Pricing European Options with Risk-Neutral Monte Carlo Simulation

Summary

This tutorial explains how to approximate European call and put prices by simulating terminal asset prices under a risk-neutral geometric Brownian motion model. Each simulated price is generated from a standard normal draw; the option payoff is calculated at maturity, averaged across simulations, and discounted at the risk-free rate. A basic C++ implementation uses the Box-Muller method to generate normal variates and applies the call or put payoff separately.

The article compares a run with ten million paths against Black-Scholes analytical prices and reports close agreement, illustrating Monte Carlo’s ability to approximate an expectation. It also points out the tradeoff between simulation count, runtime, and accuracy, along with duplicated code in the separate call and put routines. The example is deliberately introductory and not optimized; it covers European vanilla payoffs under a specified model and does not address variance reduction, error estimates, or more complex contracts.

Key ideas

• Risk-neutral pricing expresses a European option value as a discounted expected payoff at maturity.
• Monte Carlo estimates that expectation by simulating terminal prices and averaging their payoffs.
• The example generates Gaussian shocks with the Box-Muller method and prices calls and puts separately.
• The reported simulated values are close to the analytical Black-Scholes values for the stated run.
• More paths can improve accuracy while increasing execution time.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.