In 1977, Phelim Boyle published a paper called “Options: A Monte Carlo Approach” where he explored the use of Monte Carlo simulations in pricing financial instruments known as options. Options are essentially contracts that give the contract holder the right (not obligation) to buy or sell an underlying asset at an exercise price stated on the contract. These assets are commonly shares in a listed company.

A simple example of how this valuation could be done is through simulating the eventual stock price at the end of the contract. Assuming that the movements in stock price corresponds to a standard normal probability distribution function (known as volatility in academic finance), the expected value of the stock at contract expiration is then calculated using multiple simulations of the final stock price according to this simulation. The value of a call option in one trial is given by the formula Max(0, S - X), where S is the simulated stock price and X is the exercise price. This is because the contract holder can choose not to exercise the contract if S-X is negative, thus giving him a return of $0. Thus, the result is the average simulated value of the option discounted by the appropriate interest rate.

This model can be expanded to include other variables in the pricing of the option, such as interest rate fluctuations and changes in volatility. In fact, Monte Carlo simulations are most useful in option pricing when there are multiple variables where the closed form solutions are hard to solve or do not exist. This is especially so in the world of exotic options where more and more complicated options with numerous variables are created.

In the formula for performing the simulations, the error in the simulation is inversely proportional to the square root of the number of trials. Hence, a large number of trials is necessary to ensure accuracy in pricing. This makes such option pricing computationally expensive. A method known as the Antithetic Variate Technique is used to minimize runtime.

In simulating stock price movement, e = N(0,1), a random variable between 0 and 1 is used for each trial, and this is repeated over M independent trials. In the antithetic variable technique,  we generate 2M trials by calculating the payoff for -e as well. Hence, in this technique, we are able to generate twice as many trials without the corresponding cost involved.

References:
Options: A Monte Carlo Approach, Phelim P. Boyle (1976) - Seminal paper on using Monte Carlo simulations for option pricing

Option Pricing, Java Programming and Monte Carlo Simulation, Nitesh Aidasani Khyami - Includes actual examples on how to use algorithms to determine option prices for basic European options