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FNCE30007 Derivative Securities Tutorial 5: Dividends and Options on Other Instruments - Solutions

Question 1 What is the price of a European call option on a dividend-paying stock when the stock price is $65, the strike price is $75, the risk-free rate is 8% per annum, the volatility is 27% per annum, and the time to maturity is 15 months? The stock is expected to pay a dividend of $1.25 in three and six months and a dividend of $1.50 in nine and twelve months. In this case S = 65, K = 75, r = 0.08, = 0.27, T = 15/12=1.25, div1 = div2 = $1.25 and div3 = div4 = $1.50.

( 0.08)(3/12) ( 0.08)(6/12) ( 0.08)(9/12) ( 0.08)(12/12)

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1

2 1

( ) 1.25 1.25 1.50 1.50 $5.2236

ln(59.78 / 75) (0.08 0.27 / 2)1.25 0.2694 0.270.27 1.25

0.27 1.25 0.5712 0.57

PV Dividends e e e e

d

d d

= + + + =

+ += = =

= = =

The price of the European call is

(0.08)(1.25)

(0.08)(1.25)

59.78 ( 0.27) 75 ( 0.57)

59.78(0.3936) 75 (0.2843)4.24

N e N

e

=

=

=

Question 2 Consider an American call option on a stock which is expected to pay a dividend of $5.00 per share in the next 12 months. The current stock price is $120, the strike price is $100, the risk-free rate is 4% per annum, the volatility is 20% per annum and there is 2 years to maturity. Calculate the current value of the option. S = 120, K=100, r = 0.04, = 0.20, T = 2.00, D = 5, Tdiv = 1.00 PV(dividend) = 5.00e(-0.04)(1) = 4.8039 New S = 120 4.8039 = 115.1961 First find the value of the short maturity option. In this case S = 120, K=100, r = 0.04, = 0.20, T = 1.00 d1 = 1.21 N(d1) = 0.8869 d2 = 1.01 N(d2) = 0.8438 option value = $25.3566

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Then find the value of the long option New S = 115.1961, K=100, r = 0.04, = 0.20, T = 2.00 d1 = 0.92 N(d1) = 0.8212 d2 = 0.64 N(d2) = 0.7389 option value = $26.3900 And compare the two values and choose the highest one as the value of the option in the question. Option value = $26.39 Question 3 Consider a European put option on an index that is two months from maturity. The current value of the index is 310 points, the exercise price is 300 points, the risk-free rate is 8% per annum and the volatility of the index is 20% per annum. If the dividend yield on the index is 3% per annum and each contract point is worth $100, how much would you pay for one option? S = 310, K = 300, r = 0.08, = 0.20, T = 2/12 = 0.1667, q = 0.03 Se-qt = 310e-0.03(0.1667) = 308.4539 Use the Black-Scholes and substitute Se-qt instead of S. d1 = 0.54 N(-d1) = 0.2946 d2 = 0.46 N(-d2) = 0.3228 p = 300e-0.08(0.1667)(0.3228) 308.4539(0.2946) = 4.6869 points = $468.69 Question 4 Find the value of the option in Question 3 using a 2-period binomial model. Since there are two periods t = T/12 = 1/12

= = 0.2 112 = 1.0594 d = 1/u = 0.9439

= ()

= (0.080.03)( 112) 0.94391.0594 0.9439 = 0.5219

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= 0.08 112(2)0 + 0+(1 0.5219)2(300 276.1936) = 5.3696 = $536.96 Question 5 Consider an American put option on a stock which is expected to pay a dividend of $5.00 per share in one years time. A friend of yours has suggested that you use the Blacks approximation to value the option. What do you think? The Blacks approximation applies to American calls on stocks that pay one (or more) discrete dividends over the life of the option. The reason it works for a call is because Merton has showed that the only time it may be optimal to exercise the call is (in the case of a single dividend) either just prior to the ex-dividend date or at maturity. With only two possible optimal exercise dates, Black suggests that American call is like two European calls - one with maturity just prior to the ex-dividend date and one with maturity time T. The current value of the American call is then approximately equal to the maximum of the two European calls. However, the approximation cannot be used to vale an American put on a dividend paying stock. The reason is that Merton has shown that it may be optimal to exercise an American put on a stock which pays a single dividend any anytime except just prior to the ex-dividend date. Therefore, in this case there are infinite number of possible exercise dates which would need to be considered and valued. Question 6 Suppose that the spot price of the Canadian dollar is U.S. $0.85 and that the Canadian dollar/U.S. dollar exchange rate has a volatility of 4% per annum. The risk-free rates of interest in Canada and the United States are 4% and 5% per annum. Calculate the value of a European call option to buy one Canadian dollar for U.S. $0.85 in nine months. In this case S = 0.85, K = 0.85, r = 0.05, rf = 0.04, = 0.04, T = 0.75

1 = ln 0.850.85 + 0.05 0.04 + 0.00162 (0.75)0.040.75 = 0.2338 0.23 2 = 1 0.040.75 = 0.1992 0.20 N(d1) = 0.5910 N(d2) = 0.5793

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The value of the option is then = 0.850.04(0.75)(0.5910) 0.850.05(0.75)(0.5793) = $0.0132 Or 1.32 U.S. cents. Question 7 Suppose that a portfolio is worth $84 million and the market index (i.e., S&P/ASX 200) is at 5,600. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide protection against the value of the portfolio falling below $75.6 million in one years time? Each option contract is for $10 times the index in ASX. If the value of the portfolio mirrors the value of the index, the index can be expected to have dropped by 10% when the value of the portfolio drops by 10%. Hence when the value of the portfolio drops to $75.6 million the value of the index can be expected to be 5,040. This indicates that the put options with an exercise price of 5,040 should be purchased. The options should be on

$84 million / 5,600 = $15,000 times the index. Each option contract is for $10 times the index in ASX. Hence, 1,500 contracts should be purchased ($15,000/$10 = 1,500). Question 8 Consider again the situation in the previous question. Suppose that the portfolio has a beta of 2.0, the risk-free rate is 5% per annum, and the dividend yield on both the portfolio and the index is 3% per annum. What options should be purchased to provide protection against the value of the portfolio falling below $75.6 million in one years time? When the value of the portfolio falls to $75.6 million the holder of the portfolio makes a capital loss of 10%. After dividends are taken into account the loss is 7% during the year. This is 12% below the risk-free rate. According to the capital asset pricing model: Excess return of portfolio above risk-free rate = ()(excess expected return of market above risk-free rate) Thus, when the portfolio provides a return 12% below the risk-free rate, the markets expected return is 6% below the risk-free rate. As the index can be assumed to have a beta of 1.0, this is also the excess expected return (including dividends) from the index. The expected return from the index therefore is -1% per annum. Since the index provides a 3% per annum dividend yield, the expected movement in the index is -4%. Therefore, when the portfolios value is $75.6 million, the expected value of the

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index is 0.96x5600 = 5,376. Hence European put options should be purchased with an exercise price of 5,376. Their maturity date should be in one year. The number of options required is twice the number required in the previous question. This is because we wish to protect a portfolio which is twice as sensitive to changes in market conditions as the portfolio in the previous question. Hence options on $30,000 (3,000 contracts) should be purchased. To check that the answer is correct consider what happens when the value of the portfolio declined by 20% to $67.2 million. The return including dividends is -17%. This is 22% less than the risk-free rate. The index can be expected to provide a return (including dividends) which is 11% less than the risk-free rate. The index can, therefore, be expected to drop by 9% to 5,096. The payoff from the put options is (5,376 5,096)x(30,000) = $8.4 million. This is exactly what is required to restore the value of the portfolio to $75.6 million.