ࡱ>  }G_ a`bjbj 7&jA\jA\aX6 6 4K~)~)~)~)m*m*m*IKKKKKKKKKKKKK$lM"PVoKm*m*m*m*m*oK~)~)Ki.i.i.m*~)~)IKi.m*IKi.i.RF<%J~)R#8*:H5KK0KH>xP5.4xP<%JxP%Jm*m*i.m*m*m*m*m*oKoKi.m*m*m*Km*m*m*m*xPm*m*m*m*m*m*m*m*m*6 > t:  CHAPTER 13 Valuing Stock Options: The Black-Scholes-Merton Model Practice Questions Problem 13.8. A stock price is currently $40. Assume that the expected return from the stock is 15% and its volatility is 25%. What is the probability distribution for the rate of return (with continuous compounding) earned over a one-year period? In this case  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . From equation (13.4) the probability distribution for the rate of return over a one-year period with continuous compounding is:  EMBED Equation.3  i.e.,  EMBED Equation.3  The expected value of the return is 11.875% per annum and the standard deviation is 25.0% per annum. Problem 13.9. A stock price has an expected return of 16% and a volatility of 35%. The current price is $38. What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in six months will be exercised? What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised? The required probability is the probability of the stock price being above $40 in six months time. Suppose that the stock price in six months is  EMBED Equation.DSMT4 . The probability distribution of ln ST is  EMBED Equation.3  i.e.,  EMBED Equation.3  Since  EMBED Equation.DSMT4 , the required probability is  EMBED Equation.DSMT4  From normal distribution tables N(0.008) = 0.5032 so that the required probability is 0.4968. In this case the required probability is the probability of the stock price being less than $40 in six months. It is  EMBED Equation.DSMT4  Problem 13.10. Prove that, with the notation in the chapter, a 95% confidence interval for  EMBED Equation.DSMT4  is between  EMBED Equation.DSMT4  From equation (13.2), ln ST has the distribution  EMBED Equation.3  95% confidence intervals for  EMBED Equation.DSMT4  are therefore  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  95% confidence intervals for  EMBED Equation.DSMT4  are therefore  EMBED Equation.DSMT4  i.e.  EMBED Equation.DSMT4  Problem 13.11. A portfolio manager announces that the average of the returns realized in each of the last 10 years is 20% per annum. In what respect is this statement misleading? This problem relates to the material in Section 13.2 and Business Snapshot 13.1. The statement is misleading in that a certain sum of money, say $1,000, when invested for 10 years in the fund would have realized a return (with annual compounding) of less than 20% per annum. The average of the returns realized in each year is always greater than the return per annum (with annual compounding) realized over 10 years. The first is an arithmetic average of the returns in each year; the second is a geometric average of these returns. Problem 13.12. Assume that a non-dividend-paying stock has an expected return of  EMBED Equation.DSMT4  and a volatility of  EMBED Equation.DSMT4 . An innovative financial institution has just announced that it will trade a derivative that pays off a dollar amount equal to  EMBED Equation.DSMT4  at time  EMBED Equation.DSMT4 . The variables  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  denote the values of the stock price at time zero and time T. Describe the payoff from this derivative. Use risk-neutral valuation to calculate the price of the derivative at time zero. The derivative will pay off a dollar amount equal to the continuously compounded return on the security between times 0 and  EMBED Equation.DSMT4 . The expected value of  EMBED Equation.DSMT4  is, from equation (13.4),  EMBED Equation.DSMT4 . The expected payoff from the derivative is therefore  EMBED Equation.DSMT4 . In a risk-neutral world this becomes  EMBED Equation.DSMT4 . The value of the derivative at time zero is therefore:  EMBED Equation.DSMT4  Problem 13.13. 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 three months? In this case, EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  The price of the European call is  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  or $5.06. Problem 13.14. 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 six months? In this case, EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  The price of the European put is  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  or $6.40. Problem 13.15. A call option on a non-dividend-paying stock has a market price of  EMBED Equation.DSMT4 . The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is 5% per annum. What is the implied volatility? In the case  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . The implied volatility must be calculated using an iterative procedure. A volatility of 0.2 (or 20% per annum) gives  EMBED Equation.DSMT4 . A volatility of 0.3 gives  EMBED Equation.DSMT4 . A volatility of 0.4 gives  EMBED Equation.DSMT4 . A volatility of 0.39 gives  EMBED Equation.DSMT4 . By interpolation the implied volatility is about 0.396 or 39.6% per annum. The implied volatility can also be calculated using DerivaGem. Select equity as the Underlying Type in the first worksheet of DG400f.xls. Select Black-Scholes European as the Option Type. Input stock price as 15, the risk-free rate as 5%, time to exercise as 0.25, and exercise price as 13. Leave the dividend table blank because we are assuming no dividends. Select the button corresponding to call. Select the implied volatility button. Input the Price as 2.5 in the second half of the option data table. Hit the Enter key and click on calculate. DerivaGem will show the volatility of the option as 39.64%. Problem 13.16. Show that the BlackScholesMerton formula for a call option gives a price that tends to  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  As  EMBED Equation.DSMT4 , the second term on the right hand side tends to zero. The first term tends to  EMBED Equation.DSMT4  if  EMBED Equation.DSMT4  and to  EMBED Equation.DSMT4  if  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4  when  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  when  EMBED Equation.DSMT4 , it follows that  EMBED Equation.3   EMBED Equation.3  Similarly  EMBED Equation.3   EMBED Equation.3  Under the Black-Scholes-Merton formula the call price,  EMBED Equation.DSMT4  is given by:  EMBED Equation.DSMT4  From the above results, when  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4  so that  EMBED Equation.DSMT4 . Also, when  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4  so that  EMBED Equation.DSMT4 . These results show that  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4 . Problem 13.17. Explain carefully why Blacks 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 Blacks approach understate or overstate the true option value? Explain your answer. Blacks approach in effect assumes that the holder of option must decide at time zero whether it is a European option maturing at time  EMBED Equation.DSMT4  (the final ex-dividend date) or a European option maturing at time  EMBED Equation.DSMT4 . In fact, the holder of the option has more flexibility than this. The holder can choose to exercise at time  EMBED Equation.DSMT4  if the stock price at that time is above some level but not otherwise. Furthermore, if the option is not exercised at time  EMBED Equation.DSMT4 , it can still be exercised at time T. It appears that Blacks approach should understate the true option value. This is because the holder of the option has more alternative strategies for deciding when to exercise the option than the two strategies implicitly assumed by the approach. These alternative strategies add value to the option. However, this is not the whole story! The standard approach to valuing either an American or a European option on a stock paying a single dividend applies the volatility to the stock price less the present value of the dividend. (The procedure for valuing an American option is explained in Chapter 18.) Blacks approach when considering exercise just prior to the dividend date applies the volatility to the stock price itself. Blacks approach therefore assumes more stock price variability than the standard approach in some of its calculations. In some circumstances it can give a higher price than the standard approach. Problem 13.18. Consider an American call option on a stock. The stock price is $70, the time to maturity is eight 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 three months and again after six months. Use the results in the appendix to 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. With the notation in the text  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Hence  EMBED Equation.DSMT4  Also:  EMBED Equation.DSMT4  Hence:  EMBED Equation.DSMT4  It follows from the conditions established in the Appendix to Chapter 13 that the option should never be exercised early. The option can therefore be value as a European option. The present value of the dividends is  EMBED Equation.DSMT4  Also:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  and the call price is  EMBED Equation.DSMT4  or $10.94. DerivaGem can be used to calculate the price of this option. Select equity as the Underlying Type in the first worksheet of DG400f.xls. Select Black-Scholes European as the Option Type. Input stock price as 70, the volatility as 32%, the risk-free rate as 10%, time to exercise as =8/12, and exercise price as 65. In the dividend table, enter the times of dividends as 0.25 and 0.50, and the amounts of the dividends in each case as 1. Select the button corresponding to call. Hit the Enter key and click on calculate. DerivaGem will show the value of the option as $10.942. Problem 13.19. 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 BlackScholesMerton? Stock PriceMaturity = 3 monthsMaturity = 6 monthsMaturity = 12 months547.008.3010.50503.505.207.50551.602.905.10 Using DerivaGem we obtain the following table of implied volatilities Stock PriceMaturity = 3 monthsMaturity = 6 monthsMaturity = 12 months5437.7834.9934.025032.1232.7832.035531.9830.7730.45To calculate first number, select equity as the Underlying Type in the first worksheet of DG400f.xls. Select Black-Scholes European as the Option Type. Input stock price as 50, the risk-free rate as 5%, time to exercise as 0.25, and exercise price as 45. Leave the dividend table blank because we are assuming no dividends. Select the button corresponding to call. Select the implied volatility button. Input the Price as 7.0 in the second half of the option data table. Hit the Enter key and click on calculate. DerivaGem will show the volatility of the option as 37.78%. Change the strike price and time to exercise and recompute to calculate the rest of the numbers in the table. The option prices are not exactly consistent with BlackScholesMerton. If they were, the implied volatilities would be all the same. We usually find in practice that low strike price options on a stock have significantly higher implied volatilities than high strike price options on the same stock. This phenomenon is discussed in Chapter 19. Problem 13.20. Show that the BlackScholesMerton formulas for call and put options satisfy putcall parity. The BlackScholesMerton formula for a European call option is  EMBED Equation.DSMT4  so that  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  The BlackScholesMerton formula for a European put option is  EMBED Equation.DSMT4  so that  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  This shows that the putcall parity result  EMBED Equation.DSMT4  holds. Problem 13.21. 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,  EMBED Equation.DSMT4 . What is an expression for the value of a derivative that pays off $100 if the price of a stock at time T is greater than  EMBED Equation.DSMT4 ? The probability that the call option will be exercised is the probability that  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the stock price at time  EMBED Equation.DSMT4 . In a risk neutral world the probability distribution of ln ST is  EMBED Equation.3  The probability that  EMBED Equation.DSMT4  is the same as the probability that  EMBED Equation.DSMT4 . This is  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The expected value at time  EMBED Equation.DSMT4  in a risk neutral world of a derivative security which pays off $100 when  EMBED Equation.DSMT4  is therefore  EMBED Equation.DSMT4  From risk neutral valuation the value of the security is  EMBED Equation.DSMT4  Further Questions Problem 13.22. If the volatility of a stock is 18% per annum, estimate the standard deviation of the percentage price change in (a) one day, (b) one week, and (c) one month.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Problem 13.23. A stock price is currently $50. Assume that the expected return from the stock is 18% per annum and its volatility is 30% per annum. What is the probability distribution for the stock price in two years? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval. In this case  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The probability distribution of the stock price in two years,  EMBED Equation.DSMT4 , is lognormal and is, from equation (13.2), given by:  EMBED Equation.3  i.e.,  EMBED Equation.3  The mean stock price is from equation (13.3)  EMBED Equation.DSMT4  and the standard deviation is  EMBED Equation.DSMT4  95% confidence intervals for  EMBED Equation.DSMT4  are  EMBED Equation.DSMT4  i.e.,  EMBED Equation.DSMT4  These correspond to 95% confidence limits for  EMBED Equation.DSMT4  of  EMBED Equation.DSMT4  i.e.,  EMBED Equation.DSMT4  Problem 13.24. (Excel file) Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows: 30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2 Estimate the stock price volatility. What is the standard error of your estimate? The calculations are shown in the table below  EMBED Equation.DSMT4  and an estimate of standard deviation of weekly returns is:  EMBED Equation.DSMT4  The volatility per annum is therefore  EMBED Equation.DSMT4  or 20.79%. The standard error of this estimate is  EMBED Equation.DSMT4  or 3.9% per annum. WeekClosing Stock Price ($)Price Relative  EMBED Equation.DSMT4 Weekly Return  EMBED Equation.DSMT4 130.2232.01.059600.05789331.10.971880.02853430.10.967850.03268530.21.003320.00332630.31.003310.00331730.61.009900.00985833.01.078430.07551932.90.996970.003031033.01.003040.003031133.51.015150.015041233.51.000000.000001333.71.005970.005951433.50.994070.005951533.20.991040.00900 Problem 13.25. A financial institution plans to offer a derivative that pays off a dollar amount equal to  EMBED Equation.DSMT4  at time  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the stock price at time  EMBED Equation.DSMT4 . Assume no dividends. Defining other variables as necessary use risk-neutral valuation to calculate the price of the derivative at time zero. (Hint: The expected value of  EMBED Equation.DSMT4  can be calculated from the mean and variance of  EMBED Equation.DSMT4  given in Section 13.1.) From Section 13.1, if  EMBED Equation.DSMT4  denotes expected value,  EMBED Equation.DSMT4  Because  EMBED Equation.DSMT4 , it follows that  EMBED Equation.DSMT4  so that  EMBED Equation.DSMT4  In a risk-neutral world  EMBED Equation.DSMT4  so that  EMBED Equation.DSMT4  Using risk-neutral valuation, the value of the derivative security at time zero is  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Problem 13.26. Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. What is the price of the option if it is a European call? What is the price of the option if it is an American call? What is the price of the option if it is a European put? Verify that putcall parity holds. In this case  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The European call price is  EMBED Equation.DSMT4  or $2.52. The American call price is the same as the European call price. It is $2.52. The European put price is  EMBED Equation.DSMT4  or $1.05. Put-call parity states that:  EMBED Equation.DSMT4  In this case  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  and it is easy to verify that the relationship is satisfied, Problem 13.27. Assume that the stock in Problem 13.26 is due to go ex-dividend in 1.5 months. The expected dividend is 50 cents. What is the price of the option if it is a European call? What is the price of the option if it is a European put? Use the results in the Appendix to this chapter to determine whether there are any circumstances under which the option is exercised early. The present value of the dividend must be subtracted from the stock price. This gives a new stock price of:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The price of the option is therefore  EMBED Equation.DSMT4  or $2.21. Because  EMBED Equation.DSMT4  the value of the option when it is a European put is  EMBED Equation.DSMT4  or $1.22. If  EMBED Equation.DSMT4  denotes the time when the dividend is paid:  EMBED Equation.DSMT4  This is less than the dividend. Hence the option should be exercised immediately before the ex-dividend date for a sufficiently high value of the stock price. Problem 13.28. Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is six months, the volatility is 30% per annum, and the risk-free interest rate is 10% per annum. Two equal dividends of 40 cents are expected during the life of the option, with ex-dividend dates at the end of two months and five months. Use Blacks approximation and the DerivaGem software to value the option. Suppose now that the dividend is  EMBED Equation.DSMT4  on each ex-dividend date. Use the results in the Appendix to determine how high  EMBED Equation.DSMT4  can be without the American option being exercised early. We first value the option assuming that it is not exercised early, we set the time to maturity equal to 0.5. There is a dividend of 0.4 in 2 months and 5 months. Other parameters are  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . DerivaGem gives the price as 0.7947. We next value the option assuming that it is exercised at the five-month point just before the final dividend. DerivaGem gives the price as 0.7668. The price given by Blacks approximation is therefore 0.7947. (DerivaGem also shows that the American option price calculated using the binomial model with 100 time steps is 0.8243.) It is never optimal to exercise the option immediately before the first ex-dividend date when  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the size of the first dividend, and  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are the times of the first and second dividend respectively. Hence we must have:  EMBED Equation.DSMT4  that is,  EMBED Equation.DSMT4  It is never optimal to exercise the option immediately before the second ex-dividend date when:  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the size of the second dividend. Hence we must have:  EMBED Equation.DSMT4  that is,  EMBED Equation.DSMT4  It follows that the dividend can be as high as 16.6 cents per share without the American option being worth more than the corresponding European option.  ABCUVWeN ^ _ ` t u v w x } ~  Źuh_QDjhhN:EHUjwT hhN:UVhhN:EHjhhN:EHUjvT hhN:UVhhN:EHjhhN:U hhN:hhN:6]h `hN:5hkw!5CJaJh `hkw!5CJaJh `h `5CJaJh `h `h `CJ aJ h `h `5CJ \aJ h `hkw!5CJ \aJ hkw! BCWe 1 7 N # 7 $ & - ^gdN:  H$1$H$gdN:$a$gdN: & FgdN: & FgdN:$a$gdN:gdN:$a$gd$a$$a$     , - . / 0 6 7 8 9 I J K L M " # 5 ̿㨛~ugZ~j1 hhN:EHUjzT hhN:UVhhN:EHjhhN:UhhN:6]h `hN:5jhN}hN:EHUjyT hN}hN:UVhN}hN:EHj hN}hN:EHUjxT hN}hN:UVhN}hN:EH hN}hN:jhN}hN:UhN: hhN:#      ! 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