heston - Heston stochastic volatility models

Synopsis

heston = heston(Return, Speed, Level, Volatility)

heston = heston(Return, Speed, Level, Volatility, 'Name1', Value1, 'Name2', Value2, ...)

Class

Description

This constructor creates and displays heston objects, which derive from the SDEDDO (SDE from drift and diffusion objects) class. Use heston objects to simulate sample paths of two state variables. Each state variable is driven by a single Brownian motion source of risk over NPERIODS consecutive observation periods, approximating continuous-time stochastic volatility processes.

Heston models are bivariate composite models. Each Heston model consists of two coupled univariate models:

A geometric Brownian motion (GBM) model with a stochastic volatility function.

This model usually corresponds to a price process whose volatility (variance rate) is governed by the second univariate model.
A Cox-Ingersoll-Ross (CIR) square root diffusion model.

This model describes the evolution of the variance rate of the coupled GBM price process.

Input Arguments

Specify required input parameters as one of the following types:

A MATLAB array. Specifying an array indicates a static (non-time-varying) parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.
A MATLAB function. Specifying a function provides indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported via an interface, because all implementation details are hidden and fully encapsulated by the function.

Note You can specify combinations of array and function input parameters as needed.

The required input parameters are:

`Return`	If you specify `Return` as a scalar, it represents the expected (mean) instantaneous rate of return of the univariate GBM price model. If you specify it as a function, `Return` calculates the instantaneous rate of return of the GBM price model. This function generates a scalar when invoked with two inputs: A real-valued scalar observation time t. A 2-by-1 bivariate state vector X_t.
`Speed`	If you specify `Speed` as a scalar, it represents the mean-reversion speed of the univariate CIR stochastic variance model (the speed at which the CIR variance reverts to its long-run average level). If you specify it as a function, `Speed` calculates the speed of mean reversion of the CIR variance model. This function generates a scalar when invoked with two inputs: A real-valued scalar observation time t. A 2-by-1 state vector X_t.
`Level`	If you specify `Level` as a scalar, it represents the reversion level of the univariate CIR stochastic variance model. If you specify it as a function, `Level` calculates the reversion level of the CIR variance model. This function generates a scalar when invoked with two inputs: A real-valued scalar observation time t. A 2-by-1 state vector X_t.
`Volatility`	If you specify `Volatility` as a scalar, it represents the instantaneous volatility of the CIR stochastic variance model, often called the volatility of volatility or volatility of variance. If you specify it as a function, `Volatility` generates a scalar when invoked with two inputs: A real-valued scalar observation time t. A 2-by-1 state vector X_t.

Note Although the constructor does not enforce restrictions on the signs of any of these input arguments, each argument is usually specified as a positive value.

Optional Input Arguments

Specify optional input arguments as variable-length lists of matching parameter name/value pairs: 'Name1', Value1, 'Name2', Value2, ... and so on. The following rules apply when specifying parameter-name pairs:

Specify the parameter name as a character string, followed by its corresponding parameter value.
You can specify parameter name/value pairs in any order.
Parameter names are case insensitive.
You can specify unambiguous partial string matches.

The following table lists valid parameter names.

`StartTime`	Scalar starting time of the first observation, applied to all state variables. If you do not specify a value for `StartTime`, the default is `0`.
`StartState`	Scalar, 2-by-1 column vector, or 2-by-`NTRIALS` matrix of initial values of the state variables. If `StartState` is a scalar, `heston` applies the same initial value to both state variables on all trials. If `StartState` is a bivariate column vector, `heston` applies a unique initial value to each state variable on all trials. If `StartState` is a matrix, `heston` applies a unique initial value to each state variable on each trial. If you do not specify a value for `StartState`, all variables start at `1`.
`Correlation`	Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes). Specify `Correlation` as a scalar, a 2-by-2 positive semidefinite matrix, or as a deterministic function C(t) that accepts the current time t and returns an 2-by-2 positive semidefinite correlation matrix. A `Correlation` matrix represents a static condition. As a deterministic function of time, `Correlation` allows you to specify a dynamic correlation structure. If you do not specify a value for `Correlation`, the default is a 2-by-2 identity matrix representing independent Gaussian processes.
`Simulation`	A user-defined simulation function or SDE simulation method. If you do not specify a value for `Simulation`, the default method is simulation by Euler approximation (`simByEuler`).

Output Arguments

heston

Object of class heston with the following displayed parameters:

StartTime: Initial observation time
StartState: Initial state at StartTime
Correlation: Access function for the Correlation input, callable as a function of time
Drift: Composite drift-rate function, callable as a function of time and state
Diffusion: Composite diffusion-rate function, callable as a function of time and state
Simulation: A simulation function or method
Return: Access function for the input argument Return, callable as a function of time and state
Speed: Access function for the input argument Speed, callable as a function of time and state
Level: Access function for the input argument Level, callable as a function of time and state
Volatility: Access function for the input argument Volatility, callable as a function of time and state

Examples

See Creating Heston Stochastic Volatility Models.