GARCH Toolbox™

Introduction

The GARCH Toolbox™ software enables you to model dependent financial and economic variables, such as interest rates and equity prices, by performing 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.

Tasks you can perform using this functionality include:

• Running vectorized methods to simulate static univariate models.

• Simulating static, separable Geometric Brownian Motion and Brownian Motion multivariate models.

• Performing path-dependent analysis using end-of-period processing and state vector adjustments.

• Sampling SDEs at intermediate times without reporting those times, improving accuracy and reducing memory consumption.

• Approximating analytic solutions for separable geometric Brownian motion and Hull-White/Vasicek models.

• Reducing variance using antithetic sampling.

The SDE architecture also supports the following features:

• Euler approximation default simulation method

• Stochastic interpolation and Brownian bridge simulation methods

• Support for any combination of static and dynamic model parameters

• Support for state and Brownian vectors of arbitrary dimensionality

• Optional user-specified random number generation and dependence/correlation structure

• The ability to avoid storing state and noise time series to improve performance and memory efficiency

Monte Carlo Simulation of Stochastic Differential Equations

Terminology