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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 . 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 = 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 ?...
View Full Document

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.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online