SDEs

SDE Modeling

Financial Toolbox™ enables you to model dependent financial and economic variables, such as interest rates and equity prices, by performing standard Monte Carlo or Quasi-Monte Carlo simulation of stochastic differential equations (SDEs). The flexible architecture of the SDE engine provides efficient simulation methods that allow you to create new simulation and derivative pricing methods.

The following table lists tasks you can perform using the SDE functionality.

To perform this task ...	Use these types of models ...
Simulating Equity Prices	Geometric Brownian Motion (GBM) Constant Elasticity of Variance (CEV) Stochastic Differential Equation (SDE) Stochastic Differential Equations from Drift and Diffusion Objects (SDEDDO) Stochastic Differential Equations from Linear Drift (SDELD) Heston Stochastic Volatility (Heston)
Simulating Interest Rates	Hull-White-Vasicek (HWV) Cox-Ingersoll-Ross (CIR) Stochastic Differential Equation (SDE) Stochastic Differential Equations from Drift and Diffusion Objects (SDEDDO) Stochastic Differential Equations from Mean-Reverting Drift (SDEMRD) Models
Price European Stock Options Using Monte Carlo Simulation	Geometric Brownian Motion (GBM)
Stratified Sampling	All supported models
Quasi-Monte Carlo Simulation	All supported models
Performance Considerations	All supported models

Trials vs. Paths

Monte Carlo simulation literature often uses different terminology for the evolution of the simulated variables of interest, such as trials and paths. 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 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 pairwise dependence induced by antithetic sampling, and applies to more sophisticated variance reduction techniques, such as stratified sampling which is a variance reduction technique that constrains a proportion of sample paths to specific subsets (or strata) of the sample space.

NTrials, NPeriods, and NSteps

SDE functions in the Financial 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 three-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: About Solution Precision and Error and Managing Memory.