# Documentation

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# heston

Construct Heston model

## Synopsis

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

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

`heston`

## 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.

`$d{X}_{1t}=B\left(t\right){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}$`

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.

`$d{X}_{2t}=S\left(t\right)\left[L\left(t\right)-{X}_{2t}\right]dt+V\left(t\right)\sqrt{{X}_{2t}}d{W}_{2t}$`

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.

Moreover, a parameter is identified as a deterministic function of time if the function accepts a scalar time `t` as its only input argument. Otherwise, a parameter is assumed to be a function of time t and state X(t) and is invoked with both input arguments.

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. As a deterministic function of time, when `Return` is called with a real-valued scalar time `t` as its only input, `Return` must produce a scalar. If you specify it as a function of time and state, `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 Xt. `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). As a deterministic function of time, when `Speed` is called with a real-valued scalar time `t` as its only input, `Speed` must produce a scalar. If you specify it as a function time and state, `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 Xt. `Level` If you specify `Level` as a scalar, it represents the reversion level of the univariate CIR stochastic variance model. As a deterministic function of time, when `Level` is called with a real-valued scalar time `t` as its only input, `Level` must produce a scalar. If you specify it as a function time and state, `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 Xt. `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. As a deterministic function of time, when `Volatility` is called with a real-valued scalar time `t` as its only input, `Volatility` must produce a scalar. If you specify it as a function time and state, `Volatility` generates a scalar when invoked with two inputs:A real-valued scalar observation time t.A `2`-by-`1` state vector Xt.

### Note

Although the constructor does not enforce restrictions on the signs of any of these input arguments, each argument is 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 vector, 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 character vector 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 a `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

## References

Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest Rate.” The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.

Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, Vol. 54, No. 4, August 1999.

Glasserman, P. Monte Carlo Methods in Financial Engineering. New York, Springer-Verlag, 2004.

Hull, J. C. Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.

Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 2, 2nd ed. New York, John Wiley & Sons, 1995.

Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer-Verlag, 2004.