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4106-08-h6 - HMWK 6 1 A stock has an initial price of S0 =...

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HMWK 6 1. A stock has an initial price of S 0 = 40. S n denotes the price at time t = n , where we assume the binomial lattice model with parameters u = 1 . 25 d = 0 . 8 p = 0 . 60 . The interest rate is r = 0 . 05. (Note that ud = du = 1.) (a) Compute E ( S 1 ) and E ( S 2 ). (b) Compute p * , the risk-neutral probability. (c) Compute the price of a European call option with strike price K = 45 when the expiration time is T = 1, and for T = 2. 2. Recall the car replacement problem (Example 7.12 in Text, Page 434) using the renewal reward theorem to find the optimal value of T (the one that minimizes cost). For each T > 0, the long-run cost rate is given by g ( T ) = C 1 + C 2 H ( T ) T 0 H ( x ) dx , where C 1 > 0 is the cost of a new car, C 2 > 0 is the cost of breakdown, and H ( x ) = P ( V x ) is the car lifetime distribution; H ( x ) = P ( V > x ) its tail. Prove that when H is exponential, the minimum value of g ( T ) is acheived at T = ; there is no finite optimal value of T . (Recall our intuitive argument for this: If the car is alive at T , then by the memoryless property it is still as good as new, so we would keep it.) 3. Consider Example 7.13 (Train Dispatching problem) on Page 436 of the Text, but we change it as follows: Assume that the passengers arrive according to a Poisson process at
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