are 4.0% and 7.0%, respectively. The price for one million $0.00100 strike 3 year call options on Won is $40.11. What is the price in Won of a 1000 Won strike 3-year call on dollars?
6.9. C. By put-call parity, C = P + S e-δt - K e-rt. Apply parity to the dollar denominated call in order to get the premium for the similar dollar denominated put: 40.11/1,000,000 = P + (0.00097) exp[-(.07)(3)] - (0.00100) exp[-(.04)(3)]. P = $0.0001408. Now use the premium for the dollar denominated put in order to get the premium for the corresponding Won denominated call: P $ (x 0, K, T) = x 0 K C f (1/x 0, 1/K, T). Therefore, 0.0001408 = (0.00097) (0.001) Cf. C f = 145 Won.
6.9, continued Alternately, use the premium for the dollar denominated call in to get the premium for the corresponding Won denominated put: C $ (x 0, K, T) = x 0 K P f (1/x 0, 1/K, T). 40.11 / 1,000,000 = (0.00097) (0.00100) Pf. Pf = 41.35 Won. Now apply parity to the Won denominated put in order to get the premium for the similar Won denominated call. (Won money. Dollars asset.) C Won (x 0, K, T) = P Won (x 0, K, T) + x 0 exp[-t r $ ] - K exp[-t r Won ] The spot exchange rate of South Korean Won per dollars is: 1 / 0.00097 = 1030.9. C = 41.35 + 1030.9 exp[-(.04)(3)] - 1000 exp[-(.07)(3)] = 145 Won.
6.9, continued 2 Put-Call Parity $ denominated call $ denominated put Sam-Johann Won denominated put Sam-Johann Won denom. call Put-Call Parity Comment: Similar to MFE 5/09, Q.9. Note that the strike for the Won denominated options is: 1 / 0.001 = 1000 Won.
Exchange Options A call exchange option allows one to exchange one share of a Stock B (strike asset) for one share of the stock of Stock A (underlying asset). So rather than a strike price K in dollars, we have the option to use an asset, Stock B, to obtain Stock A. For example, if the call expires in 2 years, among the many possible situations: Price of Stock A Price of Stock B Exercise (Underlying Asset) (Strike Asset) Exchange 2 Years from now 2 Years from now Call? 100 90 Yes 110 130 No 120 100 Yes
A similar put exchange option would instead allow us to sell stock A in exchange for Stock B. In other words, get B in exchange for A.
Where S is the underlying asset and Q is the strike asset, a form of put call-parity for exchange options: C Eur (S 0, Q 0, T) = P Eur (S 0, Q 0, T) + F 0,T P (S0 ) - F 0,T P (Q0 ). Where F 0,T P is the prepaid forward price. We would pay F 0,T P at time 0 in order to own the asset at time T. F 0,T P (S0 ) = S 0 exp[-δ S T]. F 0,T P (Q0 ) = Q 0 exp[-δ Q T].
The current price of Stock A is $100 and the current price of Stock B is $120. Stock A pays dividends at a continuous annual rate of 1%, while Stock B pays dividends at a continuous annual rate of 2%. What is the difference in price between a 3 year European call option to exchange Stock B for Stock A, and the similar put? C Eur (S 0, Q 0, 3) - P Eur (S 0, Q 0, 3) = (100) exp[-(3)(1%)] - (120) exp[-(3)(2%)] = -15.97. In this case, the put is worth more than the similar call. Exchange Call: can use B to get A. Exchange Put: can use A to get B. Since Stock B is currently worth much more than Stock A, the put is more likely to be worth something at expiration than is the call.
7.2. A 2 year European exchange call option has underlying asset one share of MGH Shipping with a current price of $100 per share. The strike asset is 2 shares of Galactus Transport, with a per share price of $60. One will have the option to use 2 shares of Galactus in order to obtain one share of MGH. The price of this call option is $11. MGH Shipping pays dividends at a continuous rate of 2% per year. Galactus Transport pays dividends at a continuous rate of 1% per year. What is the price of the similar put?
7.2. C Eur (S 0, Q 0, T) = P P P Eur (S 0, Q 0, T) + F 0,T(S0 ) - F 0,T(Q0 ). 11 = P + 100 exp[-(2)(2%)] - (2)(60) exp[-(2)(1%)]. P = $32.54.
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. The buyer and the seller post margin.
Frank is a farmer who expects to harvest 10,000 bushels of wheat in July. On April 15, Frank enters into a futures contract for $4 per bushel agreeing to deliver 10,000 bushels of wheat on July 15 to Momʼs Bakery. Frank has sold a futures contract. Frankʼs position would be referred to as the short position; he has agreed to deliver wheat. Momʼs Bakery has bought a futures contract. Momʼs Bakery has a long position; it has agreed to accept delivery of wheat at a predetermined price.
Page 180 Options on Futures Contracts: On March 15, a 3 month $4 per bushel strike put on a futures contract on 10,000 bushels of wheat for July 15 delivery has a premium of $815. The buyer of this put would on June 15 have the option to sell a futures contract on 10,000 bushels of wheat for July 15 delivery at $4 per bushel. Frank the farmer might buy such a put. The buyer of a similar call would on June 15 have the option to buy a futures contract on 10,000 bushels of wheat for July 15 delivery at $4 per bushel. Momʼs Bakery might buy such a call.
P. 182 Parity for Options on Futures Contracts: C = P + e-rt F 0 - e-rt K. Where F 0 is the futures price at time zero for the delivery time in the futures contract underlying the options, and T is the time from when the option is bought to when it expires.
On March 15, a 3 month $4 per bushel strike put on a futures contract on 10,000 bushels of wheat for July 15 delivery has a premium of $815. On March 15, the futures price for July 15 delivery of wheat is $4.20 per bushel. If r = 5%, what is the premium for the corresponding call? C = P + e-rt F 0 - e-rt K = $815 + (10,000) ($4.20) e-(.05)(1/4) - (10,000) ($4.00) e-(.05)(1/4) = $2790. The $4.20 futures price is just another input.
The put-call parity relationship for options on futures contracts can be obtained from that for options on stocks by: S 0 F 0 = futures price δ r. C = P + e-rt F 0 - e-rt K. C = P + e-δt S 0 - e-rt K.
Synthetic Positions The law of one price states that two positions that generate the exact same cashflows should have the same cost. We used this in the discussion of put-call parity. C = P + S - PV[Div] - PV[K]. S = C - P + PV[Div] + PV[K]. Therefore, for example, one can create a synthetic share of stock by: buying a call, selling a put, lending the present value of any dividends, and lending the present value of the strike price K. This synthetic share of stock will have the same cashflows as an actual share of stock.
9.5 (2 points) Use the following information: The price of a non-dividend paying stock is $77. The price of a European call with a strike price of $80 is $4.51. The price of a European put with a strike price of $80 is $5.92. Both options expire in four months. Calculate the annual continuously compounded risk-free rate on a synthetic T-Bill created using these options. A. 3% B. 4% C. 5% D. 6% E. 7%
9.5. D. Using put-call parity, C = P - Ke-rT + Se-δT. 4.51 = 5.92-80 e-r/3 + 77 e-0/3. e-r/3 = 78.41 / 80 = 0.9801. r = 6.0%. Comment: Similar to CAS3, 5/07, Q.13. K e-rt = S e-δt + P - C. A synthetic T-Bill can be created by buying the stock, buying the put, and selling the call. If the stock paid dividends, then one would need to buy a forward contract on the stock in order to create a synthetic T-Bill.
American Options
An American style option may be exercised at any time up to the expiration date. An American option is worth at least as much as the similar European option. An American option is worth at least as much as a similar option with less time to expiration. If the stock pays no dividends, then it is never worthwhile to exercise an American call option early. Thus for a stock that pays no dividends, an American call option is worth the same as a European call option.
Page 200. The continuation value of an American option is the value of the option at a given point in time and at a given stock price, if we do not exercise the option immediately. Continuation Value > Exercise Value. Wait. Do Not Exercise Early. Continuation Value < Exercise Value. Exercise Early.
Page 201. For American calls there are two reasons to wait and one reason to exercise early: 1. The time value of the strike price is lost if one exercises early. 2. The implicit insurance protection against the stock price moving below the strike price is lost if one exercises early. 3. The value of dividends is gained if one exercises early. Reasons to wait: #1 and #2. Reasons to exercise early: #3.
If we exercise the call early, then we own the stock and get the dividends. If we exercise early, then we spend the strike price. If instead we wait to exercise, then we do not spend the strike price and we can earn interest on that money. If we exercise early, then we own the stock, and risk the stock price going down. If instead we wait to exercise the call, we avoid the possibility of capital losses on the stock.
Page 200 By Put-Call Parity: C Eur = S T - PV[Div] - Ke-rT + P Eur. C Amer C Eur. C Amer S T - K - PV[Div] + K (1 - e-rt) + P Eur. S T - K = Exercise Value of the Call. PV[Div] = Present Value of Future Dividends. K (1 - e-rt) = Time Value of Money on Strike. P Eur = Insurance against S T < K.
More generally, assume that you own a stock and wish to lock in your capital gains. If you wish to guarantee that you can sell the stock for K at time T in the future, then you could buy a K-strike European put that expires at time T. This put acts as insurance against a decrease in the stock price.
Page 203. For American puts there are two reasons to wait and one reason to exercise early: 1. The time value of the strike price is gained if one exercises early. 2. The implicit insurance protection against the stock price moving above the strike price is lost if one exercises early. 3. The value of dividends is lost if one exercises early. Reasons to wait: #2 and #3. Reasons to exercise early: #1. If we exercise the put early and use it to sell the stock, then we do not own the stock, and fail to gain if the stock price goes up. If instead we wait to exercise the put, we retain the possibility of additional capital gains on the stock.
By Put-Call Parity: P Eur = Ke-rT - S T + PV[Div] + C Eur. P Amer P Eur. P Amer K - S T + PV[Div] - K (1 - e-rt) + C Eur. K - S T = Exercise Value of the Put. PV[Div] = Present Value of Future Dividends. K (1 - e-rt) = Time Value of Money on Strike. C Eur = Insurance against S T > K.
10.1. r = 0%. Stock ABC pays dividends. You own an American call on ABC. Is it ever optimal to exercise this option before expiration? Briefly explain why.
10.1. For the American call option, dividends on the stock are the reason why we want to receive the stock earlier. With the interest rate equal to zero, we do not have the benefit of earning interest on the strike, which is normally a reason for waiting to exercise the call. However, there is another benefit to waiting to exercise the call, the insurance protection against the stock price moving below the strike price. We will not exercise the option if it is out-of-the-money. There may be circumstances in which we will early exercise, but we will not always do so.
Page 195. S 0 C Eur (PV 0,T [F 0,T ] - PV 0,T [K]) + 0. In fact, since a European call can only be exercised at expiration, it can not be worth more than the value of getting a share of stock at expiration, the prepaid forward price: PV 0,T [F 0,T ] C Eur. Since C Amer C Eur : S 0 C Amer C Eur (PV 0,T [F 0,T ] - PV 0,T [K]) +. In fact, since an American option can be exercised right away, the premium can not be less than the exercise value at time of purchase: C Amer (S 0 - K) +.
K P Eur (PV 0,T [K] - PV 0,T [F 0,T ])+ 0. In fact, since a European put can only be exercised at expiration: PV 0,T [K] P Eur. See MFE Sample Exam Q. 26. Since P Amer P Eur : K P Amer P Eur (PV 0,T [K] - PV 0,T [F 0,T ])+. In fact, since an American option can be exercised right away, the premium can not be less than the exercise value at time of purchase: P Amer (K - S 0 ) +. See MFE Sample Exam Q. 26.
10.14 (1 point) An American 95 strike 3 year call is an option on a stock with current price 90 and forward price of 100. r = 5%. Determine the width of the range of possible prices of this option. A. 84 B. 86 C. 88 D. 90 E. 92
10.14. B. S 0 C Amer (S 0, K, T) (PV 0,T [F 0,T ] - PV 0,T [K]) + 0. 90 C Amer (100 e-0.15-95 e-0.15) + = 4.30. If S 0 > K, then one could exercise the American call immediately, getting a payoff of S 0 - K. C Amer (S 0, K, T) (S 0 - K) + = (90-95) + = 0, which in this case has no effect. The width of the interval is: 90-4.30 = 85.70. Comment: PV[F 0,T ] = S0 - PV[Div]. 100 e-0.15 = 90 - PV[Div]. PV[Div] = 90-100 e-0.15 = 3.93.
Page 206 The same three properties hold for American Options as for European Options. For K 1 < K 2, C(K 1 ) C(K 2 ). For K 1 < K 2, C(K 1 ) - C(K 2 ) K 2 - K 1. C(K 1 ) - C(K 2 ) K2 - K1 C(K 2) - C(K 3 ) K 3 - K 2. For K 1 < K 2, P(K 1 ) P(K 2 ). For K 1 < K 2, P(K 2 ) - P(K 1 ) K 2 - K 1. P(K 2 ) - P(K 1 ) K 2 - K 1 P(K 3) - P(K 2 ) K 3 - K 2.