ProblemSet4_solution - 1 Suppose a 3 year zero coupon bond...

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1. Suppose a 3 year zero coupon bond with face value 1 is currently priced at 0.95. A 5 year zero coupon bond is selling for 0.92. What is the annual forward interest rate on a 2 year zero coupon bond delivered after 3 years? Forward price of bond F = 0.92 / 0.95 = 0.9684 Converting this to a forward interest rate, we get (1 / 0.9684) 1/2 – 1 = 1.62% 2. You are long a futures position with initial settlement price $1.5 and hold your position until maturity. Consider two scenarios. In the first scenario, the exchange rate increases to $3 before falling to $1.6 at maturity. In the second scenario, the exchange rate falls to $0.50 before rising to $1.6 at maturity. Is your resulting profit different in the two scenarios? Why or why not? Yes, if the exchange rate rises the futures position gains value, leading to margin account gains. These gains can be withdrawn and invested, leading to a higher profit in scenario 1 as a result of the interest earned on the reinvested gains. If there is negative correlation between exchange rates and interest rates, this effect may not necessarily hold. 3. Using the following three period binomial tree to value a European call option on 100,000 euros with strike price $0.90, maturity 0.5 (ie 6 months), domestic and foreign discount rates 5% (annual), and volatility 10%. What is the price according to the binomial model? $1 $1.05 $0.95 $1.10 $1.00 $0.90 S 0 S 0.25 S 0.5
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The quarterly interst rate = (1.05) 0.25 – 1 = 1.23% ( = r f = r d ) Delta = (Cu – Cd)/(Su-Sd)/(1+r f ), Alpha = Cu – delta * Su(1+r f ) = Cd – delta * Sd(1+r f ) Price = delta * S + alpha / (1+r d ) Using the above formulas and pricing each node backward, the call option price is $0.0976 per euro, or $9,760. What is the Black-Scholes value? N(1.5254) = 0.936 and N(1.4547) = 0.927 d1 = 1.525378 d2 = 1.454667 According to Black-Scholes the call price is $0.0995 per euro, or $9,950. 4. Explain intuitively how each of S, K, r d , r f , T, and σ affect the price of a European call option. As S increases or K decreases, the intrinsic value (value from exercising the option) increases, hence the price of the call increases. As T increases, there is more time for the underlying to move. Depending on r d and r f , the price may increase or decrease by interest rate parity. As σ increases, there is a higher probability that the option will land far in the money leading to high upside. There is a high probability of a negative movement, but in an option the downside risk is limited, so the price of the call option increases. As r d - r f increases, by interest rate parity the spot rate is expected to increase hence the price of the call option increases. 5. Draw the payoff profiles for the following strategies. a) Write a call. Write a put with the same strike K. Delta=0.9878 Alpha=-0.9 Price = $0.1482 Delta=0.9878 Alpha=-0.9 Price = $0.0494 $0.20 $0.10 $0.00 C 0 C 0.25 C 0.5 Delta=0.9760 Alpha=-0.8892 Price = $0.0976
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b) Buy a call. Sell a put with the same strike K.
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