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.