Monte Carlo Options Pricing

Brownian motion describes the motion of a particle in a fluid or gas. Such a particle bounces around “randomly” within the fluid surrounding it. The path it creates appears quite noisy and random. It can essentially be considered a limiting form of the random walk where both the time between steps and the step length approach zero (with some caveats: namely, if the time step is

, the step length must be

- this produces a process whose variance scales exactly with time). Consider flipping a coin to determine whether to take a step forward or backward. Now let the time between steps approach zero.

Monte-Carlo simulation is a statistical technique inspired by the casinos of Monaco. Much like gamblers resigning their fates to probability, we hand over the results of statistical analysis to chance. By running enough trials, we can make conclusions with statistical significance.

Consider evaluating a call option with a strike prices of $105 for this stock. What would be the expected value of the option at expiry, given a geometric Brownian model for the stock’s movement?

Monte-Carlo simulation of terminal values is a relatively simple simulation, and one can probably be completed analytically. The true power of Monte-Carlo simulation is unlocked when analysing scenarios that are more difficult to solve analytically, if not impossible. For example, if we wanted to analyse an American-style option, which can be exercised anytime, we might want to count the probability of a stock price exceeding the strike price at any time during the lifetime of that option. We could do this by counting how many trials cross the strike price boundry.