Solution of Derivative

Question 418 auric enterprises is using gold as an

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Unformatted text preview: ive full credit for tax losses, the tax code does not have an effect on the expected after-tax profits of firms that have the same expected pre-tax profits, but different cash-flow variability. Question 4.18. Auric Enterprises is using gold as an input. Therefore, it would like to hedge against price increases in gold. a) The cost of this collar today is the premium of the purchased 440-strike call ($2.49) less the premium for the sold 400-strike put. We calculate a cost of $2.49 − $2.21 = −$0.28 , which means that Auric in fact generates a revenue from entering into this collar. 32 Chapter 4 Introduction to Risk Management b) A good starting point are the values of part a). You see that both put and call are worth approximately the same, therefore start shrinking the 440 – 400 span symmetrically until you get a difference of 30, and then do some trial and error. This should bring you the following values: The call strike is 435.52, and the put strike is 405.52. Both call and put have a premium of $3.425 Question 4.20. a) Since we know that the value of a call is decreasing in the strike, and we need to sell two call options, the Black-Scholes prices of which equal the 440-strike call price, we know that we have to look for a higher strike price. Trial and error results in a strike price of 448.93. The premium of the 440-strike call is $2.4944, and indeed the Black-Scholes premium of the 448.93 strike call is $1.2472. b) Profit diagram: 33 Chapter 5 Financial Forwards and Futures Question 5.2. a) The owner of the stock is entitled to receive dividends. As we will get the stock only in one year, the value of the prepaid forward contract is today’s stock price, less the present value of the four dividend payments: 4 F0PT = $50 − ∑ $1e , 3 − 0.06× i 12 = $50 − $0.985 − $0.970 − $0.956 − $0.942 = $50 − $3.853 = $46.147 i =1 b) The forward price is equivalent to the future value of the prepaid forward. With an interest rate of 6 percent and an expiration of the forward in one year, we thus have: F0,T = F0PT × e 0 .06×1 = $46.147 × e 0 .06×1 = $46.147 × 1.0618 = $49.00 , Question 5.4. This question asks us to familiarize ourselves with the forward valuation equation. a) We plug the continuously compounded interest rate and the time to expiration in years into the valuation formula and notice that the time to expiration is 6 months, or 0.5 years. We have: F0,T = S 0 × e r×T = $35 × e 0.05×0.5 = $35 × 1.0253 = $35.886 b) The annualized forward premium is calculated as: annualized forward premium = 1 F0,T ln T S0 1 $35.50 = 0.5 ln $35 = 0.0284 c) For the case of continuous dividends, the forward premium is simply the difference between the risk-free rate and the dividend yield: S × e (r −δ )T = 1 ln 0 T S0 1 1 = ln (e (r −δ )T ) = (r − δ )T T T = r −δ annualized forward premium = 1 F0,T ln T S0 34 Chapter 5 Financial Forwards and Futures Therefore, we can solve: 0.0284 ⇔ δ = 0.05 − δ = 0.0216 The annualized dividend yield is 2.16 percent. Question 5.6. a) We plug the continuously compounded interest rate, the dividend yield and the time to expiration in years into the valuation formula and notice that the time to expiration is 9 months, or 0.75 years. We have: F0,T = S 0 × e (r −δ )×T = $1,100 × e (0.05−0.015 )×0.75 = $1,100 × 1.0266 = $1,129.26 b) We engage in a reverse cash and carry strategy. In particular, we do the following: Description Today Long forward, resulting 0 from customer purchase Sell short tailed position + S 0 e −δT of the index − S 0 e −δT Lend S 0 e −δT TOTAL 0 In 9 months S T − F0,T − ST S 0 × e (r −δ )T S 0 × e (r −δ )T − F0,T Specifically, we have: Description Today Long forward, resulting 0 from customer purchase Sell short tailed position $1,100 × .9888 of the index = 1087.69 − $1,087.69 Lend $1,087.69 TOTAL 0 In 9 months ST − $1,129.26 − ST $1,087.69 × e 0.05×0.75 = $1,129.26 0 Therefore, the market maker is perfectly hedged. He does not have any risk in the future, because he has successfully created a synthetic short position in the forward contract. 35 Part 2 Forwards, Futures, and Swaps c) Description Today Short forward, resulting 0 from customer purchase Buy tailed position in − S 0 e −δT index Borrow S 0 e −δT S 0 e − δT TOTAL 0 In 9 months F0,T − ST ST − S 0 × e (r −δ )T F0,T − S 0 × e (r −δ )T Specifically, we have: Description Today Short forward, resulting 0 from customer purchase Buy tailed position in − $1,100 × .9888 index = −$1,087.69 $1,087.69 Borrow $1,087.69 TOTAL 0 In 9 months $1,129.26 − S T ST − $1,087.69 × e 0.05×0.75 = −$1,129.26 0 Again, the market maker is perfectly hedged. He does not have any index price risk in the future, because he has successfully created a synthetic long position in the forward contract that perfectly offsets his obligation from the sold forward contract. Question 5.8. First, we need to find the fair value of the forward price. We plug the continuously compounded interest rate, the dividend yield and th...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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