# roughheston

## Description

Creates and displays a `roughheston`

object.

Use `roughheston`

to create a rough Heston stochastic volatility
model. Rough Heston models are bivariate composite models, composed of two coupled and
dissimilar univariate models, each driven by a single Brownian motion source of risk
over `NPERIODS`

consecutive observation periods, approximating
continuous-time rough Heston stochastic volatility processes.

The first univariate model is a geometric Brownian motion (GBM) model with a stochastic volatility function, and usually corresponds to a price process whose variance rate is governed by the second univariate model. The second model incorporates the fractional Brownian motion (fBm) into a Cox-Ingersoll-Ross (CIR) square root diffusion process that describes the evolution of the variance rate of the coupled GBM price process.

## Creation

### Syntax

### Description

creates a default `RoughHeston`

= roughheston(`Return`

,`Speed`

,`Level`

,`Volatility`

,`Alpha`

)`RoughHeston`

object with Properties for these required
arguments.

Specify the 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 through 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.

creates a `RoughHeston`

= roughheston(___,`Name=Value`

)`RoughHeston`

object with additional options
specified by one or more name-value arguments to set the properties.

### Input Arguments

## Properties

## Object Functions

`simByEuler` | Simulate `RVM` , `roughbergomi` , or
`roughheston` sample paths by Euler approximation |

## Examples

## More About

## Algorithms

The rough Heston model is a type of stochastic volatility model, which means it assumes that the volatility of the underlying asset is not constant but varies over time and is not necessarily correlated with the asset price.

$$\begin{array}{l}d{X}_{t}=B(t){X}_{t}dt+\sqrt{{Y}_{t}}{X}_{t}d{W}_{t}\\ {Y}_{t}={Y}_{0}+\frac{1}{\Gamma (\alpha +1)}{\displaystyle {\int}_{0}^{t}{(t-s)}^{\alpha}\kappa (\theta -{Y}_{s)}ds+\frac{1}{\Gamma (\alpha +1)}{\displaystyle {\int}_{0}^{t}{(t-s)}^{\alpha}\sigma}\sqrt{{Y}_{s}}d{W}_{s}}\end{array}$$

where ɑ = – ½ and *H* is the Hurst exponent.

The first equation is a geometric Brownian motion model with a stochastic volatility function. By adding the Volterra kernel, the Heston stochastic volatility model allows for a rough behavior of the volatility.

The Brownian semistationary process
(*Y*_{t} in the general
rough volatility model) has *W* as a two-sided Brownian motion
providing the fundamental noise innovations where the amplitude is modulated by a
stochastic volatility process that depends on *W*. This driving noise
is then convolved with a deterministic kernel function *g* that
specifies the dependence structure of
*Y*_{t}. The process
*Y*_{t} is also viewed as a
moving average of volatility modulated Brownian noise and, when setting the volatility
of volatility = `1`

, the stationary Brownian moving averages are nested
in this class of processes.

## References

[1] Euch, O. E. and M. Rosenbaum.
“The Characteristic Function of Rough Heston.” *Mathematical
Finance*. Vol. 29, No. 1 , 2019, pp. 3–38.

## Version History

**Introduced in R2024b**

## See Also

`rvm`

| `roughbergomi`

| `simByEuler`

| `nearcorr`

### Topics

- 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