bkmsol_ch21

# Inputs outputs standard deviation annual 03566 d1

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INPUTS OUTPUTS Standard deviation (annual) 0.3566 d1 -0.0484 Maturity (in years) 0.5 d2 -0.3006 Risk-free rate (annual) 0.05 N(d1) 0.4807 Stock Price 98 N(d2) 0.3819 Exercise price 105 B/S call value 8.0000 Dividend yield (annual) 0 B/S put value 12.4075 21-5

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23. a. The delta of the collar is calculated as follows: Position Delta Buy stock 1.0 Buy put, X = \$45 N(d 1 ) – 1 = –0.40 Write call, X = \$55 –N(d 1 ) = –0.35 Total 0.25 If the stock price increases by \$1, then the value of the collar increases by \$0.25. The stock will be worth \$1 more, the loss on the purchased put will be \$0.40, and the call written represents a liability that increases by \$0.35. b. If S becomes very large, then the delta of the collar approaches zero. Both N(d 1 ) terms approach 1. Intuitively, for very large stock prices, the value of the portfolio is simply the (present value of the) exercise price of the call, and is unaffected by small changes in the stock price. As S approaches zero, the delta also approaches zero: both N(d 1 ) terms approach 0. For very small stock prices, the value of the portfolio is simply the (present value of the) exercise price of the put, and is unaffected by small changes in the stock price. 24. Statement a: The hedge ratio (determining the number of futures contracts to sell) ought to be adjusted by the beta of the equity portfolio, which is 1.20. The correct hedge ratio would be: 400 , 2 2 . 1 000 , 2 β 2,000 β 500 100 \$ million 100 \$ = × = × = × × Statement b: The portfolio will be hedged, and should therefore earn the risk-free rate, not zero, as the consultant claims. Given a futures price of 100 and an equity price of 100, the rate of return over the 3-month period is: (100 99)/99 = 1.01% = approximately 4.1% annualized 25. Put X Delta A 10 0.1 B 20 0.5 C 30 0.9 26. a. Choice A: Calls have higher elasticity than shares. For equal dollar investments, a call’s capital gain potential is greater than that of the underlying stock. b. Choice B: Calls have hedge ratios less than 1.0, so the shares have higher profit potential. For an equal number of shares controlled, the dollar exposure of the shares is greater than that of the calls, and the profit potential is therefore greater. 21-6
27. a. The value of the call option is expected to decrease if the volatility of the underlying stock price decreases. The less volatile the underlying stock price, the less the chance of extreme price movements and the lower the probability that the option expires in the money. This makes the participation feature on the upside less valuable. The value of the call option is expected to increase if the time to expiration of the option increases. The longer the time to expiration, the greater the chance that the option will expire in the money resulting in an increase in the time premium component of the option’s value. b. i. When European options are out of the money, investors are essentially saying that they are willing to pay a premium for the right, but not the obligation, to buy or sell the underlying asset. The out-of-the-money option has no intrinsic value, but, since options require little capital (just the premium paid) to obtain a