金融工程习题课(四)

在线测验

练习

期权市场

  1. An investor buys a European put on a share for $3. The stock price is $42 and the strike price is $40. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor’s profit with the stock price at the maturity of the option.


  2. An investor sells a European call on a share for $4. The stock price is $47 and the strike price is $50. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor’s profit with the stock price at the maturity of the option.


  3. An investor sells a European call option with strike price of KK and maturity TT and buys a put with the same strike price and maturity. Describe the investor’s position.


  4. A company declares a 2-for-1 stock split. Explain how the terms change for a call option with a strike price of $60.


  5. “Employee stock options issued by a company are different from regular exchange-traded call options on the company’s stock because they can affect the capital structure of the company.” Explain this statement.


  6. Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the option depends on the stock price at maturity of the
    option.


  7. Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option.


  8. Consider the following portfolio: a newly entered-into long forward contract on an asset and a long position in a European put option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up. Show that it has the same value as a European call option with the same strike price and maturity as the European put option. Deduce that a European put option has the same value as a European call option with the same strike price and maturity when the strike price for both options is the forward price.


  9. A trader buys a call option with a strike price of $45 and a put option with a strike price of $40. Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagram showing the variation of the trader’s profit with the asset price.


  10. Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date.


  11. Explain why an American option is always worth at least as much as its intrinsic value.


  12. The treasurer of a corporation is trying to choose between options and forward contracts to hedge the corporation’s foreign exchange risk. Discuss the advantages and disadvantages of each.


  13. Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 and maturity in 4 months. Explain how the terms of the option contract change when there is: (a) a 10% stock dividend; (b) a 10% cash dividend; and (c) a 4-for-1 stock split.


  14. “If most of the call options on a stock are in the money, it is likely that the stock price has risen rapidly in the last few months.” Discuss this statement.


  15. What is the effect of an unexpected cash dividend on (a) a call option price and (b) a put option price?


  16. Options on a stock are on a March, June, September, and December cycle. What options trade on (a) March 1, (b) June 30, and (c) August 5?


  17. Explain why the market maker’s bid–ask spread represents a real cost to options investors.


  18. A U.S. investor writes five naked call option contracts. The option price is $3.50, the strike price is $60.00, and the stock price is $57.00. What is the initial margin requirement?


  19. Calculate the intrinsic value and time value from the mid market (average of bid and ask) prices for the September call options in Table 1.2. Do the same for the September put options in Table 1.3. Assume in each case that the current mid market stock price is $316.00.

    Table 1.2 Prices of call options on Apple, May 21, 2020; stock price: bid $316.23, ask $316.50 (Source: CBOE).

    Stike Price ($) June 2020 September 2020 December 2020
    bid ask bid ask bid ask
    290 29.80 30.85 39.35 40.40 46.20 47.60
    300 21.55 22.40 32.50 33.90 40.00 41.15
    310 14.35 15.30 26.35 27.25 34.25 35.65
    320 8.65 9.00 20.45 21.70 28.65 29.75
    330 4.20 5.00 15.85 16.25 23.90 24.75
    340 1.90 2.12 11.35 12.00 19.50 20.30

    Table 1.3 Prices of put options on Apple, May 21, 2020; stock price: bid $316.23, ask $316.50 (Source: CBOE).

    Stike Price ($) June 2020 September 2020 December 2020
    bid ask bid ask bid ask
    290 3.00 3.30 12.70 13.65 20.05 21.30
    300 4.80 5.20 15.85 16.85 23.60 24.90
    310 7.15 7.85 19.75 20.50 28.00 28.95
    320 11.25 12.05 24.05 24.80 32.45 33.35
    330 17.10 17.85 28.75 29.85 37.45 38.40
    340 24.40 25.45 34.45 35.65 42.95 44.05


  20. A trader writes 5 naked put option contracts with each contract being on 100 shares. The option price is $10, the time to maturity is 6 months, and the strike price is $64.
    (a) What is the margin requirement if the stock price is $58?
    (b) How would the answer to (a) change if the rules for index options applied?
    (c) How would the answer to (a) change if the stock price were $70?
    (d) How would the answer to (a) change if the trader is buying instead of selling the options?


  21. “If a company does not do better than its competitors but the stock market goes up, executives do very well from their stock options. This makes no sense.” Discuss this viewpoint. Can you think of alternatives to the usual employee stock option plan that take the viewpoint into account.


  22. On July 20, 2004, Microsoft surprised the market by announcing a $3 dividend. The ex-dividend date was November 17, 2004, and the payment date was December 2, 2004. Its stock price at the time was about $28. It also changed the terms of its employee stock options so that each exercise price was adjusted downward to
    Predividend exercise price×Closing price - $3.00Closing price\text{Predividend exercise price}\times\frac{\text{Closing price - \$3.00}}{\text{Closing price}}
    The number of shares covered by each stock option outstanding was adjusted upward to
    Number of shares predividend×Closing priceClosing price - $3.00\text{Number of shares predividend}\times\frac{\text{Closing price}}{\text{Closing price - \$3.00}}
    “Closing Price” means the official NASDAQ closing price of a share of Microsoft common stock on the last trading day before the ex-dividend date. Evaluate this adjustment.



期权性质

  1. What is a lower bound for the price of a 4-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per annum?


  2. What is a lower bound for the price of a 1-month European put option on a non-dividend-paying stock when the stock price is $12, the strike price is $15, and the risk-free interest rate is 6% per annum?


  3. Give two reasons why the early exercise of an American call option on a non-dividend-paying stock is not optimal. The first reason should involve the time value of money. The second should apply even if interest rates are zero.


  4. “The early exercise of an American put is a trade-off between the time value of money and the insurance value of a put.” Explain this statement.


  5. Why is an American call option on a dividend-paying stock always worth at least as much as its intrinsic value. Is the same true of a European call option? Explain your answer.


  6. The price of a non-dividend-paying stock is $19 and the price of a 3-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a 3-month European put option with a strike price of $20?


  7. Explain why the arguments leading to put–call parity for European options cannot be used to give a similar result for American options.


  8. What is a lower bound for the price of a 6-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% per annum?


  9. What is a lower bound for the price of a 2-month European put option on a non-dividend-paying stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is 5% per annum?


  10. A 4-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in 1 month. The risk-free interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur?


  11. A 1-month European put option on a non-dividend-paying stock is currently selling for $2.50. The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum. What opportunities are there for an arbitrageur?


  12. Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases.


  13. The price of a European call that expires in 6 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 2 months and again in 5 months. Risk-free interest rates (all maturities) are 10%. What is the price of a
    European put option that expires in 6 months and has a strike price of $30?


  14. Explain the arbitrage opportunities in Problem 13 if the European put price is $3.


  15. The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in 3 months. The risk-free interest rate is 8%. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date.


  16. Explain carefully the arbitrage opportunities in Problem 15 if the American put price is greater than the calculated upper bound.


  17. Prove the result in equation (S0KCPS0KerTS_0-K\leq C-P\leq S_0-Ke^{-rT}). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to KK, and (b) a portfolio consisting of an American put option plus one share.)


  18. Prove the result in equation (S0DKCPS0KerTS_0-D-K\leq C-P\leq S_0-Ke^{-rT}). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to D+KD + K, and (b) a portfolio consisting of an American put option plus one share.)


  19. Consider a 5-year call option on a non-dividend-paying stock granted to employees. The option can be exercised at any time after the end of the first year. Unlike a regular exchange-traded call option, the employee stock option cannot be sold. What is the likely impact of this restriction on the early-exercise decision?


  20. What is the impact (if any) of negative interest rates on:
    (a) The put–call parity result for European options
    (b) The result that American call options on non-dividend-paying stocks should never be exercised early
    (c) The result that American put options on non-dividend-paying stocks should sometimes be exercised early.
    Assume that holding cash earning zero interest is not possible.


  21. Calls were traded on exchanges before puts. During the period of time when calls were traded but puts were not traded, how would you create a European put option on a non-dividend-paying stock synthetically.


  22. The prices of European call and put options on a non-dividend-paying stock with an expiration date in 12 months and a strike price of $120 are $20 and $5, respectively. The current stock price is $130. What is the implied risk-free rate?


  23. A European call option and put option on a stock both have a strike price of $20 and an expiration date in 3 months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in 1 month. Identify the arbitrage opportunity open to a trader.


  24. Suppose that c1c_1, c2c_2, and c3c_3 are the prices of European call options with strike prices K1K_1, K2K_2, and K3K_3, respectively, where K3>K2>K1K_3>K_2>K_1 and K3K2=K2K1K_3 - K_2 = K_2 - K_1. All options have the same maturity. Show that c20.5×(c1+c3)c_2\leq0.5\times(c_1 + c_3)
    (Hint: Consider a portfolio that is long one option with strike price K1K_1, long one option with strike price K3K_3, and short two options with strike price K2K_2.)


  25. What is the result corresponding to that in Problem 24 for European put options?



期权交易策略

  1. Call options on a stock are available with strike prices of $15, $1712\$17\frac{1}{2}, and $20, and expiration dates in 3 months. Their prices are 4,4,2, and $12\$\frac{1}{2}, respectively. Explain how
    the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.


  2. A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?


  3. Use put–call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.


  4. Explain how an aggressive bear spread can be created using put options.


  5. Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payoff for both spreads.


  6. Use put–call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.


  7. A call with a strike price of 60costs60 costs6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?


  8. Construct a table showing the payoff from a bull spread when puts with strike prices K1K_1 and K2K_2, with K2>K1K_2> K_1, are used.


  9. An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction. Identify six different strategies the investor can follow and explain the differences among them.


  10. How can a forward contract on a stock with a particular delivery price and delivery date be created from options?


  11. “A box spread comprises four options. Two can be combined to create a long forward position and two to create a short forward position.” Explain this statement.


  12. What is the result if the strike price of the put is higher than the strike price of the call in a strangle?


  13. A foreign currency is currently worth $0.64. A 1-year butterfly spread is set up using European call options with strike prices of $0.60, $0.65, and $0.70. The risk-free interest rates in the United States and the foreign country are 5% and 4% respectively, and the volatility of the exchange rate is 15%. Use the DerivaGem software to calculate the cost of setting up the butterfly spread position. Show that the cost is the same if European put options are used instead of European call options.


  14. A trader creates a bear spread by selling a 6-month put option with a $25 strike price for $2.15 and buying a 6-month put option with a $29 strike price for $4.75. What is the initial investment? What is the total payoff (excluding the initial investment) when the stock price in 6 months is (a) $23, (b) $28, and (c) $33.


  15. A trader sells a strangle by selling a 6-month European call option with a strike price of $50 for $3 and selling a 6-month European put option with a strike price of $40 for $4. For what range of prices of the underlying asset in 6 months does the trader make a profit?


  16. Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy. For what range of stock prices would the butterfly spread lead to a loss?


  17. A diagonal spread is created by buying a call with strike price K2K_2 and exercise date T2T_2 and selling a call with strike price K1K_1 and exercise date T1T_1, where T2>T1T_2>T_1. Draw a diagram showing the profit from the spread at time T1T_1 when (a) K2>K1K_2>K1 and (b) K2<K1K_2<K_1.


  18. Draw a diagram showing the variation of an investor’s profit and loss with the terminal stock price for a portfolio consisting of :
    (a) One share and a short position in one call option
    (b) Two shares and a short position in one call option
    (c) One share and a short position in two call options
    (d) One share and a short position in four call options.
    In each case, assume that the call option has an exercise price equal to the current stock price.


  19. Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30%, and the risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the following positions:
    (a) A bull spread using European call options with strike prices of $25 and $30 and a maturity of 6 months
    (b) A bear spread using European put options with strike prices of $25 and $30 and a maturity of 6 months
    (c) A butterfly spread using European call options with strike prices of $25, $30, and $35 and a maturity of 1 year
    (d) A butterfly spread using European put options with strike prices of $25, $30, and $35 and a maturity of 1 year
    (e) A straddle using options with a strike price of $30 and a 6-month maturity
    (f) A strangle using options with strike prices of $25 and $35 and a 6-month maturity.
    In each case provide a table showing the relationship between profit and final stock price. Ignore the impact of discounting.


二叉树

  1. A stock price is currently $100. Over each of the next two 6-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-year European call option with a strike price of $100?


  2. For the situation considered in Problem 1, what is the value of a 1-year European put option with a strike price of $100? Verify that the European call and European put prices satisfy put–call parity.


  3. Consider a situation where stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock and the option that remains riskless for the whole of the life of the option.


  4. A stock price is currently $50. It is known that at the end of 2 months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a 2-month European call option with a strike price of $49? Use no-arbitrage arguments.


  5. A stock price is currently $80. It is known that at the end of 4 months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $80? Use no-arbitrage arguments.


  6. A stock price is currently $40. It is known that at the end of 3 months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a 3-month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments
    give the same answers.


  7. A stock price is currently $50. Over each of the next two 3-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a 6-month European call option with a strike price of $51?


  8. For the situation considered in Problem 7, what is the value of a 6-month European put option with a strike price of $51? Verify that the European call and European put prices satisfy put–call parity. If the put option were American, would it ever be optimal to exercise it early at any of the nodes on the tree?


  9. A stock price is currently $25. It is known that at the end of 2 months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose STS_T is the stock price at the end of 2 months. What is the value of a derivative that pays off ST2S^2_T at this time?


  10. Calculate uu, dd, and pp when a binomial tree is constructed to value an option on a foreign currency. The tree step size is 1 month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.


  11. The volatility of a non-dividend-paying stock whose price is $78, is 30%. The risk-free rate is 3% per annum (continuously compounded) for all maturities. Calculate values for uu, dd, and pp when a 2-month time step is used. What is the value a 4-month European call option with a strike price of $80 given by a two-step binomial tree. Suppose a trader sells 1,000 options (10 contracts). What position in the stock is necessary to hedge the trader’s position at the time of the trade?


  12. A stock index is currently 1,500. Its volatility is 18%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2.5%. Calculate values for uu, dd, and pp when a 6-month time step is used. What is the
    value a 12-month American put option with a strike price of 1,480 given by a two-step binomial tree.


  13. The futures price of a commodity is $90. Use a three-step tree to value (a) a 9-month American call option with strike price $93 and (b) a 9-month American put option with strike price $93. The volatility is 28% and the risk-free rate (all maturities) is 3% with continuous compounding.


  14. The current price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The risk-free rate is 4%. For a 3-month time step:
    (a) What is the percentage up movement?
    (b) What is the percentage down movement?
    (c) What is the probability of an up movement in a risk-neutral world?
    (d) What is the probability of a down movement in a risk-neutral world?
    Use a two-step tree to value a 6-month European call option and a 6-month European put option. In both cases the strike price is $150.


  15. In Problem 14, suppose a trader sells 10,000 European call options and the two-step tree describes the behavior of the stock. How many shares of the stock are needed to hedge the 6-month European call for the first and second 3-month period? For the second time period, consider both the case where the stock price moves up during the first period and the case where it moves down during the first period.


  16. A stock price is currently $50. It is known that at the end of 6 months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. alculate the value of a 6-month European call option on the stock with an exercise price of $48. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.


  17. A stock price is currently $40. Over each of the next two 3-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. (a) What is the value of a 6-month European put option with a strike price of $42? (b) What is the value of a 6-month American put option with a strike price of $42?


  18. Using a “trial-and-error” approach, estimate how high the strike price has to be in Problem 17 for it to be optimal to exercise the option immediately.


  19. Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is 6 months.
    (a) Calculate uu, dd, and pp for a two-step tree.
    (b) Value the option using a two-step tree.
    (c) Verify that DerivaGem gives the same answer.
    (d) Use DerivaGem to value the option with 5, 50, 100, and 500 time steps.


  20. Repeat Problem 19 for an American put option on a futures contract. The strike price and the futures price are $50, the risk-free rate is 10%, the time to maturity is 6 months, and the volatility is 40% per annum.



维纳过程与伊藤引理

  1. Can a trading rule based on the past history of a stock’s price ever produce returns that are consistently above average? Discuss.


  2. A company’s cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.5 per quarter and a variance rate of 4.0 per quarter. How high does the company’s initial cash position have to be for the company to have a less than 5% chance of a negative cash position by the end of 1 year?


  3. Variables X1X_1 and X2X_2 follow generalized Wiener processes, with drift rates μ1\mu_1 and μ2\mu_2 and variances σ12\sigma^2_1 and σ12\sigma^2_1. What process does X1+X2X_1 + X_2 follow if:
    (a) The changes in X1X_1 and X2X_2 in any short interval of time are uncorrelated?
    (b) There is a correlation r between the changes in X1X_1 and X2X_2 in any short time interval?


  4. Consider a variable SS that follows the process
    dS=μdt+σdzdS = \mu dt + \sigma dz
    For the first three years, μ=2\mu = 2 and σ=3\sigma = 3; for the next three years, μ=3\mu = 3 and σ=4\sigma = 4. If the initial value of the variable is 5, what is the probability distribution of the value of the variable at the end of year 6?


  5. Suppose that GG is a function of a stock price SS and time. Suppose that σS\sigma_S and σG\sigma_G are the volatilities of SS and GG. Show that, when the expected return of SS increases by λσS\lambda\sigma_S, the growth rate of GG increases by λσG\lambda\sigma_G, where λ\lambda is a constant.


  6. Stock A and stock B both follow geometric Brownian motion. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of one of stock A and one of stock B follow geometric Brownian motion? Explain your answer.

  7. The process for the stock price in equation (14.8) is
    ΔS=μSΔt+σSϵΔt\Delta S = \mu S \Delta t + \sigma S\epsilon\sqrt{\Delta t}
    where μ\mu and σ\sigma are constant. Explain carefully the difference between this model and each of the following:
    ΔS=μΔt+σϵΔtΔS=μSΔt+σϵΔtΔS=μΔt+σSϵΔt\begin{align} &\Delta S = \mu \Delta t + \sigma\epsilon\sqrt{\Delta t}\nonumber\\ &\Delta S = \mu S \Delta t + \sigma \epsilon\sqrt{\Delta t}\nonumber\\ &\Delta S = \mu \Delta t + \sigma S\epsilon\sqrt{\Delta t}\nonumber \end{align}
    Why is the model in equation (14.8) a more appropriate model of stock price behavior than any of these three alternatives?


  8. It has been suggested that the short-term interest rate r follows the stochastic process
    dr=a(br)dt+rcdzdr = a(b - r) dt + rc dz
    where aa, bb, cc are positive constants and dzdz is a Wiener process. Describe the nature of this process.


  9. Suppose that a stock price SS follows geometric Brownian motion with expected return μ\mu and volatility σ\sigma:
    dS=μSdt+σSdzdS = \mu S dt + \sigma S dz
    What is the process followed by the variable SnS^n? Show that SnS^n also follows geometric Brownian motion.


  10. Suppose that xx is the yield to maturity with continuous compounding on a zero-coupon bond that pays off $1 at time TT. Assume that xx follows the process
    dx=a(x0x)dt+sxdzdx = a(x_0 - x) dt + s x dz
    where aa, x0x_0, and ss are positive constants and dzdz is a Wiener process. What is the process followed by the bond price?


  11. A stock whose price is 、$30 has an expected return of 9% and a volatility of 20%. In Excel, simulate the stock price path over 5 years using monthly time steps and random samples from a normal distribution. Chart the simulated stock price path. By hitting F9, observe how the path changes as the random samples change. (How to do it with python?)


  12. Suppose that a stock price has an expected return of 16% per annum and a volatility of 30% per annum. When the stock price at the end of a certain day is $50, calculate the following:
    (a) The expected stock price at the end of the next day
    (b) The standard deviation of the stock price at the end of the next day
    (c) The 95% confidence limits for the stock price at the end of the next day.


  13. Suppose that xx is the yield on a perpetual government bond that pays interest at the rate of $1 per annum. Assume that xx is expressed with continuous compounding, that interest is paid continuously on the bond, and that x follows the process
    dx=a(x0x)dt+sxdzdx = a(x_0 - x) dt + sx dz
    where aa, x0x_0, and ss are positive constants, and dzdz is a Wiener process. What is the process followed by the bond price? What is the expected instantaneous return (including interest and capital gains) to the holder of the bond?


  14. Stock A, whose price is $30, has an expected return of 11% and a volatility of 25%. Stock B, whose price is $40, has an expected return of 15% and a volatility of 30%. The processes driving the returns are correlated with correlation parameter ρ\rho. In Excel, simulate the two stock price paths over 3 months using daily time steps and random samples from normal distributions. Chart the results and by hitting F9 observe how the paths change as the random samples change. Consider values for ρ\rho equal to 0.25, 0.75, and 0.95.


  15. Consider whether markets are efficient in each of the following two cases: (a) a stock price follows fractional Brownian motion and (b) a stock price volatility follows fractional Brownian motion.



Black-Scholes-Merton模型

  1. What does the Black–Scholes–Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the probability distribution of the ontinuously compounded rate of return on the stock during the year?


  2. The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?


  3. Calculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum.


  4. What difference does it make to your calculations in Problem 3 if a dividend of $1.50 is expected in 2 months?


  5. A stock price is currently $40. Assume that the expected return from the stock is 15% and that its volatility is 25%. What is the probability distribution for the rate of return (with continuous compounding) earned over a 2-year period?


  6. A stock price follows geometric Brownian motion with an expected return of 16% and a volatility of 35%. The current price is $38.
    (a) What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in 6 months will be exercised?
    (b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised?


  7. Using the notation in this chapter, prove that a 95% confidence interval for STS_T is between S0e(μσ2/2)T1.96σTS_0e^{(\mu-\sigma^2/2)T-1.96\sigma\sqrt{T}} and S0e(μσ2/2)T+1.96σTS_0e^{(\mu-\sigma^2/2)T+1.96\sigma\sqrt{T}}.


  8. A portfolio manager announces that the average of the returns realized in each year of the last 10 years is 20% per annum. In what respect is this statement misleading?


  9. Assume that a non-dividend-paying stock has an expected return of μ\mu and a volatility of σ\sigma. An innovative financial institution has just announced that it will trade a security that pays off a dollar amount equal to ln STS_T at time TT, where STS_T denotes the value of the stock price at time TT.
    (a) Use risk-neutral valuation to calculate the price of the security at time tt in terms of the stock price, SS, at time tt. The risk-free rate is rr.
    (b) Confirm that your price satisfies the differential equation (15.16).
    ft+rSfS+12σ2S22fS2=rf\frac{\partial f}{\partial t}+rS \frac{\partial f}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}=rf


  10. Consider a derivative that pays off STnS^n_T at time TT, where STS_T is the stock price at that time. When the stock pays no dividends and its price follows geometric Brownian motion, it can be shown that its price at time t(tT)t (t\leq T) has the form h(t,T)Snh(t,T)S^n, where SS is the stock
    price at time tt and hh is a function only of tt and TT.
    (a) By substituting into the Black–Scholes–Merton partial differential equation, derive an ordinary differential equation satisfied by h(t,T)h(t, T).
    (b) What is the boundary condition for the differential equation for h(t,T)h(t, T)?
    (c) Show that h(t,T)=e[0.5σ2n(n1)+r(n1)](Tt)h(t, T) = e^{[0.5\sigma^2n(n-1)+r(n-1)](T-t)}, where rr is the risk-free interest rate and σ\sigma is the stock price volatility.


  11. What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is 3 months?


  12. What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is 6 months?


  13. Consider an American call option on a stock. The stock price is $70, the time to maturity is 8 months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the volatility is 32%. A dividend of $1 is expected after 3 months and again after 6 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Use DerivaGem to calculate the price of the option.


  14. A call option on a non-dividend-paying stock has a market price of $212\$2\frac{1}{2}. The stock price is $15, the exercise price is $13, the time to maturity is 3 months, and the risk-free interest rate is 5% per annum. What is the implied volatility?


  15. With the notation used in this chapter:
    (a) What is N(x)N'(x)?
    (b) Show that SN(d1)=Ker(Tt)N(d2)SN'(d_1) = Ke^{-r(T-t)}N'(d_2), where SS is the stock price at time tt and
    d1=ln(S/K)+(r+σ2/2)(Tt)σTtd2==ln(S/K)+(rσ2/2)(Tt)σTtd_1=\frac{\ln(S/K)+(r+\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}\text{, }d_2==\frac{\ln(S/K)+(r-\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}
    (c) Calculate d1/S\partial d_1/\partial S and d2/S\partial d_2/\partial S.
    (d) Show that when c=SN(d1)Ker(Tt)N(d2)c = SN(d_1) - Ke^{-r(T-t)}N(d_2), it follows that
    ct=rKer(Tt)N(d2)SN(d1)σsTt\frac{\partial c}{\partial t} = -rKe^{-r(T-t)} N(d_2) - SN'(d_1)\frac{\sigma}{s\sqrt{T - t}}
    where cc is the price of a call option on a non-dividend-paying stock.
    (e) Show that c/S=N(d1)\partial c/\partial S = N(d_1).
    (f) Show that cc satisfies the Black–Scholes–Merton differential equation.
    (g) Show that cc satisfies the boundary condition for a European call option, i.e., that c=max(SK,0)c = \max(S - K, 0) as tSTt\rightarrow S_T.


  16. Show that the Black–Scholes–Merton formulas for call and put options satisfy put–call parity.


  17. A stock price is currently $50 and the risk-free interest rate is 5%. Use the DerivaGem software to translate the following table of European call options on the stock into a table of implied volatilities, assuming no dividends. Are the option prices consistent with the assumptions underlying Black–Scholes–Merton?

    Stike Price ($) Maturity (months)
    3 6 12
    45 7.0 8.3 10.5
    50 3.7 5.2 7.5
    55 1.6 2.9 5.1



  1. Explain carefully why Black’s approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer given by Black’s approach understate or overstate the true option value? Explain your answer.


  2. Consider an American call option on a stock. The stock price is $50, the time to maturity is 15 months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and the volatility is 25%. Dividends of $1.50 are expected in 4 months and 10 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Calculate the price of the option.


  3. Show that the probability that a European call option will be exercised in a risk-neutral world is, with the notation introduced in this chapter, N(d2)N(d_2). What is an expression for the value of a derivative that pays off $100 if the price of a stock at time TT is greater than KK?


  4. A company has an issue of executive stock options outstanding. Should dilution be taken into account when the options are valued? Explain your answer.


  5. A company’s stock price is $50 and 10 million shares are outstanding. The company is considering giving its employees 3 million at-the-money 5-year call options. Option exercises will be handled by issuing more shares. The stock price volatility is 25%, the 5-year risk-free rate is 5%, and the company does not pay dividends. Estimate the cost to the company of the employee stock option issue.


  6. If the volatility of a stock is 18% per annum, estimate the standard deviation of the percentage price change in (a) 1 day, (b) 1 week, and (c) 1 month.


  7. A stock price is currently $50. Assume that the expected return from the stock is 18% and its volatility is 30%. What is the probability distribution for the stock price in 2 years? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval.


  8. Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows:
    30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2
    Estimate the stock price volatility. What is the standard error of your estimate?


  9. The appendix derives the key result
    E[max(VK,0)]=E(V)N(d1)KN(d2)E[\max(V - K, 0)] = E(V)N(d_1) - KN(d_2)
    Show that
    E[max(VK,0)]=KN(d2)E(V)N(d1)E[\max(V - K, 0)] = KN(-d_2) - E(V)N(-d_1)
    and use this to derive the Black–Scholes–Merton formula for the price of a European put option on a non-dividend-paying stock.