# heston

`Heston`

model

## Description

Creates and displays `heston`

objects, which derive from the
`sdeddo`

(SDE from drift and diffusion
objects).

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

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

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

## Creation

### Description

creates a default `heston`

= heston(`Return`

,`Speed`

,`Level`

,`Volatility`

)`heston`

object.

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.

constructs a `heston`

= heston(___,`Name,Value`

)`heston`

object with additional options
specified by one or more `Name,Value`

pair
arguments.

`Name`

is a property name and `Value`

is
its corresponding value. `Name`

must appear inside single
quotes (`''`

). You can specify several name-value pair
arguments in any order as
`Name1,Value1,…,NameN,ValueN`

The `heston`

object has the following Properties:

`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

### Input Arguments

## Properties

## Object Functions

`interpolate` | Brownian interpolation of stochastic differential equations (SDEs) for
`SDE` , `BM` , `GBM` ,
`CEV` , `CIR` , `HWV` ,
`Heston` , `SDEDDO` , `SDELD` , or
`SDEMRD` models |

`simulate` | Simulate multivariate stochastic differential equations (SDEs) for
`SDE` , `BM` , `GBM` ,
`CEV` , `CIR` , `HWV` ,
`Heston` , `SDEDDO` , `SDELD` ,
`SDEMRD` , `Merton` , or `Bates`
models |

`simByEuler` | Euler simulation of stochastic differential equations (SDEs) for
`SDE` , `BM` , `GBM` ,
`CEV` , `CIR` , `HWV` ,
`Heston` , `SDEDDO` , `SDELD` , or
`SDEMRD` models |

`simByQuadExp` | Simulate `Bates` , `Heston` , and
`CIR` sample paths by quadratic-exponential discretization
scheme |

`simByTransition` | Simulate `Heston` sample paths with transition
density |

`simByMilstein` | Simulate `Heston` sample paths by Milstein
approximation |

## Examples

## More About

## Algorithms

When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.

Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these parameters with no inputs, they behave like simple properties and allow you to test the data type (double vs. function, or equivalently, static vs. dynamic) of the original input specification. This is useful for validating and designing methods.

When you invoke these parameters with inputs, they behave like functions, giving the
impression of dynamic behavior. The parameters accept the observation time
*t* and a state vector
*X _{t}*, and return an array of appropriate
dimension. Even if you originally specified an input as an array,

`heston`

treats it as a static function of time and state, by that
means guaranteeing that all parameters are accessible by the same interface.## References

[1] Aït-Sahalia, Yacine. “Testing
Continuous-Time Models of the Spot Interest Rate.” *Review of Financial
Studies*, vol. 9, no. 2, Apr. 1996, pp. 385–426.

[2] Aït-Sahalia, Yacine.
“Transition Densities for Interest Rate and Other Nonlinear Diffusions.” *The
Journal of Finance*, vol. 54, no. 4, Aug. 1999, pp.
1361–95.

[3] Glasserman, Paul.
*Monte Carlo Methods in Financial Engineering*. Springer,
2004.

[4] Hull, John.
*Options, Futures and Other Derivatives*. 7th ed, Prentice
Hall, 2009.

[5] Johnson, Norman Lloyd, et al.
*Continuous Univariate Distributions*. 2nd ed, Wiley,
1994.

[6] Shreve, Steven E.
*Stochastic Calculus for Finance*. Springer,
2004.

## Version History

**Introduced in R2008a**

## See Also

`drift`

| `diffusion`

| `sdeddo`

| `simulate`

| `interpolate`

| `simByEuler`

| `nearcorr`

### Topics

- Creating Heston Stochastic Volatility Models
- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Price American Basket Options Using Standard Monte Carlo and Quasi-Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Quasi-Monte Carlo Simulation
- Performance Considerations