# Math Modeling

### Question Description:

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Due: Thu Apr 25 2013 MTH 437 Homework 3 Problem 1 (10 points) Consider the random variable X with the probability density function given by p(x) = x e??x; x 2 [0;1) with > 0. 1. Find as a function . 2. Find the expectation of X and also its variance. 3. Compute the characteristic function of X. Problem 2 (10 points) Write a Matlab program that implements the following stochastic dierence equation: Sn+1 ?? Sn = SnW where the W is the Brownian increment dened by W = rpt and r N(0; 1). Document Preview: Due: Thu Apr 25 2013 MTH 437 Homework 3 Problem 1 (10 points) Consider the random variable X with the probability density function given by p(x) = x e??x; x 2 [0;1) with > 0. 1. Find as a function . 2. Find the expectation of X and also its variance. 3. Compute the characteristic function of X. Problem 2 (10 points) Write a Matlab program that implements the following stochastic dierence equation: Sn+1 ?? Sn = SnW where the W is the Brownian increment dened by W = rpt and r N(0; 1). Using a Monte-Carlo simulation, compute the approximate value ofE((ST ?? K)+); and show that this expectation approximates the value of a European call option given by the Black-Scholes formula. Problem 3 (10 points) A stock starts at time t = 0 with a value \$100 and, at each time step, the stock can go up \$40 or down \$20. Consider three time steps and a European call with a strike price \$150. 1. Sketch the stock process. 2. Sketch the option process and nd the value of the option. 3. Assume that the stock rst goes up and then down. Explain your hedging strategy and show that you are risk-free. Attachments: Mth437HW3.pdf