Monte Carlo Simulation

Monte Carlo simulation is a computational method that estimates the outcome of an uncertain process by generating a large number of random scenarios and summarising the results. Instead of solving a problem with a formula, the analyst builds a model of how the uncertain inputs behave, draws random values for them thousands or millions of times, calculates the result for each draw, and reads off the average, the spread or the frequency of particular outcomes.

The method was developed in the 1940s by Stanislaw Ulam and John von Neumann during work on nuclear weapons at Los Alamos, and was named after the casino district of Monaco. In finance it is used to value derivatives that have no pricing formula, to measure portfolio risk, to test the robustness of trading strategies and to plan long-term investments. Its use for option pricing was introduced by Phelim Boyle in 1977.

A simulation has four steps. First, a model is specified for each uncertain quantity, usually as a probability distribution or a stochastic process. Second, a computer generates random numbers and converts them into one possible scenario. Third, the quantity of interest, such as an option's payoff or a portfolio's loss, is computed for that scenario. Fourth, the first three steps are repeated many times and the results are collected. The justification is the law of large numbers: the average over many independent trials converges on the true expected value.

For share prices, the standard model is geometric Brownian motion. One step of a simulated path is:

• $S_t$ is the price at time $t$ and $\Delta t$ is the length of the step in years.
• $\mu$ is the annual drift and $\sigma$ is the annual volatility.
• $Z$ is a random draw from the standard normal distribution.

Repeating the step builds a path, and each path ends at a price drawn from a lognormal distribution.

A Monte Carlo result is an estimate with a margin of error. The standard error of the estimated mean is:

Here $s$ is the standard deviation of the simulated results and $N$ is the number of trials. Suppose an option is valued by simulating 10,000 price paths, and the discounted payoffs average 5.20 with a standard deviation of 8.00. The standard error is $8.00 / \sqrt{10{,}000} = 0.08$, so a 95 percent confidence interval for the value runs from about 5.04 to 5.36. Because the error falls with the square root of $N$, halving it to 0.04 requires 40,000 paths, four times as many. This slow convergence is the method's main cost. Variance reduction techniques, such as pairing each random draw with its mirror image, improve accuracy without more trials.

• Derivatives pricing. Under the risk-neutral approach, an option's value is the average simulated payoff discounted at the risk-free rate. Simulation is the natural tool where the payoff depends on the whole price path, as with an Asian option, or on several assets at once. Options with early exercise are harder, and are handled by regression-based extensions of the method.
• Risk measurement. Banks simulate changes in market factors to produce a distribution of portfolio gains and losses, from which value at risk and related measures are read.
• Strategy testing. A backtest shows one historical path. Reshuffling or resampling its trades or returns thousands of times shows the range of outcomes the same strategy might have produced, including the distribution of its maximum drawdown.
• Financial planning. Advisers simulate decades of investment returns to estimate the chance that savings will last through retirement.

Where a closed-form solution exists, as the Black-Scholes-Merton model provides for European options, a formula is faster and exact. Lattice methods such as the binomial option pricing model handle early exercise easily but become unwieldy when there are several sources of uncertainty. Monte Carlo is slower than both for simple problems, but its cost grows only modestly as dimensions are added, which makes it the practical choice for complex products and large portfolios. Modern implementations run on graphics processing units, which compute many paths in parallel.

A simulation is only as good as the model behind it. If returns are assumed to be normally distributed and independent, the simulation will reproduce that assumption faithfully and understate the frequency of extreme moves, since real markets have fat tails and volatility that clusters. Correlations between assets, which matter greatly for portfolio risk, are difficult to estimate and tend to change in a crisis. A large number of trials gives a precise answer to the question the model poses, which is not the same as an accurate answer about the world.

The output can also create a false sense of certainty. A statement that a plan has a 92 percent chance of success conveys a precision that the inputs do not justify. Careful users vary the assumptions, test alternative distributions and treat the results as a guide to the range of outcomes, not as a forecast.