Monte Carlo Simulation

A simulation begins with a model that maps inputs to an output: a valuation formula, a cash flow projection, or a portfolio return equation. The analyst then specifies which inputs are uncertain and assigns each a probability distribution with its parameters, for example a normal distribution for asset returns, a lognormal distribution for prices, or a triangular distribution for a cost with a plausible minimum, mode and maximum. Correlations among inputs must be specified as well, since drawing variables independently when they move together systematically misstates the spread of results.

Each trial proceeds identically: a pseudorandom number generator produces uniform values between zero and one, each uniform value is converted into a draw from the assigned distribution through the inverse of its cumulative distribution function, the model is evaluated, and the output is stored. After many thousands of trials the stored outputs form an empirical frequency distribution from which the mean, standard deviation, percentiles and tail probabilities are read directly.

Simulation earns its cost when the payoff depends on the path rather than only the terminal value. Asian options settle against an average price, lookback and barrier options depend on extremes reached along the way, and structured credit tranches depend on the timing of defaults. In fixed income, the value of embedded prepayment and call features depends on the sequence of short rates, not merely their final level. None of these admit tidy analytical formulas under realistic assumptions.

Simulation is equally common where the analytical problem is tractable but the question is distributional. Firms simulate multi-year capital projects to estimate the probability that net present value falls below zero, risk teams generate portfolio loss distributions to obtain value at risk and expected shortfall, and wealth managers project retirement wealth paths to estimate the probability of depleting a portfolio before a target horizon.

The dispersion of the estimated mean across independent runs narrows roughly in proportion to one over the square root of the number of trials, so quadrupling the trial count halves the sampling noise. That improvement applies only to the statistical error of the simulation itself. It says nothing about whether the assumed distributions, parameters and correlations describe the real world, and it cannot correct a misspecified model.

This is why simulation output should be treated as conditional. A stated probability of loss is the probability implied by the assumptions, not an objective property of the security. Sound practice includes sensitivity testing of key parameters, stressing correlations toward one to reflect crisis behaviour, checking whether tail-heavy distributions are more appropriate than normal ones, and validating the model against observable prices where they exist.

Examinable competence is conceptual and procedural rather than computational. Be able to describe the sequence: specify the model and its uncertain inputs, choose distributions and parameters, generate random draws, run repeated trials, and summarise the resulting distribution. Be able to explain the role of the inverse cumulative distribution function in converting uniform random numbers into draws from a chosen distribution.

Also be able to state the applications and the boundaries. Simulation values path-dependent instruments, estimates distributions of project or portfolio outcomes, and tests the sensitivity of a model to input uncertainty. It does not establish causal relationships, does not substitute for economic reasoning, and produces results no more reliable than the assumptions supplied. Compare it with historical simulation, which resamples observed data instead of drawing from an assumed distribution, and with scenario analysis.