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# 014 - 14 Option Sensitivities and Option Hedging Answers to...

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119 Answers to Questions and Problems 1. Consider Call A, with: X \$70; r 0.06; T t 90 days; 0.4; and S \$60. Compute the price, DELTA, GAMMA, THETA, VEGA, and RHO for this call. 2. Consider Put A, with: X \$70; r 0.06; T t 90 days; 0.4; and S \$60. Compute the price, DELTA, GAMMA, THETA, VEGA, and RHO for this put. 3. Consider a straddle comprised of Call A and Put A. Compute the price, DELTA, GAMMA, THETA, VEGA, and RHO for this straddle. RHO 3.5985 13.4083 9.8098 VEGA 9.9144 9.9144 19.8288 THETA 8.9173 4.47790 13.6963 GAMMA 0.0279 0.0279 0.0558 DELTA 0.2735 0.7265 0.4530 price c p \$12.61 RHO 13.4083 VEGA 9.9144 THETA 4.7790 GAMMA .0279 DELTA .7265 p \$10.79 RHO 3.5985 VEGA 9.9144 THETA 8.9173 GAMMA .0279 DELTA .2735 c \$1.82 14 Option Sensitivities and Option Hedging

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120 CHAPTER 14 OPTION SENSITIVITIES AND OPTION HEDGING 4. Consider Call A. Assuming the current stock price is \$60, create a DELTA-neutral portfolio consisting of a short position of one call and the necessary number of shares. What is the value of this portfolio for a sudden change in the stock price to \$55 or \$65? As we saw for this call, DELTA 0.2735. The DELTA-neutral portfolio, given a short call component, is 0.2735 shares 1 call, costs: .2735 (\$60) \$1.82 \$14.59 If the stock price goes to \$55, the call price is \$.77, and the portfolio will be worth: .2735 (\$55) \$.77 \$14.27 With a stock price of \$65, the call is worth \$3.55, and the portfolio value is: .2735 (\$65) \$3.55 \$14.23 Notice that the portfolio values are lower for both stock prices of \$55 and \$65, reflecting the negative GAMMA of the portfolio. 5. Consider Call A and Put A from above. Assume that you create a portfolio that is short one call and long one put. What is the DELTA of this portfolio? Can you find the DELTA without computing? Explain. Assume that a share of stock is added to the short call/long put portfolio. What is the DELTA of the entire position? The DELTA of the portfolio is 1.0 0.2735 0.7265. This is necessarily true, because the DELTA of the call is N ( d 1 ), the DELTA of the put is N ( d 2 ), and N ( d 1 ) N ( d 2 ) 1.0. If a long share of stock is added to the portfolio, the DELTA will be zero, because the DELTA of a share is always 1.0. 6. What is the GAMMA of a share of stock if the stock price is \$55 and a call on the stock with X \$50 has a price c \$7 while a put with X \$50 has a price p \$4? Explain. The GAMMA of a share of stock is always zero. All other information in the question is irrelevant. The GAMMA of a share is always zero because the DELTA of a share is always 1.0. As GAMMA measures how DELTA changes, there is nothing to measure for a stock, since the DELTA is always 1.0. 7. Consider Call B written on the same stock as Call A with: X \$50; r 0.06; T t 90 days; 0.4; and S \$60. Form a bull spread with calls from these two instruments. What is the price of the spread? What is its DELTA? What will the price of the spread be at expiration if the terminal stock price is \$60? From this information, can you tell whether THETA is positive or negative for the spread? Explain. As observed in problem 1, for Call A, c \$1.82, DELTA 0.2735, and THETA 8.9173. For Call B, c \$11.64, DELTA 0.8625, and THETA 7.7191. The long bull spread with calls consists of buying the call with the lower exercise price (Call B) and selling the call with the higher exercise price (Call A). The spread costs \$11.64 \$1.82 \$9.82. The DELTA of the spread equals DELTA B
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