GARCH Toolbox™

Terminology

Trials vs. Paths

Monte Carlo simulation literature often uses different terminology for the evolution of the simulated variables of interest, such as trials, paths, realizations, or replications. The following sections use the terms trial and path interchangeably.

However, there are situations where you should distinguish between these terms. Specifically, the term trial often implies the result of an independent random experiment (for example, the evolution of the price of a single stock or portfolio of stocks). Such an experiment computes the average or expected value of a variable of interest (for example, the price of a derivative security) and its associated confidence interval.

By contrast, the term path implies the result of a random experiment that is different or unique from other results, but that may or may not be independent.

The distinction between these terms is usually unimportant. It may, however, be useful when applied to variance reduction techniques that attempt to increase the efficiency of Monte Carlo simulation by inducing dependence across sample paths. A classic example involves pair-wise dependence induced by antithetic sampling, and applies to more sophisticated variance reduction techniques, such as stratified sampling.

For more information, about antithetic sampling, see Antithetic Sampling. For more information about stratified sampling, see User-Specified Random Number Generation: Stratified Sampling.

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NTRIALS, NPERIODS, and NSTEPS

SDE methods in the GARCH Toolbox™ software use the parameters NTRIALS, NPERIODS, and NSTEPS as follows:

• The input argument NTRIALS specifies the number of simulated trials or sample paths to generate. This argument always determines the size of the third dimension (the number of pages) of the output 3-dimensional time-series array Paths. Indeed, in a traditional Monte Carlo simulation of one or more variables, each sample path is independent and represents an independent trial.

• The parameters NPERIODS and NSTEPS represent the number of simulation periods and time steps, respectively. Both periods and time steps are related to time increments that determine the exact sequence of observed sample times. The distinction between these terms applies only to issues of accuracy and memory management. For more information, see Optimizing Accuracy of Solutions and Managing Memory.

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Introduction

Behavior and Syntax of SDE Objects