Section 4, Put-Call Parity

Transcription

1 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 49 Section 4, ut-call arity 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 rice 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 = ut + 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. V[F 0,T ] = S 0 - V[Div]. Setting equal the values of Adamʼs position and Eveʼs position minus any dividends we have: Call + K e -rt = ut + V[F 0,T ].

2 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 50 ut-call arity formula: Thus we have the following put-call parity formula: 77 C Eur (K, T) = Eur (K, T) + V[F 0,T ] - V[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 + ut - Call = K. Value of Call at Time T - Value of ut at Time T = Value of Stock at Time T - Strike rice. Stocks With Discrete Dividends: If dividends are paid at discrete times, then V[F 0,T ] = S 0 - V[Div]. 78 C Eur (K, T) = Eur (K, T) + S 0 - V[Div] - K e -rt. 79 Stocks With Continuous Dividends: If dividends are paid continuously, then V[F 0,T ] = S 0 e -δt. C Eur (K, T) = Eur (K, T) + S 0 e -δt - K e -rt See Equation 3.1 in Derivatives Markets by McDonald. 78 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. 79 See equation 9.2 in Derivatives Markets by McDonald. 80 See below equation 9.2 in Derivatives Markets by McDonald. In the case of no dividends, δ = 0.

3 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 51 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 -.03 = $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 -.03 = $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) - 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 rice Equal to Current Stock rice: 81 If the stock pays no dividends, then C Eur (K, T) - Eur (K, T) = S 0 - K e -rt. Let us assume S 0 = K = $100. Then C - = $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. 81 See Example 9.1 in Derivative Markets by McDonald. When S = K, the option is said to be at the money.

4 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 52 Derivation of ut-call arity: 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 - = F 0,T [S T ] - F 0,T [K] = V[F 0,T ] - V[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: 82 C Eur (K, T) = Eur (K, T) + B 0 - V[Coupons] - K e -rt. Summary: 83 As will be discussed in subsequent sections, there are similar relationships for other assets. Asset Stock, No Dividends arity Relationship S 0 = C - + e -rt K Stock, Discrete Dividends S 0 - V[Div] = C - + e -rt K Stock, Continuous Dividends e - δt S 0 = C - + e -rt K Bond B 0 - V[Coupons] = C - + e -rt K. Futures Contract e -rt F 0,T = C - + e -rt K Currency exp[-r f T] x 0 = C - + e -rt K Exchange Options F 0,T (S0 ) = C - + F 0,T (Q0 ) exp[-δ S T] S 0 = C - + exp[-δ Q T] Q 0 82 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. 83 See Table 9.9 at page 305 of Derivatives Markets by McDonald.

5 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 53 roblems: 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, maturing 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 = exercise 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

6 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 54 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 ut Bid ut 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

7 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (2 points) The prices for otherwise similar European puts and calls are: Strike Call premium ut 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 rice Call rice ut rice $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 = exercise 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

8 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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 rice Call rice ut rice $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 ut Bear Spread. A B C D E. 4.75

9 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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 rice Call remium ut remium 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 (5B, 5/99, Q.1) (1 point) You sell a 1-year European call option on Greystokes Inc. with an exercise price of 110 and buy a 1-year European put option with the same exercise 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) 0.079

10 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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 (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 S index currently sells for 1,000. The S 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

11 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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

12 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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

13 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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 ut 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) (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 (45, T) - C(50, T) + S 55e -rt (III) 45e -rt (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

14 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 62 Section 5, Bounds on remiums of European Options Option premiums must stay within certain bounds. Maximum and Minimum rices of Calls: 84 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 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 0,T (S0 ) C Eur (S 0, K, T) See page 292 and 293 of Derivative Markets by McDonald. 85 McDonald does not mention this in Derivative Markets. In the absence of dividends, the prepaid forward price is just S0.

15 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 63 hyllis can buy a forward contract to buy a share of stock at T in the future. At time T in the future hyllis 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 V 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. hyllis 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 hyllisʼs. C(S 0, K, T) + V 0,T [K] V 0,T [F 0,T ]. C(S 0, K, T) V 0,T [F 0,T ] - V 0,T [K]. 86 Therefore, recalling that an option can never have a negative value: S 0 C(S 0, K, T) (V 0,T [F 0,T ] - V 0,T [K]) However, V 0,T [F 0,T ] = S 0 - V[Div]. Therefore, C(S 0, K, T) (S 0 - V[Div] - V 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: V 0,T [F 0,T ] - V 0,T [K] = S 0 e -δt - K e -rt = S 0 e e -.12 =.961 S S 0 C(S 0, K, T) (.961 S ) +. Comment:.961 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 86 From put-call parity: C = + V[F0,T] - V[K]. Also 0. C V[F0,T] - V[K]. 87 See Equation 9.9 in Derivative Markets by McDonald.

16 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 64 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: V 0,T [F 0,T ] - V 0,T [K] = S 0 e -δt - K e -rt = 90 e K e -.12 = 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 rices of uts: 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. 88 K (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: V[K] (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 V 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. 88 Eur(S0, K, T) = E[(K - ST)+] e -Tr E[(K - ST)+] K.

17 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 65 Therefore, (S 0, K, T) + V 0,T [F 0,T ] V 0,T [K]. (S 0, K, T) V 0,T [K] - V 0,T [F 0,T ]. 89 Therefore, recalling that an option can never have a negative value: K (S 0, K, T) (V 0,T [K] - V 0,T [F 0,T ]) However, V 0,T [F 0,T ] = S 0 - V[Div]. Therefore, (S 0, K, T) (V 0,T [K] - S 0 + V[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: V 0,T [K] - V 0,T [F 0,T ] = K e -rt - S 0 e -δt = 80 e S 0 e -.03 = S (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 : 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: V 0,T [K] - V 0,T [F 0,T ] = K e -rt - S 0 e -δt = K e e -.03 =.861 K K (S 0, K, T) (.861 K ) +. Comment:.861 K = 0, for K = 78.9 = S 0 e (r-δ)t.] 89 From put-call parity: = C + V[K] - V[F0,T]. Also C 0. V[K] - V[F0,T]. 90 See Equation 9.10 in McDonald.

18 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 66 For these inputs, here is a graph of the possible put premiums as a function of K: 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 (S 0, K, T). 91 Relationship to ut-call arity: From ut-call arity, we have: C Eur (K, T) = Eur (K, T) + V[F 0,T ] - V[K]. Since, Eur (K, T) 0, C Eur (K, T) V[F 0,T ] - V[K], as discussed previously. Since, C Eur (K, T) 0, Eur (K, T) + V[F 0,T ] - V[K] 0. Therefore, Eur (K, T) V[K] - V[F 0,T ], as discussed previously. 91 McDonald does not mention this in Derivative Markets. See for example, Options, Futures, and Other Derivatives by John C. Hull, not on the syllabus.

19 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 67 An Increasing Strike rice: 92 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. 93 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. 94 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.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: (V[K] - V[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.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 - = Ke -rt - Se -δt. Thus, in this situation with no dividends and K = S 0 e rt : C - = 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. 93 This is analogous to keeping a insurance policy deductible or limit up with inflation. 94 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. 95 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.

20 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 68 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 maturity. [Solution: K T = Ke rt. Take similar two calls, with 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: (V[Forward contract on the stock] - V[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.] ortfolio Insurance for the Long Run: 96 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. 97 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. 96 See page 299 of Derivatives Markets by McDonald. See also page 603, to be discussed subsequently. 97 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.

21 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 69 roblems: 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]

22 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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? A S B S C S D S E S0

23 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (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

24 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 72 Section 6, Options on Currency For example, one may be able today to buy one Euro for 1.25 U.S. Dollars. 1 = $ = $1. Instead one could buy one U.S. Dollar for 0.8 Euros. Let r $ be the interest rate on dollars and r be the interest rate on Euros. Then the forward price in dollars for one Euro in the future is: (current exchange rate)exp[(r $ - r )T]. 98 The forward price in Euros for one dollar in the future is: (current exchange rate)exp[(r - r $ )T]. 99 In general, one can exchange one currency for another currency at an exchange rate. 100 However, exchange rates change over time. Therefore, it is useful for businesses engaged in international trade to buy and sell options related to currencies. Dollar Denominated Options: For example, one could purchase a call option to buy one euro in 2 years for $1.20. The owner of this call would have the option in 2 years to use $1.20 to obtain one Euro. This is an example of a dollar-denominated option, since the strike price and premium (cost) are in dollars. 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. If the price of one euro in two years is more than $1.20, then one would exercise the above option. For example if the exchange rate two years from now is $1.50 per Euro, then one would make $ $1.20 = $0.30 by exercising this call. 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, a 2 year $1.20 strike put on Euros would give its owner the option in 2 years to sell one Euro for $1.20. The owner would exercise this put if 2 years from now the exchange rate is less than $1.20 per Euro. The payoff of the call option is: (x 2 - $1.20) +, where x 2 is the exchange rate in two years. The payoff of the put option is: ($ x 2 ) +, where x 2 is the exchange rate in two years. The payoff on a call on currency has the same mathematical expression as for a call on stocks, with x T substituted for S T : (x T - K) +. Similarly, the payoff on a put on currency is: (K - x T ) See Equation 5.18 in Derivative Markets by McDonald, on the syllabus of an earlier exam. 99 Since the current exchange rates are reciprocals of each other, so are the forward prices. 100 For current exchange rates see for example

25 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 73 ut-call arity for Dollar Denominated Options: 101 The same put-call parity relationship holds as was discussed previously: C(K, T) = (K, T) + V[F 0,T ] - V[K]. If the option is dollar-denominated, then the strike price K is in dollars. We discount it back to the present using the interest rate for dollars r $, usually just written as r in the United States. The present value of one euro at time T, V[F 0,T ], is the amount we would need to invest now in a euro-denominated risk free bond in order to have 1 euro at time t. A euro-denominated risk free bond earns interest continuous at a rate of r. 102 If we spend x 0 exp[-tr ] dollars to buy exp[-tr ] euros at time 0, and invest in a euro-denominated risk free bond, we will have 1 euro at time T. Therefore, V[F 0,T ] = x 0 exp[-tr ]. Therefore, for dollar-denominated options on euros: C $ (x 0, K, T) = $ (x 0, K, T) + x 0 exp[-tr ] - K e -Tr. 103 x 0 is the exchange rate for Euros in terms of dollars, $1.25 = 1 Euro, and K is in dollars. This is analogous to what we had for options on stock: C(K, T) = (K, T) + S 0 e -Tδ - K e -Tr. x 0 S 0. Dollars act as money: r $ r. Euros act as the asset: r δ. Exercise: 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. [Solution: $ ($1.25)e -(2)(.04) - ($1.20)e -(2)(.06) = $ Comment: Note how every term in the equation is in dollars. This is one way to check your work.] 101 See page 286 of Derivative Markets by McDonald. 102 We assume that r$ will usually differ from r. Similarly, each currency will have its own associated interest rate. 103 See equation 9.4 of Derivative Markets by McDonald. This is similar to the parity equation for stocks paying continuous dividends, with r taking the place of δ.

26 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 74 Foreign Denominated Options: 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. For example, a 2 year 0.70 strike call on dollars would give its owner the option in 2 years to buy one dollar for The owner would exercise this call if 2 years from now the exchange rate is more than 0.70 per dollar. 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 example, a 2 year 0.60 strike put on dollars would give its owner the option in 2 years to sell one dollar for The owner would exercise this put if 2 years from now the exchange rate is less than 0.60 per dollar. ut-call arity for Foreign Denominated Options: For Euro-denominated options on dollars: C (x 0, K, T) = (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. This is analogous to what we had for options on stock: C(K, T) = (K, T) + S 0 e -Tδ - K e -Tr. x 0 S 0. Euros act as money: r r. Dollars act as the asset: r $ δ. Exercise: x 0 = 0.8 /$. T = 2 years. K = /$. The dollar-denominated interest rate is 6% and the euro-denominated interest rate is 4%. The option premium for a euro-denominated put on one dollar is 0.08 euro. Determine the premium for a euro-denominated call on one dollar. [Solution: 0.08 euros + (.8 euros)e -(2)(.06) - (.833 euros)e -(2)(.04) = euro. Comment: Since the options are euro-denominated, the roles of the two interest rates are reversed. Here euros act as the currency, while dollars act as the asset. Therefore, the interest rate on dollars is analogous to the dividend rate on a stock. Note how every term in the equation is in euros. This is one way to check your work.]

27 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 75 Different oints of View: Assume that currently 1 = $ = $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. Exercise: In one year, 1 = $1.4. What is the payoff on Samʼs calls? [Solution: $1.4 - $1.30 = $0.10. For 1000 calls: (1000)($0.10) = $100.] From Samʼs point of view, the future value of each of his calls in dollars is: (x ) +. Johann is in Germany. Johann has 1300 Euro denominated puts with strike price: 1/1.30 =.7692, giving him the option one year from now to sell 1300 dollars for: (1300)(.7692) = 1000 Euros. Exercise: In one year, 1 = $1.4. What is the payoff on Johannʼs puts? [Solution: $1 = 1/1.4 = euros = euros. (1300)(0.0549) = 71.4 euros. Comment: At this future exchange rate, 71.4 euros = (71.4 )($1.4/ ) = $100.] From Johannʼs point of view, the payoff on each of his puts in euros is: ( /x 1 ) +, where x 1 is the exchange rate to buy euros with dollars, and therefore 1/x 1 is the exchange rate to buy dollars with euros. The payoff in dollars on each of Johannʼs puts is: x 1 ( /x 1 ) + = (.7692x 1-1) + = (x ) + /1.30. Samʼs calls have value if x 1 > Johannʼs puts have value if.7682 > 1/x 1 x 1 > Their options have value under the same conditions. If x 1 = 1.4, then as calculated above, Samʼs calls are worth $100 and Johannʼs puts are worth 71.4 euros or $100 at this future exchange rate. Johannʼs puts are worth the same as Samʼs calls. Both Samʼs and Johannʼs positions give them the option one year from now to use 1300 dollars to obtain 1000 Euros. Therefore, their positions must be worth the same amount.

28 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 76 Exercise: If each of Johannʼs euro-denominated puts for one dollar with a strike price of euros has a price of euros, determine the price of Samʼs dollar denominated one year call for one euro, with a strike price of $1.30. [Solution: Johannʼs position costs (1300)(0.0523) = 68 euros. Thus each of Samʼs calls should cost: 68 euros/1000 = euros. At the current exchange rate, this is: (0.068 euros)($1.25/ euro) = $ Comment: Samʼs position is worth: (1000)($0.085) = $85.] In general, in order to replicate the dollar denominated call, one needs to buy K euro denominated puts. Thus the value of the call is K times the value of the put. In order to put the price of the call in dollars rather than euros, one must multiply by the current exchange rate x 0. C $ (x 0, K, T) = x 0 K f (1/x 0, 1/K, T). 104 Exercise: Use the above formula to solve the previous exercise. [Solution: C $ (1.25, 1.3, T) = (1.25) (1.3) (1/1.25, 1/1.3, T) = (1.25)(1.3)(0.0523) = $0.085.] Calls from Samʼs point of view was equivalent to puts from Johannʼs point of view. A call from one point of view is a put from an opposite point of view. Similarly, $ (x 0, K, T) = x 0 K C f (1/x 0, 1/K, T). For a dollar denominated call, we have the option to use dollars to receive another currency. For a dollar denominated put, we have the option to receive dollars by using another currency. For a Euro denominated call, we have the option to use Euros to receive another currency. For a Euro denominated put, we have the option to receive Euros by using another currency. 104 See equation 9.7 in Derivative Markets by McDonald.

29 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 77 Derivation of the Relationship of Dollar and Foreign Denominated Currency Options: Let x(t) be the exchange rate at time t; x(t) is the value in dollars of one of the foreign currency at time t. 105 The payoff at expiration of a K-strike dollar denominated call on a foreign currency is in dollars: (x(t) - K) +. Now at time T, this payoff would be equal to in the foreign currency: (x(t) - K) + /x(t) = (1 - K/x(T)) + = K(1/K - 1/x(T)) +. The payoff at expiration of a 1/K-strike foreign denominated put on dollars is in that foreign currency: (1/K - 1/x(T)) +. Therefore, both in the foreign currency at expiration: K(payoff of 1/K-strike foreign denominated put on dollars) = (payoff of K-strike dollar denominated call on foreign currency). Therefore, K(premium of 1/K-strike foreign denominated put on dollars) = (premium in foreign currency of K-strike dollar denominated call on foreign currency) = (premium in dollars of K-strike dollar denominated call on foreign currency)/x In other words, C $ (x 0, K, T) / x 0 = K f (1/x 0, 1/K, T). C $ (x 0, K, T) = x 0 K f (1/x 0, 1/K, T). 105 For example, the current exchange rate could be $0.8 per Euro. 106 For example, if one could pay 0.10 Euros to purchase the dollar denominated call on Euros, and the current exchange rate were $0.8 per Euro, then one could also pay 0.10/0.8 = $0.125 to purchase this call.

30 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 78 Examples of Using Currency Options for Hedging: Aidan Math is an American consulting actuary. On January 1, Aidan takes a three month assignment consulting for the Seguros opular, a Mexican Insurance Company. Aidan will be paid a lump sum for his work on March 31 at the completion of his assignment. Aidan will be doing all the work at his office in the United States. The exchange rate on January 1 is 10 pesos per U.S. dollar, or $0.10 U.S. per Mexican eso. If Seguros opular agrees to pay Aidan 600,000 esos on March 31, then Aidan faces a risk of the exchange rate changing. On January 1, 600,000 esos is equivalent to $60,000. If on March 31 the exchange rate were 12 pesos per U.S. dollar, then Aidan would receive only the equivalent of 600,000/12 = $50,000. Aidan would have the choice of just accepting this risk, or buying an option to hedge this risk. Aidan could buy 60,000 peso denominated 3 month 10 peso European call options to buy dollars. Then on March 31, if the exchange rate is more than 10 peso per dollar, he can exercise his options and use the 600,000 esos he is paid by Seguros opular to buy $60,000. If instead on March 31 the exchange rate were 8 pesos per U.S. dollar, then Aidan would receive the equivalent of 600,000/8 = $75,000. In this case, Aidan would benefit from the movement in the exchange rate. If on March 31 the exchange rate were less than 10 pesos per U.S. dollar, then the 600,000 pesos Aidan receives would be the equivalent of more than $60,000. Instead of accepting this possibility of being lucky, Aidan could have sold 600,000 peso denominated 3 month 10 peso European put options to buy dollars. 109 In that case, if the exchange rate were less than 10 pesos per U.S. dollar, then the person who bought these puts from Aidan would exercise them, and buy the 600,000 esos that Aidan receives from Seguros opular for $60, Aidan would no longer benefit if the exchange rate were less than 10 pesos per U.S. dollar, but in any case he would have the money from selling the puts. This money would help to offset the cost of the calls he bought. 107 Equivalently, Aidan could buy 600,000 dollar denominated 3 month $0.1 European put options to sell pesos. Then on March 31, if the exchange rate is more than 10 peso per dollar, in other words less than $0.1 per peso, he can exercise his option and sell the 600,000 esos he is paid by Seguros opular for $60, As will be discussed subsequently, such a call could be priced using the Black-Scholes formula. If σ = 12%, the interest rate in pesos is 6%, and the interest rate in dollars is 5%, then the premium of each such call would be about 0.25 pesos. Thus 60,000 such calls would cost Aidan about 15,000 pesos on January 1, the equivalent of $ Each such put would cost about pesos, and Aidan would receive about $1460. Thus if Aidan both buys the calls and sells the puts, then for a net cost of about $ $1460 = $40, he would be unaffected by changes in currency exchange rates. In this case, the calls cost more than the puts, since I have assumed the interest rate in pesos is greater than the interest rate in dollars. 110 For example, if the exchange rate on March 31 were 8 pesos per U.S. dollar, obtaining 600,000 pesos would normally cost 600,000/8 = $75,000. Using the options bought from Aidan, this person could instead obtain 600,000 pesos for only $60,000.

31 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 79 Since Aidan will be doing all the work at his office in the United States, he would prefer to be paid in dollars. If Seguros opular agrees to pay Aidan $60,000 on March 31, then Seguros opular rather than Aidan would face a risk of the exchange rate changing. In this case, if on March 31 the exchange rate were 12 pesos per U.S. dollar, then Seguros opular would have to use: (12)(60000) = 720,000 pesos in order to pay $60,000 to Aidan. Similar to the previous situation, Seguros opular would have the choice of just accepting this risk, or buying an option to hedge this risk. Seguros opular could buy 60,000 peso denominated 3 month 10 peso European call options to buy dollars. 111 Then on March 31, if the exchange rate is more than 10 peso per dollar, Seguros opular can exercise its options and use the 600,000 esos to buy $60,000, with which to pay Aidan. If on March 31 the exchange rate is less than 10 peso per dollar, then Seguros opular would benefit from the exchange rate movement; they would need to use less than 600,000 pesos in order to pay Aidan $60,000. Rosetta Stone is a Mexican pension actuary. On January 1, Rosetta takes a six month assignment consulting for the Grate American Cheese Company, which is based in the United States. Rosetta will be paid a lump sum for her work on June 30 at the completion of her assignment. Rosetta will be doing all the work at her office in Mexico. The exchange rate on January 1 is 10 pesos per U.S. dollar, or $0.10 U.S. per Mexican eso. Exercise: Grate American Cheese Company has agreed to pay Rosetta $100,000. Briefly discuss the currency exchange risk faced by Rosetta and how she can hedge it using options. [Solution: If the exchange rate is less than 10 pesos per dollar, then the $100,000 Rosetta will be paid is worth less than the 1 million pesos she expected. Rosetta could buy 100,000 peso denominated 6 month 10 peso European put options to sell dollars. Alternately, Rosetta could buy 1 million dollar denominated 6 month $0.1 European call options to buy pesos.] Exercise: Grate American Cheese Company has agreed to pay Rosetta 1 million pesos. Briefly discuss the currency exchange risk faced by the company and how to hedge it using options. [Solution: If the exchange rate is less than 10 pesos per dollar, then the 1 million pesos Rosetta will be paid will cost the company more than the $100,000 it expected. The company could buy 1 million dollar denominated 6 month $0.1 peso European call options to buy pesos. Alternately, it could buy 100,000 peso denominated 6 month 10 peso European put options to sell dollars. Comment: In this and the previous exercise, Rosetta and the company each face the same risk, and can hedge that risk the same way. If the company had agreed to pay Rosetta $50,000 and 500,000 pesos, then Rosetta and the company would share the currency exchange risk.] 111 Equivalently, Seguros opular could buy 600,000 dollar denominated 3 month $0.1 European put options to sell pesos.

32 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 80 roblems: 6.1 (2 points) The spot exchange rate of dollars per euro is 1.1. Dollar and euro interest rates are 5.0% and 6.0%, respectively. The price of a $0.93 strike 18-month call option is $0.28. What is the price of a similar put? A. Less than $0.05 B. At least $0.05, but less than $0.10 C. At least $0.10, but less than $0.15 D. At least $0.15, but less than $0.20 E. At least $ (2 points) Currently one can buy one dollar for 120 yen. Dollar and yen interest rates are 4.5% and 3.0%, respectively. The price of a 100 yen strike 2 year put option is 13 yen. What is the price in yen of a similar call? A. Less than 15 B. At least 15, but less than 20 C. At least 20, but less than 25 D. At least 25, but less than 30 E. At least (2 points) Currently one can buy one Swiss Franc for 0.40 British ounds ( 0.4). The interest rate for Swiss Francs is 5%. The interest rate for British ounds is 3%. The price of a 0.5 strike 3 year call option is Determine the price in Swiss Francs of a 2 Swiss Franc strike 3 year put option. A B C D E (2 points) The price of a $0.02 strike 1 year call option on an Indian Rupee is $ The price of a $0.02 strike 1 year put option on an Indian Rupee is $ Dollar and rupee interest rates are 4.0% and 7.0%, respectively. How many dollars does it currently take to purchase one rupee? A B C D E (2 points) The current exchange rate is $1.90 per British ound. A $2 strike 6 month call on the British ound currently costs $ What is the price in pounds of a 0.5 strike 6 month put on United States dollars? A B C D E

33 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (2 points) Currently one can buy 1 Brazilian Real for $0.47. Dollar and Real interest rates are 4.0% and 7.0%, respectively. The price of a $0.50 strike 6-month call option to buy 1 Brazilian Real is $0.09. What is the price of a similar put? A. Less than $0.05 B. At least $0.05, but less than $0.10 C. At least $0.10, but less than $0.15 D. At least $0.15, but less than $0.20 E. At least $ (2 points) The current exchange rate is $0.15 per Chinese Yuan Renminbi. An 8 yuan strike 2 year call on the U.S. Dollar currently costs yuan. What is the price in dollars of a $0.125 strike 2 year put on yuan? A. Less than $0.006 B. At least $0.006, but less than $0.008 C. At least $0.008, but less than $0.010 D. At least $0.010, but less than $0.012 E. At least $ (2 points) Currently one can buy 1 U.S. Dollar for 11 Mexican esos. Dollar and eso interest rates are 4.0% and 6.0%, respectively. The price of a 12 eso strike 1-year put option is 1.36 esos. What is the price in esos of a similar call? A. Less than 0.60 B. At least 0.60, but less than 0.70 C. At least 0.70, but less than 0.80 D. At least 0.80, but less than 0.90 E. At least (3 points) The spot exchange rate of dollars per South Korean Won is Dollar and Won interest rates are 4.0% and 7.0%, respectively. The price for one million $ strike 3 year call options on Won is $ What is the price in Won of a 1000 Won strike 3 year call on dollars? A. 135 B. 140 C. 145 D. 150 E. 155

34 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (3 points) The spot exchange rate of yen per Euro is 125. You are given the following premiums for five year Euro denominated calls on yen. Strike remium Determine the premium for a five year 100 yen strike put on Euros. A. 5 B. 6 C. 9 D. 13 E (3 points) You are given: (i) The current exchange rate is 1.16 /. (ii) A two-year European call option on Euros with a strike price of 0.80 sells for (iii) The continuously compounded risk-free interest rate on British ounds is 5%. (iv) The continuously compounded risk-free interest rate on Euros is 3%. Calculate the price of 1000 two-year European call options on British ounds with a strike price of (A) 44 (B) 46 (C) 48 (D) 50 (E) (5B, 5/99, Q.14) (1 point) An American manufacturer has contracted to sell a large order of widgets for one million Canadian dollars, with payment due on delivery in six months. How could the American company reduce foreign exchange risk? 1. Borrow Canadian currency against its receivables, convert to U.S. dollars at the spot rate, and invest proceeds in the U.S. 2. Buy an option to sell Canadian dollars in six months at a specific price. 3. Buy Canadian currency forward. A. 1 B. 1, 2 C. 1, 3 D. 2, 3 E. 1, 2, (CAS3, 11/07, Q.15) (2.5 points) A nine-month dollar-denominated call option on euros with a strike price of $1.30 is valued at $0.06. A nine-month dollar-denominated put option on euros with the same strike price is valued at The current exchange rate is $1.2/euro and the continuously compounded risk-free rate on dollars is 7%. What is the continuously compounded risk-free rate on euros? A. Less than 7.5% B. At least 7.5%, but less than 8.5% C. At least 8.5%, but less than 9.5% D. At least 9.5%, but less than 10.5% E. At least 10.5%

35 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age (MFE/3F, 5/09, Q.9) (2.5 points) You are given: (i) The current exchange rate is 0.011$/. (ii) A four-year dollar-denominated European put option on yen with a strike price of $0.008 sells for $ (iii) The continuously compounded risk-free interest rate on dollars is 3%. (iv) The continuously compounded risk-free interest rate on yen is 1.5%. Calculate the price of a four-year yen-denominated European put option on dollars with a strike price of 125. (A) 35 (B) 37 (C) 39 (D) 41 (E) 43

36 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 84 Section 7, Exchange Options 112 One can buy an option to exchange one asset for another. For example, Steve is given the option to exchange one share of a competitorʼs Stock B for one share of the stock of his employer Stock A. This is a call option with underlying asset Stock A and strike asset Stock B. 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, in order to obtain Stock A. Assuming a European call, then at time T, Steve would make the exchange if a share of Stock A is worth more than a share of Stock B. The future value of the call is (S T - Q T ) +, where S is price of the underlying asset, Stock A, and Q is the price of the strike asset, Stock B. Whether one will exercise the call depends on the relative value of the underlying asset and the strike asset, both of which are random variables. If at expiration the underlying asset is worth more than the strike asset, then it would be worthwhile to exercise the call. For example, if the call expires in 2 years, among the many possible situations: rice of Stock A rice of Stock B Exercise (Underlying Asset) (Strike Asset) Exchange 2 Years from now 2 Years from now Call? Yes No Yes No 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. 112 See Section 9.2 of Derivative Markets by McDonald. Also called an outperformance option.

37 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 85 ut-call arity: Buy one European call, sell a similar put, sell a prepaid forward contract on Stock A, and buy a prepaid forward contract on Stock B. 113 At Time T, you will get a share of Stock B due to the forward contract you bought. If S T Q T, you will exercise your call and exchange your Share of Stock B for a Share of Stock A. Then you will provide this share of Stock A to the person who bought a forward contract from you. If instead S T < Q T, then the person who bought your put, will exchange a share of Stock A for your share of Stock B. You will then provide this share of Stock A to the person who bought a forward contract from you. In either case, at time T you end up with nothing after all of the transactions. Therefore, the correct price for this position is zero. Let F 0,T (S0 ) be the prepaid forward price of a share of Stock A. Let F 0,T (Q0 ) be the prepaid forward price of a share of Stock B. We have shown that: 0 = C Eur (S 0, Q 0, T) - Eur (S 0, Q 0, T) - F 0,T (S0 ) + F 0,T (Q0 ). C Eur (S 0, Q 0, T) = Eur (S 0, Q 0, T) + F 0,T (S0 ) - F 0,T (Q ). If the current price of Stock A is $100 and the current price of Stock B is $120, and neither one pays dividends, then F 0,T (S0 ) = 100 and F 0,T (Q0 ) = 120. Therefore, C Eur (S 0, Q 0, T) = Eur (S 0, Q 0, T) = Eur (S 0, Q 0, T) If instead Stock A pays dividends at a continuous rate δ S and Stock B pays dividends at a continuous rate δ Q, then F 0,T (S0 ) = S 0 exp[-tδ S ] and F 0,T (Q0 ) = Q 0 exp[-tδ Q ]. 116 Therefore: C Eur (S 0, Q 0, T) = Eur (S 0, Q 0, T) + S 0 exp[-tδ S ] - Q 0 exp[-tδ Q ]. 113 Under a prepaid forward contract, you pay now but receive the asset at a fixed point of time in the future. 114 See Equation 9.6 in Derivative Markets by McDonald. 115 This put-call parity follows from the fact that: (S(T) - Q(T))+ + Q(T) = (Q(T) - S(T))+ + S(T). See remark (iii) on MFE Sample Exam Q While you pay for the prepaid forward contract now, you do not get the dividends that are paid on the stock between now and time T, since you do not own the stock until time T.

38 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 86 Exercise: 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? [Solution: C Eur (S 0, Q 0, 3) - Eur (S 0, Q 0, 3) = (100)exp[-(3)(1%)] - (120)exp[-(3)(2%)] = Comment: In this case, the put is worth more than the similar call. Exchange Call: can use B to get A. Exchange ut: 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.] C Eur (S 0, Q 0, T) = Eur (S 0, Q 0, T) + F 0,T (S0 ) - F 0,T (Q0 ). Therefore, the premium of the put is equal to the premium of the similar call, if and only if the two stocks have the same prepaid forward price. Note that this form of put-call parity includes the situations previously discussed as special cases. For example, if the strike asset is money, specifically K dollars, then the dividend on money is interest at rate r, and C Eur (S 0, K, T) = Eur (S 0, K, T) + S 0 exp[-tδ S ] - K exp[-tr]. arity is the observation that buying a European call and selling a European put with the same strike price and time to expiration is equivalent ot making a leveraged investment in the underlying asset, less the value of cash payments to the underlying asset over the life of the option See page 305 of Derivatives Markets by McDonald.

39 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 87 oints of View: As discussed previously, a call from one point of view is a put from another point of view. Neither Stock A nor Stock B pay dividends. Adam has a 2 year European call with underlying asset Stock A and strike asset Stock B. Adam has the option 2 years from now to buy a share of Stock A in exchange for a share of Stock B. He will do so if two years from now Stock A is worth more than Stock B. Eve has a 2 year European put with underlying asset Stock B and strike asset Stock A. Eve has the option 2 years from now to sell a share of Stock B in exchange for a share of Stock A. She will do so if two years from now Stock A is worth more than Stock B. Two years from now they will each own whichever of the two stocks is worth more. Adam and Eve have options with the same outcome and therefore the same value. In the absence of dividends, C(S 0, Q 0, T) = (Q 0, S 0, T). Call to buy Stock A for price Stock B: A is the underlying asset and B is the strike asset. ut to sell Stock B for price Stock A: B is the underlying asset and A is the strike asset. Despite the different language, both give you the option to use Stock B to obtain Stock A.

40 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 88 roblems: 7.1 (2 points) Consider the case of a European exchange option in which the underlying stock is Widget Co. with a current price of $65 per share. The strike asset is Gadget Inc., with a per share price of $62. Neither stock will pay dividends within the next 3 months. Interest rates are 5% and the 3 month call option is trading for $8. What is the price of the similar put? A. 3 B. 5 C. 7 D. 9 E (2 points) 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? A. Less than $20 B. At least $20, but less than $25 C. At least $25, but less than $30 D. At least $30, but less than $35 E. At least $ (2 points) A 3 year European exchange option has underlying asset one share of each Computer Company with a current price of $200 per share. The strike asset is the Silicon Valley Stock Index, with a current price of $210. The price of this call option is $13. each Computer Company pays dividends at a continuous rate of 3% per year. Silicon Valley Stock Index pays dividends at a continuous rate of 1% per year. What is the price of the similar put? A. Less than $20 B. At least $20, but less than $25 C. At least $25, but less than $30 D. At least $30, but less than $35 E. At least $ (2 points) Consider four-year European exchange options involving the stocks of Global Dynamics and Deon International. The price of an option to exchange a share of Global Dynamics for a share of Deon International is the same as the price of an option to exchange a share of Deon International for a share of Global Dynamics. Global Dynamics has a current price of 92 and pays dividends at a continuous rate of 2% per year. Deon International pays dividends at a continuous rate of 0.9% per year. What is the current price of a share of Deon International? A. 85 B. 86 C. 87 D. 88 E. 89

41 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 89 Section 8, Futures Contracts 118 As discussed previously, a forward contract is an agreement that sets the terms today, including price and quantity, at which one will buy or sell an asset or commodity at a specific time in the future. A futures contract is similar, except that the buyer and seller post margin, the contract is marked to market, and the contract is typically traded on an exchange. 119 An Example of a Futures Contract: Frank is a farmer. He 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. 120 Frank has sold a futures contract. 121 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. 122 Frank has guarded against the risk of the price of wheat falling between April and July. Note that Frank has not guarded against for example his crop being destroyed by hail prior to harvest. Momʼs Bakery has guarded against the risk of the price of wheat rising between April and July. There is no investment required to enter a futures contract; however, Frank and Momʼs Bakery must each post margin with the broker who arranged the contract. The total value is $40,000, so for example, 10% margin would require that they each post a $4000 margin with the broker. Let us assume that on April 16, the futures price on July 15 delivery of wheat is $3.99. Then Frank has made money on his contract, since he would in essence get $4.00 per bushel for his wheat, instead of the $3.99 he would expect to get in the absence of his futures contract. The contract is marked to market daily. 123 Thus Frankʼs margin account will be increased by: ($.01)(10,000) = $100, while Momʼs margin account will be decreased by $100. Each of them earns one day of interest. Assuming r = 5%, they earn $4000(e.05/365-1) = $ See Section 5.4 of Derivative Markets by McDonald, on the syllabus of a previous exam. 119 One can think of mark to market as follows, as the for ward price moves up or down the owner of the futures contract either makes or loses money right away. In contrast, for a forward contract the profit or loss is realized at expiration. 120 Futures contracts can be on other assets like gold, stock indices such as the S& 500, currency such as yen, etc. 121 A contract would be for 5000 bushels, so Frank has actually sold two contracts. 122 Some futures contracts are settled by delivery, but most are closed out prior to the delivery date. When you enter into a futures contract you are obligated to make payments if the forward price moves in the wrong direction for you. Thus the effect of the movement in the forward price is captured monetarily on an ongoing basis. 123 Therefore, Frank will take his profit now, and the new futures price is $3.99.

42 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 90 Thus Frankʼs margin account is now: $ $100 + $0.55 = $ Momʼs margin account is now: $ $100 + $0.55 = $ Exercise: On April 17 the futures price on July 15 delivery of wheat is $4.02. What happens to Frankʼs and Momʼs margin accounts? [Solution: The one day move in price is an increase of $.03. Frankʼs margin account has ($.03)(10,000) = $300 subtracted. He earns one day of interest of: $ (e.05/365-1) = $0.56. Frankʼs margin account is now: $ $300 + $0.56 = $ Similarly, Momʼs margin account is now: $ $300 + $0.54 = $ ] Options on Futures Contracts: One can buy or sell calls and puts on futures contracts. If a call is exercised, then the owner of the call acquires a long position in a futures contract with futures price equal to the strike price of the call. The owner of the call if and when he exercises it, will enter into an agreement to accept delivery of the commodity such as corn. If a call is exercised, then the writer (seller) of the call acquires a short position in a futures contract with futures price equal to the strike price of the call. The writer of the call if and when it is exercised, will enter into an agreement to deliver the commodity such as gold. If a put is exercised, then the owner of the put acquires a short position in a futures contract with futures price equal to the strike price of the put. The owner of the put if and when he exercises it, will enter into an agreement to deliver the commodity such as oil. If a put is exercised, then the writer (seller) of the put acquires a long position in a futures contract with futures price equal to the strike price of the put. The writer of the put if and when it is exercised, will enter into an agreement to accept delivery of the commodity such as cattle. An Example of Using Options on Futures: We have seen how Frank can use a futures contract to guard against the risk of the price of wheat falling between April and July. Frank could instead buy an option on such a futures contract. 124 For example, on March 15 Frank could purchase a 3 month put to sell a futures contract on 10,000 bushels of wheat for July delivery. This put would give Frank the option to sell in June such a futures contract at a given strike price By buying an option on a futures contract Frank would get price protection without limiting his profit potential. 125 There are two important dates: when the option expires, May, and the delivery date in the futures contract, July. One could instead of a European option have an American option, to be discussed subsequently. In that case, one could exercise the option at any date up to expiration. Options on futures contracts are often American, with expiration date one month before the delivery date of the commodity.

43 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 91 Assume for example the strike price were $4.00 per bushel. Then in June, Frank would have the option to exercise his put and enter into an agreement to deliver wheat in July at $4.00 per bushel. If the futures price in June were more than $4.00 per bushel, then Frank would not exercise his put. If instead the futures price in June were less than $4.00 per bushel, then Frank would exercise his put. For example, if the futures price in June were $3.80 per bushel, then Frank would exercise his put. 126 He would now have locked in a price of $4.00 per bushel for his wheat, which is better than the $3.80 he could get by entering into a futures contract in June. 127 The payoff to Frank from the put would be: $ $3.80 = $0.20 per bushel. The payoff on his put is: (Strike rice - Futures rice) +. This is the same payoff formula as for a put on stock, with the futures price taking the place of the stock price. Thus even though there is no money required to invest in a futures contract, one can apply the same mathematics to options on futures contracts as to options on stocks, with appropriate modifications. 128 Such a put would protect Frank against the price of wheat declining between March and June. If such a price decline occurs, Frankʼs wheat is worth less, but his put is worth more. Similarly, on March 15, Momʼs Bakery could purchase a 3 month $4 per bushel strike call to buy a futures contract on 10,000 bushels of wheat for July delivery. This call would give Momʼs Bakery the option to buy in June such a futures contract. In June, Momʼs would have the option to exercise its call and enter into an agreement to accept delivery of wheat in July at $4.00 per bushel. If the futures price in June were less than $4.00 per bushel, then Momʼs Bakery would not exercise this call. If the futures price in June were more than $4.00 per bushel, then Momʼs Bakery would exercise this call. If for example the futures price in June were $4.10 per bushel, then Momʼs would exercise its call and now have entered into a futures contract to receive wheat in July for $4.00 a bushel. Momʼs would now have locked in a price of $4.00 per bushel for buying wheat, which is better than the $4.10 it could get via a futures contract. The payoff to Momʼs from the call would be: = $0.10 per bushel. The payoff on this call is: (Futures rice - Strike rice) +. This is the same payoff formula as for a call on stock, with the futures price taking the place of the stock price. Such a call would protect Momʼs Bakery against the price of wheat increasing between March and June. If such a price increase occurs, Momʼs Bakery would have to pay more for wheat, but the call is worth more. 126 Alternately, Frank could make money by selling his put which would have increased in value. 127 In June, Frank could enter into a futures contract for July delivery of wheat with a $3.80 per bushel price. I Instead using his put option on a futures contract, Frank can enter into a futures contract in June for July delivery of wheat with a $4.00 per bushel price. 128 A form of put-call parity for options on future contracts will be discussed subsequently. In subsequent sections, will be discussed how to price options on futures contracts, using either Binomial Trees or the Black-Scholes formula, in a manner parallel to pricing options on stocks.

44 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 92 Advantages of Using Options on Futures rather than Options on the Underlying Asset: 129 Options on futures contracts are preferable to options on the underlying asset when it is cheaper and/or more convenient to deliver/receive the futures contract on the asset rather than the asset itself. It is easier to deliver or receive a futures contract on cattle than it is to deliver or receive the cattle. The futures contract will usually be closed out prior to delivery, and therefore, options on futures contracts are usually settled in cash. In many situations options on futures contracts tend to have lower transaction costs than options on the underlying asset. Trading on futures and options on futures are usually conducted side by side on the same exchange, which makes it easier to hedge, etc., and makes the markets more efficient. ut-call arity for Options on Futures Contracts: V[F] = F 0,T e -rt. Therefore, the put-call parity relationship for options on futures contracts is: C = + F 0,T e -rt - Ke -rt. The put-call parity relationship for options on futures contracts can be obtained from that for options on stocks by: S 0 F 0,T, and δ r. Exercise: 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, to the person who wrote the put, a futures contract on 10,000 bushels of wheat for July 15 delivery at $4 per bushel.) Currently, 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? [Solution: C = + F 0,T e -rt - Ke -rt = $815 + (10,000)($4.20) e -(.05)(1/4) - (10,000)($4.00)e -(.05)(1/4) = $2790.] 129 See Options, Futures, and Other Derivatives by Hull, not on the syllabus.

45 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 93 roblems: 8.1 (2 points) A futures contract provides for the delivery of 1000 barrels of Light Sweet Crude Oil in September. In January, a 4 month $100 per barrel strike call on this futures contract has a premium of $3400. In January, the futures price for delivery of oil in September is $95 per barrel. If r = 6%, determine the premium for the corresponding put. A. $7900 B. $8000 C. $8100 D. $8200 E. $ (2 points) The premium for a 6 month 1500 strike put on a futures contract is 53. The current futures price is If r = 4%, determine the premium for the corresponding call. A. 130 B. 140 C. 150 D. 160 E (3 points) On January 1, Saul enters into a futures contract on 5000 bushels of soybeans at $12.40 per bushel for delivery in August. Saul has a long position; he has agreed to accept delivery of soybeans in August. Saul post a margin of 10%. The interest rate is 6%. The futures prices for August delivery are on the following dates: January 1 $12.40 January 2 $12.60 January 3 $12.55 January 4 $12.30 January 5 $12.40 Determine the amount in Saulʼs margin account on each of these days. 8.4 (1 point) The premium for an at-the money call on a futures contract is $5. What is the premium for an at-money put on the same futures contract with the same time until maturity?

46 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 94 Section 9, Synthetic ositions 130 ut-call parity followed from creating a synthetic position which matched the cashflows of another investment. We made use of the law of one price which states that two positions that generate the exact same cashflows should have the same cost. Creating synthetic positions can be useful to derive results, help ones understanding, and may also have practical applications in investing. Synthetic Share of Stock: We can create a synthetic share of stock by buying a call, selling a put, lending the present value of the strike price K, and lending the present value of any dividends. Exercise: Show that this position is equivalent to buying the stock now and holding it until time T. [Solution: If we buy and hold the stock, then we will receive the dividend payments and own the stock at time T. In the above position, we will receive the dividend payments as that loan is repaid. Also we will be paid back K at time T. If S T K, then we use our call to buy a share for K. If instead S T < K, then we buy a share of Stock for K from the person to whom we sold the put. In either case, we own a share of stock at time T, as well as having received the dividend payments.] This shows that S 0 = C Eur (K, T) - Eur (K, T) + K e -rt + V[Div], one of the put-call parity relationships discussed previously. One way to remember how to create a given synthetic position is to write the put-call parity relationship with the desired item alone on the righthand side of the equation. If the stocks pays dividends continuously, then S 0 = e δt {C Eur (K, T) - Eur (K, T) + K e -rt }. We can create a synthetic share of stock by: buying e δt calls, selling e δt puts, lending K e -(r-δ)t. 130 See page 285 of Derivative Markets by McDonald.

47 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 95 Synthetic Treasury Bill: K e -rt + V[Div] = S 0 - C Eur (K, T) + Eur (K, T). Therefore, we can create a synthetic T-Bill by buying the stock, selling a call, and buying the put. This is called a conversion. One is lending money. If S T > K, then we sell our share for K to the person who bought the call. If instead S T K, then we sell a share of stock for K using our put. In either case, we have K at time T. Thus for an investment now, we get K at time T. This is equivalent to a risk free Treasury Bill that returns K at time T, plus we get the present value of any dividends on the stock. Alternately, one could also borrow the present value of dividends. This is equivalent to a risk free Treasury Bill that returns K at time T, since: Ke -rt = S 0 - C Eur (K, T) + Eur (K, T) - V[Div]. Alternately, if dividends are paid continuously, then one could instead buy exp[-δt] shares of stock, and reinvest the dividends. Then we would end up with one share of stock at time T. This is equivalent to a risk free Treasury Bill that returns K at time T, since Ke -rt = S 0 e -δt - C Eur (K, T) + Eur (K, T). One can create a synthetic short T-Bill position by shorting the stock, buying a call, and selling the put. This is called a reverse conversion. One is borrowing money. Synthetic Options: We can create a synthetic call by: buying the stock, buying a put, borrowing the present value of the strike price, and borrowing the present value of any dividends. Exercise: Show that this position is equivalent to buying the call. [Solution: We repay one loan as we receive the dividend payments. If S T K, then we sell our share for S T, repay the loan by paying K, and are left with S T - K. If instead S T < K, then we use our put to sell our share for K, repay the loan by paying K, and are left with 0. These are the same payoffs as if we had purchased the call.] This shows that C Eur (K, T) = S 0 + Eur (K, T) - K e -rt - V[Div], one of the put-call parity relationships discussed previously.

48 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 96 We can create a synthetic put by buying a call, shorting the stock, lending the present value of the strike price, and lending the present value of any dividends. Exercise: Show that this position is equivalent to buying the put. [Solution: We borrow a share of stock, sell it for S 0, and will give this person a share of stock when the option expires. We also must pay this person the stock dividends they would have gotten on the stock, when they would have gotten them. We pay the dividend payments from one of the loans. If S T K, then we buy a share for K, using our call and the money from the other loan. After returning the share to the person we borrowed it from, we are left with nothing. If instead S T < K, then we buy a share for S T. After returning the share to person we borrowed it from, we are left with K - S T. These are the same payoffs as if we had purchased the put.] This shows that Eur (K, T) = C Eur (K, T) - S 0 + K e -rt + V[Div], one of the put-call parity relationships discussed previously. In the case of continuous dividends, Eur (K, T) = C Eur (K, T) - e -δt S 0 + K e -rt. Thus we can create a synthetic put by buying a call, shorting e -δt shares of stock, and lending the present value of the strike price. C Eur (K, T) = Eur (K, T) + e -δt S 0 - K e -rt. Thus we can create a synthetic call by buying a put, buying e -δt shares of stock, and borrowing the present value of the strike price. Synthetic Forward Contract: 131 We can create a synthetic forward contract by buying a call and selling a put. If S T K, then we buy a share for K, using our call. If instead S T < K, then we buy a share for K, from the person to whom we sold the put. Thus we have a forward contract to buy the stock at price K. This has present value V 0,T [F 0,T - K]. Therefore, V 0,T [F 0,T - K] = C Eur (K, T) - Eur (K, T), as discussed previously. 131 See page 282 of Derivative Markets by McDonald.

49 HCMSA-F10-3-3B, Financial Economics, 6/30/10, age 97 roblems: 9.1 (2 points) The price of a stock is $135. The stock pays dividends at the continuous rate of 2%. The price of a European call with a strike price of $140 is $17.39 and the price of a European put with a strike price of $140 is $ Both options expire in eight months. Calculate the annual continuously compounded risk-free rate on a synthetic T-Bill created using these options. A. Less than 3% B. At least 3%, but less than 3% C. At least 4%, but less than 4% D. At least 5%, but less than 6% E. At least 6% 9.2 (2 points) Consider options on a non-dividend paying stock. The price of a 3 year European call with a strike price of $100 is $ The price of a 3 year European put with a strike price of $100 is $8.23. r = 6%. Calculate the price of a synthetic share of stock created using these options. A. 100 B. 105 C. 110 D. 115 E (3 points) The stock of Rich Resorts has a current price of 150, and pays no dividends. The stock of Cereal Growers has a current price of 80, and pays no dividends. This coming summer will either be dry or wet. If the summer is dry, then 4 months from now Rich Resorts stock price will be 180, and Cereal Growers stock price will be 70. If the summer is wet, then 4 months from now Rich Resorts stock price will be 130, and Cereal Growers stock price will be 90. What is the continuously compounded risk free rate? A. 3.25% B. 3.50% C. 3.75% D. 4.00% E. 4.25% 9.4 (CAS3, 5/07, Q.13) (2.5 points) The price of a non-dividend paying stock is $85. The price of a European call with a strike price of $80 is $6.70 and the price of a European put with a strike price of $80 is $1.60. Both options expire in three months. Calculate the annual continuously compounded risk-free rate on a synthetic T-Bill created using these options. A. Less than 1% B. At least 1%, but less than 2% C. At least 2%, but less than 3% D. At least 3%, but less than 4% E. At least 4%