6. Consider a stock that pays no dividends, has a volatility of 25% per annum and an expected return of 13% per annum. Suppose that the current share price of the stock, S0, is USD 30. You decide to model the stock price behavior using a discrete-time version of geometric Brownian motion and to simulate paths of the stock price using Monte Carlo simulation. Let Δt denote the time interval used and let St denote the stock price at time interval t. So, according to your model,

St+1=St(1+0.13Δt + 0.25√Δt ε)

Where ε is a standard normal variable.

To implement this simulation, you generate a path of the stock price by starting at t = 0, generating a sample for ε, updating the stock price according to the model, incrementing t by 1, and repeating this process until the end of the horizon is reached.

Which of the following strategies for generating a sample for Δ will implement this simulation properly?

A. Generate a sample for ε by using the inverse of the standard normal cumulative distribution of a sample value drawn from a uniform distribution between 0 and 1.

B. Generate a sample for ε by sampling from a normal distribution with mean 0.13 and standard deviation 0.25.

C. Generate a sample for ε by using the inverse of the standard normal cumulative distribution of a sample value drawn from a uniform distribution between 0 and 1.  Use Cholesky decomposition to correlate this sample with the sample from the previous time interval.

D. Generate a sample for ε by sampling from a normal distribution with mean 0.13 and standard deviation 0.25.  Use Cholesky decomposition to correlate this sample with the sample from the previous time interval.

Correct answer is A

Monte Carlo Simulation assumes independence across time so there is no need to correlate samples from time period to time period, eliminating C & D.  Choice A describes a valid method for generating a sample from a standard normal distribution.

Reference: Hull, Jorion.

Type of Question: Quantitative Analysis

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