# Chapter 21 Valuing Options

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1 Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher standard deviation of return C) A higher beta and a lower standard deviation of return D) A lower beta and a lower standard deviation of return A Type: Difficult Page: A call option has an exercise price of \$100. At the final exercise date, the stock price could be either \$50 or \$150. Which investment would combine to give the same payoff as the stock? A) Lend PV of \$50 and buy two calls B) Lend PV of \$50 and sell two calls C) Borrow \$50 and buy two calls D) Borrow \$50 and sell two calls A Type: Difficult Page: 566 Response: Value of two calls: 2 ( ) = 100 or value of two calls =: 2(0) = 0 (not exercised); payoff = = 150 or payoff = = The option delta is calculated as the ratio: A) (the spread of possible share prices) / (the spread of possible option prices) B) (the share price) / (the option price) C) (the spread of possible option prices) / (the spread of possible share prices) D) (the option price) / (the share price) C Type: Medium Page: Suppose Ralph's stock price is currently \$50. In the next six months it will either fall to \$30 or rise to \$80. What is the option delta of a call option with an exercise price of \$50? A) B) C) D) 0.75 C Type: Medium Page: 567 Response: Option delta = (30-0)/(80-30) = 30/50= 0.6 Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 1

2 5. Suppose Waldo's stock price is currently \$50. In the next six months it will either fall to \$40 or rise to \$60. What is the current value of a six-month call option with an exercise price of \$50? The six-month risk-free interest rate is 2% (periodic rate). A) \$5.39 B) \$15.00 C) \$8.25 D) \$8.09 A Type: Difficult Page: 567 Response: Replicating portfolio method: Call option payoff = = 10 and zero; (60)(A) + (1.02)(B) = 10, (40)(A) (B) = 0; A = 0.5 (option delta) & B = ; call option price (current) = 0. 5(50) = \$5.39 Risk- neutral valuation: 50 = [x(60) + (1-x)40]/1.02) ; x = 0. 55; (1-x )= 0.45; Call option value = [(0.55)(10) + (0.45)(0)]/(1.02) = \$ Suppose Waldo's stock price is currently \$50. In the next six months it will either fall to \$40 or rise to \$80. What is the current value of a six-month call option with an exercise price of \$50? The six-month risk-free interest rate is 2% (periodic rate). A) \$2.40 B) \$15.00 C) \$8.25 D) \$8.09 D Type: Difficult Page: 567 Response: Replicating portfolio method: Call option payoff = = 30 and zero; (80)(A) + (1.02)(B) = 30, (40)(A) (B) = 0; A = 0.75 (option delta) & B = ; call option price (current) = 0.75(50) = \$8.09 Risk- neutral valuation: 50 = [x(80) + (1-x)40]/1.02) ; x = 0.275; (1-x )= 0.725; Call option value = [(0.275)(30) + (0.725)(0)]/(1.02) = \$ Suppose Ann's stock price is currently \$25. In the next six months it will either fall to \$15 or rise to \$40. What is the current value of a six-month call option with an exercise price of \$20? The six-month risk-free interest rate is 5% (periodic rate). [Use the risk-neutral valuation method] A) \$20.00 B) \$8.57 C) \$9.52 D) \$13.10 B Type: Difficult Page: 567 Response: Method I: 25 = [x(40) + (1-x)15]/1.05 ; x = x =0.55 Call option price = [(0.45)(20) + ( )(0)]/(1.05) = \$ Test Bank, Chapter 21

3 8. Suppose Ann's stock price is currently \$25. In the next six months it will either fall to \$15 or rise to \$40. What is the current value of a six-month call option with an exercise price of \$20? The six-month risk-free interest rate is 5% (periodic rate). [Use the replicating portfolio method] A) \$20.00 B) \$8.57 C) \$9.52 D) \$13.10 B Type: Difficult Page: 567 Response: 40(A) (B) = 20; 15(A) +1.05(B) = 0; A = 0.8; B = Call option price = 25(0.8) = \$ Suppose Carol's stock price is currently \$20. In the next six months it will either fall to \$10 or rise to \$40. What is the current value of a six-month call option with an exercise price of \$12? The six-month risk-free interest rate (periodic rate) is 5%. [Use the risk-neutral valuation method] A) \$9.78 B) \$10.28 C) \$16.88 D) \$13.33 A Type: Difficult Page: 567 Response: 20 = [x(40) + (1-x)(10)]/ 1.05 ; x = 0.367; (1-x) = Call option price = [(0.367)(28) + (0.633)(0)]/(1.05) = \$ Suppose Carol's stock price is currently \$20. In the next six months it will either fall to \$10 or rise to \$40. What is the current value of a six-month call option with an exercise price of \$12? The six-month risk-free interest rate (periodic rate) is 5%. [Use the replicating portfolio method] A) \$9.78 B) \$10.28 C) \$16.88 D) \$13.33 A Type: Difficult Page: 567 Response: 40(A) (B) = 28; 10(A) (B) = 0; A = ; B = Call option price = (0.9333)(25) = Suppose Victoria's stock price is currently \$20. In the next six months it will either fall to \$10 or rise to \$30. What is the current value of a put option with an exercise price of \$12? The six-month risk-free interest rate is 5% (periodic rate). A) \$9.78 B) \$2.00 C) \$0.86 D) \$9.43 C Type: Difficult Page: 568 Response: Replicating portfolio method: 30(A) (B) = 0; & 10 (A) (B) = 2; A = - 0.1(option delta) & B = 2.86; Put option price = -(0.1)(20) = 0.86; Risk-neutral valuation: 20 = [x(30) + (1-x)(10)]/1.05; x = 0.55 and (1-x) = 0.45 Put option price = [(0.55)(0) + (0.45)(2)]/(1.05) = 0.86 Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 3

4 12. The delta of a put option is always equal to: A) The delta of an equivalent call option B) The delta of an equivalent call option with a negative sign C) The delta of an equivalent call option minus one D) None of the above C Type: Difficult Page: If the delta of a call option is 0.6, calculate the delta of an equivalent put option A) 0.6 B) 0.4 C) 0.4 D) 0.6 C Type: Difficult Page: 569 Response: = Suppose Victoria's stock price is currently \$20. Six-month call option on the stock with an exercise price of \$12 has a value of \$9.43. Calculate the price of an equivalent put option if the six-month risk-free interest rate is 5% (periodic rate). A) \$0.86 B) \$9.43 C) \$8.00 D) \$12.00 A Type: Difficult Page: 570 Response: Value of Put = /(1.05) = \$ If e is the base of natural logarithms, and (ó) is the standard deviation of the continuously compounded annual returns on the asset, and h is the interval as a fraction of a year, then the quantity (1 + upside change) is equal to: A) e^[(ó)?*sqrt(h)] B) e^[h*sqrt(ó)] C) (ó)?e^[sqrt(h)] D) none of the above. A Type: Difficult Page: If u equals the quantity (1+ upside change), then the quantity (1 + downside change) is equal to: A) u B) 1/u C) 1/u D) none of the above. C Type: Medium Page: Test Bank, Chapter 21

5 17. If the standard deviation of the continuously compounded annual returns (ó ) on the asset is 40%, and the time interval is a year, then the upside change is equal to: A) 88.2% B) 8.7% C) 63.2% D) 49.18% D Type: Difficult Page: 574 Response: (1 + u) = e^(0.4) = or upside change = 49.18% 18. If the standard deviation for continuously compounded annual returns on the asset is 40% and the interval is a year, then the downside change is equal to: A) 27.4% B) 53.6% C) 32.97% D) 38.7% C Type: Difficult Page: 574 Response: (1 + u) = e^[(0.4*1)] = ; downside change = 1/ = 1 + d = d = ; down % = 32.97% 19. If the standard deviation of continuously compounded annual returns on the asset is 20% and the interval is half a year, then the downside change is equal to: A) 37.9% B) 19.3% C) 20.1% D) 13.2% D Type: Difficult Page: 574 Response: (1+ u) = e^[(0.2)*] = or d= 1/1.152 =.868 or downside change = 13.2% 20. If the standard deviation of continuously compounded annual returns on the asset is 40% and the interval is a year, then the downside change is equal to : A) 27.4% B) 53.6% C) 33.0% D) 38.7% C Type: Difficult Page: 574 Response: (1 + u)=e^(0.4) = or (1+d)= 1/1.492 = 0.67 or downside change = 33% Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 5

6 21. A stock is currently selling for \$50. The stock price could go up by 10% or fall by 5% each month. The monthly interest rate is 1% (periodic rate). Calculate the price of a European call option on the stock with an exercise price of \$50 and a maturity of two months. (use the two-stage binomial method) A) \$5.10 B) \$2.71 C) \$4.78 D) \$3.62 B Type: Difficult Page: 574 Response: 50 = [55(x) (1-x)]/(1.01) ; x = 0.4; 1-x = 0.6 Payoffs at the end of month-2: \$10.5; \$2.25 ; and zero; [Exercise price = \$50] End of one month values: [(10.5)(0.4) (0.6)]/1.01 = and [2.25(0.4) + 0]/1.01 = Price of the call option = [5.495(0.4) + (0.891)(0.6)]/1.01 = An European call option with an exercise price of \$50 has a maturity (expiration) of six months, stock price of \$54 and the instantaneous variance of the stock returns The risk-free rate is 9.2%. Calculate the value of d 2 (approximately). A) B) C) D) B Type: Medium Page: 576 Response: d 2 = (0.5^0.5) = An European call option with an exercise price of \$50 has a maturity (expiration) of six months, stock price of \$54 and the instantaneous variance of the stock returns The risk-free rate is 9.2%. Then [d1] has a value of (approximately): A) 0.3 B) 0.7 C) 0.7 D) 0.5 D Type: Difficult Page: 576 Response: d 1 = [ln(54/50) + ( (0.64/2))(0.5)]/[(0.8)(0.7071)] = Cola Company has call options on its common stock traded in the market. The options have an exercise price of \$45 and expire in 156 days. The current price of the Cola stock is \$ The risk-free rate is \$7% per year and the standard deviation of the stock returns is The value of [d 1 ] is (approximately): A) B) 0.18 C) D) B Type: Difficult Page: 576 Response: d 1 = [ln(44.375/45) + [0.07 +(0.096/2)][165/365]/[(0.31)(0.6538)] = Test Bank, Chapter 21

7 25. Cola Company has call options on its common stock traded in the market. The options have an exercise price of \$45 and expire in 156 days. The current price of the Cola stock is \$ The risk-free rate is \$7% per year and the standard deviation of the stock returns is Calculate the value of d 2 : (approximately) A) B) C) D) B Type: Medium Page: 576 Response: d 2 = d 1 - [(σ )(SQRT( t)]; d2 = 0.18 (0.31)[ (156/365)^0.5] = If the value of d 1 is 1.25, then the value of N(d 1 ) is equal to: A) B) 1.25 C) 0.25 D) D Type: Medium Page: 578 Response: Use the Cumulative Probabilities of the Standard Normal Distribution Table 27. If the value of d 2 is -0.5, then the value of N(d 2 ) is: A) B) C) D) C Type: Medium Page: 578 Response: Use the Cumulative Probabilities of the Standard Normal Distribution Table 28. If the value of d is 0.75, calculate the value of N(d): A) B) C) D) A Type: Medium Page: If the strike price increases then the: [Assume everything else remaining the same] A) Value of the put option increases and that of the call option decreases B) Value of the put option decreases and that of the call option increases C) Value of both the put option and the call option increases D) Value of both the put option and the call option decreases E)None of the above A Type: Medium Page: 578 Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 7

8 30. If the volatility (variance) of the underlying stock increases then the: [Assume everything else remaining the same] A) Value of the put option increases and that of the call option decreases B) Value of the put option decreases and that of the call option increases C) Value of both the put option and the call option increases D) Value of both the put option and the call option decreases E)None of the above C Type: Medium Page: The Black-Scholes OPM is dependent on which five parameters? A) Stock price, exercise price, risk free rate, beta, and time to maturity B) Stock price, risk free rate, beta, time to maturity, and variance C) Stock price, risk free rate, probability, variance and exercise price D) Stock price, exercise price, risk free rate, variance and time to maturity D Type: Difficult Page: The value of N(d) in the Black-Scholes model can take any value between: A) 1 and + 1 B) 0 and +1 C) 1 and 0 D) None of the above B Type: Medium Page: N(d 1 ) in the Black-Scholes model represents (I) call option delta (II) hedge ratio (III) probability A) I only B) II only C) III only D) I, II, and III D Type: Medium Page: The term [N(d 2 )*PV(EX)] in the Black-Scholes model represents: A) call option delta B) bank loan C) put option delta D) none of the above B Type: Medium Page: Test Bank, Chapter 21

9 35. Which of the following statements regarding American puts is/are true? A) An American put can be exercised any time before expiration B) An American put is always more valuable than an equivalent European put C) Multi-period binomial model can be used to value an American put D) All of the above D Type: Medium Page: A stock is currently selling for \$50. The stock price could go up by 10% or fall by 5% each month. The monthly interest rate is 1% (periodic rate). Calculate the price of an American put option on the stock with an exercise price of \$55 and a maturity of two months. (Use the two-stage binomial method) A) \$5.10 B) \$3.96 C) \$4.78 D) None of the above A Type: Difficult Page: 582 Response: p=1-(-5)/(10-(-5)) = p = 0.6 End of one month values: [(0)(0.4) + (2.75)(.6)]/1.01= or zero And [(2.75)(.4)+(9.875)(0.6)]/1.01 = \$ or \$7.5 if exercised; P = [0.4(1.6337) +(7.5)(0.6)]/1.01 = \$ Suppose the exchange rate between US dollars and British pound is US\$1.50 = BP1.00. If the interest rate is 6% per year what is the adjusted price of the British pound when valuing a foreign currency option with an expiration of one year? (Approximately) A) \$1.905 B) \$ C) \$ D) None of the above B Type: Difficult Page: 583 Response: Adjusted price = 1.5/1.06 = If the interest rate is 10%, the upside change is +25% and the downside change is 20%. Calculate the risk-neutral probability of upside change. A) 0.5 B) C) 0.75 B Type: Difficult Page: 584 Response: p = 10-(-20)/(25-(-20)) = Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 9

10 39. If the interest is 8%, the upside change is +40% and the downside change is 40% calculate the risk-neutral probability of upside change. A) 0.50 B) 0.80 C) 0.60 C Type: Difficult Page: 584 Response: p= 8-(-40)/(40-(-40)) = If the interest rate is 12%, the upside change is 20% and the downside change is 10%, calculate the risk-neutral probability of upside change. A) B) 0.5 C) 0.1 A Type: Difficult Page: 584 Response: p=12-(-10)/(20-(-10)) = A stock is currently selling for \$100. One year from today the stock price could go up by 30% or go down by 20%. The risk-free interest rate is 10%[APR]. Calculate the price of a one-year European call option on the stock with an exercise price of \$100 using the binomial method. A) \$30.00 B) \$16.36 C) \$15.67 B Type: Difficult Page: 584 Response: p=10-(-20)/(30-(-20)) = 0.60; C = [(0.6)(30)-(.4)(0)]/1.1 = \$ A stock is currently selling for \$50. One month from today the stock price could go up by 10% or fall by 50%. If the monthly interest rate is 1% (periodic rate) calculate the price of a European call option on the stock with an exercise price of \$48 and a maturity of one month (use the binomial method). A) \$2.77 B) \$2.00 C) \$1.98 A Type: Difficult Page: 584 Response: p = [1-(-5)]/(10-(-5)) = 0.40; C = [(0.4)(7) + (0.6)(0)]/1.01 = \$ Test Bank, Chapter 21

11 43. A stock is currently selling for \$50. The stock price could go up by 10% or fall by 5% each month. The monthly interest rate is 1% (periodic rate). Calculate the price of a European call option on the stock with an exercise price of \$48 and a maturity of two months. (use the two-stage binomial method) A) \$3.96 B) \$2.77 C) \$1.98 A Type: Difficult Page: 584 Response: p=1-(-5)/(10-(-5)) = p = 0.6 End of one month values: [12.5(0.4) + (4.25)(.6)]/1.01= And [(4.25)(.4)+ (0.6)(0)]/1.01 = \$1.6832; C = [0.4(7.4752) + (0.6)(1.6832)]/1.01 = \$ A stock is currently selling for \$50. The stock price could go up by 10% or fall by 5% each month. The monthly interest rate is 1% (periodic rate). Calculate the price of a American put option on the stock with an exercise price of \$55 and a maturity of two months. (Use the two-stage binomial method) A) \$5.10 B) \$3.96 C) \$4.78 A Type: Difficult Page: 584 Response: p=1-(-5)/(10-(-5)) = p = 0.6 End of one month values: [(0)(0.4) + (2.75)(.6)]/1.01= or zero And [(2.75)(.4)+(9.875)(0.6)]/1.01 = \$ or \$7.5 if exercised; P = [0.4(1.6337) +(7.5)(0.6)]/1.01 = \$ A stock is currently selling for \$50. The stock price could go up by 10% or fall by 5% each month. The monthly interest rate is 1% (periodic rate). Calculate the price of a European put option on the stock with an exercise price of \$55 and a maturity of two months. (use the two-stage binomial method) A) \$5.10 B) \$2.77 C) \$4.78 C Type: Difficult Page: 584 Response: p=1-(-5)/(10-(-5)) = p = 0.6 End of one month values: [(0)(0.4) + (2.75)(.6)]/1.01= or zero And [(2.75)(.4)+(9.875)(0.6)]/1.01 = \$ or \$7.5 if exercised ; P = [0.4(1.6337) +(6.9555)(0.6)]/1.01 = \$4.78 True/False Questions T F 46. An option can be valued using the standard discounted cash flow procedure. False Type: Medium Page: 565 Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 11

12 12 Test Bank, Chapter 21

13 T F 47. It is possible to replicate an investment in a call option by a levered investment in the underlying asset. True Type: Medium Page: 566 T F 48. When risk-neutral probabilities are used, the cost of capital is the discount rate False Type: Medium Page: 567 T F 49. Option delta for a put option is always positive False Type: Difficult Page: 569 T F 50. For an European option: Value of put = Value of call) - share price + PV(exercise price) True Type: Medium Page: 570 T F 51. The binomial model is a discrete time model True Type: Medium Page: 570 T F 52. The Black-Scholes model is a discrete time model False Type: Medium Page: 575 T F 53. N(d 1 ) and N(d 2 ) are probabilities and therefore take values between 0 and 1. True Type: Medium Page: 578 T F 54. Multi-period binomial model can be used to evaluate an American put option True Type: Medium Page: 582 Essay Questions 55. Briefly explain why discounted cash flow method (DCF) does not work for valuing options? Type: Difficult Page: 565 Discounted cash flow method has two steps. First, estimating cash flows; second, discounting these cash flows using an appropriate opportunity cost of capital. In case of options, the first part is quite difficult. But the second step is almost impossible as the risk of an option changes with changes in stock price. Also, the risk changes over time even without a change in the stock price. Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 13

14 56. Briefly explain why an option is always riskier than the underlying stock? Type: Difficult Page: 566 An investment in option can be replicated by borrowing a fraction of the cost of the stock at the risk-free rate and investing in the stock. Therefore, investing in an option is always riskier than investing only in the stock. The beta and the standard deviation of returns of an option is always higher than that of investing in the underlying stock 57. Briefly explain risk-neutral valuation in the context of option valuation. Type: Difficult Page: 567 The valuation method that uses risk-neutral probabilities to evaluate the current price of an option is known as risk-neutral valuation. The risk-neutral upside probability is calculated as: p = [risk-free rate downside change]/[upside change downside change] 58. Briefly explain what is meant by risk-neutral probability? Type: Difficult Page: 567 Risk-neutral probability provides certainty equivalent expected payoffs. It assumes that the investor does not need a higher rate of return to invest in a risky asset. The present value of the payoffs are obtained by discounting at the risk-free rate of return. 59. Briefly explain the term "option delta." Type: Medium Page: 567 Option delta is the ratio of spread of possible option prices to spread of possible share prices. This is also the hedge ratio. 60. Give an example of an option equivalent investment using common stock and borrowing. Type: Difficult Page: 567 The current price of a stock is \$40. Stock price, one period from today, could be either \$50 or \$30. The exercise price is also \$40. The risk-free rate per period is 2%. The option payoff is \$10 in the up state and is zero in the down state. By borrowing \$14.74 today and buying 0.5 of a share of stock, one can replicate the payoffs from the option. The payoff in the upstate is, the value of the stock \$25 minus the payoff amount of the loan \$15, equal to \$10. The payoff in the downstate is, the value of the stock is \$15 minus the payoff amount of \$15, equal to zero. This is called a replicating portfolio. 61. Briefly explain put-call parity. Type: Medium Page: 570 The relationship between the value of an European option and the value of our equivalent put option is called put-call parity. If this does not hold, then investors have arbitrage opportunity. 14 Test Bank, Chapter 21

15 62. Briefly explain how to choose the up and down values for the binomial method? Type: Medium Page: 574 [1 + upside change] = e ó (SQRT (h)) and [1 + downside change] = 1/[1 + upside change] Where: ó = standard deviation of the annual returns of the underlying stock. h = time to expiration expressed in years. Brealey/Myers/Allen, Principles of Corporate Finance, 8/e 15

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