W09hw3 - Math 623 F 2011 Homework 3 For full credit your...

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Math 623, F 2011: Homework 3. For full credit, your solutions must be clearly presented and all code included. (1) In the Black-Scholes 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 0 . The Black-Scholes 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 0 = 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 Monte-Carlo 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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