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Unformatted text preview: Math 623, F 2011: Homework 3. For full credit, your solutions must be clearly presented and all code included. (1) In the BlackScholes method for pricing of options it is assumed that the stock price evolves according to geometric Brownian motion: dS t S t = r dt + σ dW t , (1) where W ( t ) is Brownian motion, r is the risk free rate of interest and σ is the volatility. (a) Suppose t = 0 is today and the current stock price is S . The BlackScholes price of the option is the expectation value of a function of S ( T ), where T > 0 is the expiration date of the option. The random variable log S ( T ) is known to be Gaussian. Write down formulas for its mean and variance. (b) Suppose S = 20 , r = 0 . 045 , σ = 0 . 28 , T = 0 . 5. Draw histograms for the distribution of S ( T ) with numbers of simulations given by the values N = 10 4 , 10 5 , 10 6 . You can use the MATLAB function hist to do this. (c) With the numerical values given in (b) use the MonteCarlo method to compute the value of a European call option with strike price K = 21. If V N is the value of the option based on N simulations and N is the standard error for the N simulations, plot the graphs of N against V N (convergence diagram), and N against N for 1 ≤ N ≤ 10 4 . Report the values of V N and N /V N for N = 10 6 . What is the significance of the reported value of N /V N ?...
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This note was uploaded on 12/08/2011 for the course MATH 623 taught by Professor Conlon during the Fall '08 term at University of Michigan.
 Fall '08
 CONLON
 Math

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