# homework1 - 1 Homework Problems(Part I for “Derivative...

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Unformatted text preview: 1 Homework Problems (Part I) for “Derivative Securities and Difference Methods” Problems 3 1 Introduction Problems 1. What is the difference between taking a long position in a forward contract and in a call option? 2. Suppose the futures price of gold is currently \$324 per ounce. An investor takes a short position in a futures contract for the delivery of 1,000 ounces. How much does the investor gain or lose if the price of gold at the end of the contract is (a) \$310 per ounce; (b) \$340 per ounce? 3. An investor holds a European call option on a stock with an exercise price of \$88 and the option costs \$3.50. For what value of the stock at maturity will the investor exercise the option, and for what value of the stock at maturity will the investor make a profit? 4. An investor holds a European put option for a stock with an exercise price of \$88 and the option costs \$3.50. Find the gain or loss to the investor if the stock price at maturity is (a) \$93.50; (b) \$81.50. 5. A company will receive a certain amount of foreign currency in one year. To reduce the risk of the changes in the exchange rate, what type of contract is appropriate for hedging? 6. Suppose a fund manager holds 10 million shares of IBM stock and would like to use options to reduce risk. What action is suitable for reducing the risk of decline of the stock price in the next three months? 7. A stock price is \$67 just before a dividend of \$1.50 is paid. What is the stock price immediately after the payment? Problems 5 2 Basic Options Problems 1. a) Show integraldisplay ∞ −∞ 1 √ 2 π e − x 2 / 2 dx = 1 . b) Show that integraldisplay ∞ −∞ 1 b √ 2 π e − ( x − a ) 2 / 2 b 2 dx = 1 holds for any a and b . (Because this is true and the integrand is always possitive, it can be a probability density function.) c) If the probability density function of a random variable x is 1 b √ 2 π e − ( x − a ) 2 / 2 b 2 , then it is called a normal random variable. Show E[ x ] = a and Var[ x ] = b 2 . 2. Define dX = φ √ dt , where φ is a standardized normal random variable and its probability density function is 1 √ 2 π e − φ 2 / 2 , −∞ < φ < ∞ . Find E [ dX ], Var [ dX ], E bracketleftbig dX 2 bracketrightbig , and Var bracketleftbig dX 2 bracketrightbig . 3. a) Suppose that S 1 and S 2 are two independent normal random variables. The mean and variance of S 1 are μ 1 and σ 2 1 and for S 2 they are μ 2 and σ 2 2 . Find the probability density function of the random variable S 1 + S 2 and using this function, show that S 1 + S 2 is a normal random variable with mean μ 1 + μ 2 and variance σ 2 1 + σ 2 2 . 6 2 Basic Options b) Suppose Δt = t/n and φ i , i = 1 , 2 , ··· ,n are independent standardized normal random variables. Show X ( t ) = lim n →∞ parenleftBig φ 1 √ Δt + φ 2 √ Δt + ··· + φ n √ Δt parenrightBig is a normal random variable with mean zero and variance t ....
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homework1 - 1 Homework Problems(Part I for “Derivative...

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