GARCH Toolbox™

Parametric Specification

General Parametric Specification

The SDE engine allows the simulation of generalized multivariate stochastic processes, and provides a flexible and powerful simulation architecture. The framework also provides you with utilities and model classes that offer a variety of parametric specifications and interfaces. The architecture is fully multidimensional in both the state vector and the Brownian motion, and offers both linear and mean-reverting drift-rate specifications.

You can specify most parameters as MATLAB^® arrays or as functions accessible by a common interface, that support general dynamic/nonlinear relationships common in SDE simulation. Specifically, you can simulate correlated paths of any number of state variables driven by a vector-valued Brownian motion of arbitrary dimensionality. This simulation approximates the underlying multivariate continuous-time process using a vector-valued stochastic difference equation.

Back to Top

General SDEs

Consider the following general stochastic differential equation:

(5-1)

where:

• X is an NVARS-by-1 state vector of process variables (for example, short rates or equity prices) to simulate.

• W is an NBROWNS-by-1 Brownian motion vector.

• F is an NVARS-by-1 vector-valued drift-rate function.

• G is an NVARS-by-NBROWNS matrix-valued diffusion-rate function.

The drift and diffusion rates, F and G, respectively, are general functions of a real-valued scalar sample time t and state vector X_t. Also, static (non-time-variable) coefficients are simply a special case of the more general dynamic (time-variable) situation, just as a function can be a trivial constant; for example, f(t,X_t) = 4. The SDE in Equation 5-1 is useful in implementing derived classes that impose additional structure on the drift and diffusion-rate functions.

Back to Top

Drift and Diffusion Specifications

For example, an SDE with a linear drift rate has the form:

(5-2)

where A is an NVARS-by-1 vector-valued function and B is an NVARS-by-NVARS matrix-valued function.

As an alternative, consider a drift-rate specification expressed in mean-reverting form:

(5-3)

where S is an NVARS-by-NVARS matrix-valued function of mean reversion speeds (that is, rates of mean reversion), and L is an NVARS-by-1 vector-valued function of mean reversion levels (that is, long run average level).

Similarly, consider the following diffusion-rate specification:

(5-4)

where D is an NVARS-by-NVARS diagonal matrix-valued function. Each diagonal element of D is the corresponding element of the state vector raised to the corresponding element of an exponent Alpha, which is also an NVARS-by-1 vector-valued function. V is an NVARS-by-NBROWNS matrix-valued function of instantaneous volatility rates. Each row of V corresponds to a particular state variable, and each column corresponds to a particular Brownian source of uncertainty. V associates the exposure of state variables with sources of risk.

The parametric specifications for the drift and diffusion-rate functions associate parametric restrictions with familiar models derived from the general SDE class, and provide coverage for many popular models.

As discussed in the following sections, the class system and hierarchy of the SDE engine use industry-standard technology to provide simplified interfaces for many models by placing user-transparent restrictions on drift and diffusion specifications. This design allows you to mix and match existing models, and customize drift or diffusion-rate functions.

For example, the following popular models are simply special cases of the general SDE model.

Popular Models

Back to Top

Behavior and Syntax of SDE Objects

Using SDE Objects to Create Models