# Monte Carlo Simulation

Monte Carlo (MC) simulation is a widely used computerized method that allows dealing with multiple sources of uncertainty in many different fields.

The heart of the MC simulation is its ability to generate random draws of a random variable with a certain distribution.

While MC simulation has long been used in mathematics, statistics and physics, its use in option pricing has considerably increased since the publication of the working paper of Boyle in 1977.

This method is suitable for pricing option in a framework with various sources of uncertainty.

This method is very useful to price exotic options that do not have a closed-form solution.

In this case, it is appropriate to simulate the stock price process to get an approximation of E [f(S(t)].

This estimation is achieved via the simulation of many trajectories of the underlying stock price process.

Every simulation will result in the calculation of an estimated payoff and therefore an estimated option price.

By the law of large numbers, the average of these option prices will converge to the true option price given by the expectation E [f(S(t)].

The use of MC simulation in stock option pricing generally follows five steps.

1. Express the option value as a conditional expectation of the discounted terminal option payoff.

2. Compute the potential future stock prices through a MC simulation.

3. Compute the option payoff for each of the simulated stock prices.

4. Discount this payoff back to today to obtain the estimated option price.

5. Average all the estimated option prices to obtain a fairly accurate estimation of the option price.

To achieve MC simulations, one has to discretize stochastic processes --> converting stochastic processes from continuous time to discrete time.

While the idea is straightforward, a lot of questions will arise during the process. In particular, which discretization method to implement.