MATH 3589

Math 3589 Introduction to Financial Mathematics Homework Assignment #11 1. Prove that a symmetric random walk is a martingale. Proof: Let {Mn} ∞ n=0 be a symmetric random walk. Then En [Mn+1] = 1 2 (Mn + 1) + 1 2 (Mn − 1) = Mn. 2. Prove that a symmetric random walk is a Markov process. Proof: Let {Mn} ∞ n=0 be a symmetric random walk and let f(x) be any function of a dummy variable x. Then En [f(Mn+1)] = En [f(Mn + Xn+1)] = 1 2 f(Mn + 1) + 1 2 f(Mn − 1) = g(Mn). 3. ...

Introduction to Financial Mathematics Homework Assignment #8 Solutions Exercise 16. Consider the two period binomial model, with the stock price at time t = 0, S0 = 4, the “up factor” u = 2, “down factor” d = 1/2, and risk free interest rate r = 1/4 so that ˜p = 1/2. Assume in each period the probability P[H] = 3/4. Solve the two-period investors problem for ∆i , i = 0, 1, if U(x) = −2e −x and X0 = 10. Solution. Let f(x1, x2, x3, x4) = 9 16 −2e −x4
+ 3 16 ...

Introduction to Financial Mathematics Homework Assignment #8 Solutions Exercise 16. Consider the two period binomial model, with the stock price at time t = 0, S0 = 4, the “up factor” u = 2, “down factor” d = 1/2, and risk free interest rate r = 1/4 so that ˜p = 1/2. Assume in each period the probability P[H] = 3/4. Solve the two-period investors problem for ∆i , i = 0, 1, if U(x) = −2e −x and X0 = 10. Solution. Let f(x1, x2, x3, x4) = 9 16 −2e −x4
+ 3 16 ...