# 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.

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

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.

## Creation

### Syntax

``heston = heston(Return,Level,Speed,Volatility)``
``heston = heston(___,Name,Value)``

### Description

example

````heston = heston(Return,Level,Speed,Volatility)` creates a default `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. NoteYou 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. ```

example

````heston = heston(___,Name,Value)` constructs a `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

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`Return` represents the parameter μ, specified as an array or deterministic function of time.

If you specify `Return` as an array, it must be an `NVars`-by-`NVars` matrix representing the expected (mean) instantaneous rate of return.

As a deterministic function of time, when `Return` is called with a real-valued scalar time `t` as its only input, `Return` must produce an `NVars`-by-`NVars` matrix. If you specify `Return` as a function of time and state, it must return an `NVars`-by-`NVars` matrix when invoked with two inputs:

• A real-valued scalar observation time t.

• An `NVars`-by-`1` state vector Xt.

Data Types: `double` | `function_handle`

`Level` represents the parameter L, specified as an array or deterministic function of time.

If you specify `Level` as an array, it must be an `NVars`-by-`1` column vector of reversion levels.

As a deterministic function of time, when `Level` is called with a real-valued scalar time `t` as its only input, `Level` must produce an `NVars`-by-`1` column vector. If you specify `Level` as a function of time and state, it must generate an `NVars`-by-`1` column vector of reversion levels when called with two inputs:

• A real-valued scalar observation time t.

• An `NVars`-by-`1` state vector Xt.

Data Types: `double` | `function_handle`

`Speed` represents the parameter S, specified as an array or deterministic function of time.

If you specify `Speed` as an array, it must be an `NVars`-by-`NVars` matrix of mean-reversion speeds (the rate at which the state vector 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 an `NVars`-by-`NVars` matrix. If you specify `Speed` as a function of time and state, it calculates the speed of mean reversion. This function must generate an `NVars`-by-`NVars` matrix of reversion rates when called with two inputs:

• A real-valued scalar observation time t.

• An `NVars`-by-`1` state vector Xt.

Data Types: `double` | `function_handle`

`Volatility` (often called the volatility of volatility or volatility of variance) represents the instantaneous volatility of the CIR stochastic variance model, specified as a scalar, or deterministic function of time.

If you specify`Volatility` as a scalar, it represents the instantaneous volatility of the CIR stochastic variance model.

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.

Data Types: `double` | `function_handle`

Note

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

## Properties

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Starting time of first observation, applied to all state variables, specified as a scalar

Data Types: `double`

Initial values of state variables, specified as a scalar, column vector, or matrix.

If `StartState` is a scalar, `heston` applies the same initial value to all state variables on all trials.

If `StartState` is a 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.

Data Types: `double`

Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes), specified as an `NBrowns`-by-`NBrowns` positive semidefinite matrix, or as a deterministic function C(t) that accepts the current time t and returns an `NBrowns`-by-`NBrowns` positive semidefinite correlation matrix. If `Correlation` is not a symmetric positive semidefinite matrix, use `nearcorr` to create a positive semidefinite matrix for a correlation matrix.

A `Correlation` matrix represents a static condition.

As a deterministic function of time, `Correlation` allows you to specify a dynamic correlation structure.

Data Types: `double`

User-defined simulation function or SDE simulation method, specified as a function or SDE simulation method.

Data Types: `function_handle`

Drift rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The drift rate specification supports the simulation of sample paths of `NVars` state variables driven by `NBrowns` Brownian motion sources of risk over `NPeriods` consecutive observation periods, approximating continuous-time stochastic processes.

The `drift` class allows you to create drift-rate objects using `drift` of the form:

`$F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}$`

where:

• `A` is an `NVars`-by-`1` vector-valued function accessible using the (t, Xt) interface.

• `B` is an `NVars`-by-`NVars` matrix-valued function accessible using the (t, Xt) interface.

The displayed parameters for a `drift` object are:

• `Rate`: The drift-rate function, F(t,Xt)

• `A`: The intercept term, A(t,Xt), of F(t,Xt)

• `B`: The first order term, B(t,Xt), of F(t,Xt)

`A` and `B` enable you to query the original inputs. The function stored in `Rate` fully encapsulates the combined effect of `A` and `B`.

When specified as MATLAB double arrays, the inputs `A` and `B` are clearly associated with a linear drift rate parametric form. However, specifying either `A` or `B` as a function allows you to customize virtually any drift rate specification.

Note

You can express `drift` and `diffusion` classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components `A` and `B` as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: ```F = drift(0, 0.1) % Drift rate function F(t,X)```

Data Types: `struct` | `double`

Diffusion rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The diffusion rate specification supports the simulation of sample paths of `NVars` state variables driven by `NBrowns` Brownian motion sources of risk over `NPeriods` consecutive observation periods, approximating continuous-time stochastic processes.

The `diffusion` class allows you to create diffusion-rate objects using `diffusion`:

`$G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)$`

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 an `NVars`-by-`1` vector-valued function.

• `V` is an `NVars`-by-`NBrowns` matrix-valued volatility rate function `Sigma`.

• `Alpha` and `Sigma` are also accessible using the (t, Xt) interface.

The displayed parameters for a `diffusion` object are:

• `Rate`: The diffusion-rate function, G(t,Xt).

• `Alpha`: The state vector exponent, which determines the format of D(t,Xt) of G(t,Xt).

• `Sigma`: The volatility rate, V(t,Xt), of G(t,Xt).

`Alpha` and `Sigma` enable you to query the original inputs. (The combined effect of the individual `Alpha` and `Sigma` parameters is fully encapsulated by the function stored in `Rate`.) The `Rate` functions are the calculation engines for the `drift` and `diffusion` objects, and are the only parameters required for simulation.

Note

You can express `drift` and `diffusion` classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components `A` and `B` as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: ```G = diffusion(1, 0.3) % Diffusion rate function G(t,X) ```

Data Types: `struct` | `double`

## Object Functions

 `interpolate` Brownian interpolation of stochastic differential equations `simulate` Simulate multivariate stochastic differential equations (SDEs) `simByEuler` Euler simulation of stochastic differential equations (SDEs) `simByQuadExp` Simulate Bates, Heston, and CIR sample paths by quadratic-exponential discretization scheme `simByTransition` Simulate Heston sample paths with transition density

## Examples

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The Heston (`heston`) class derives directly from SDE from Drift and Diffusion (`sdeddo`). Each Heston model is a bivariate composite model, consisting of two coupled univariate models:

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

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

Create a `heston` object to represent the model:

`$d{X}_{1t}=0.1{X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}$`

`$d{X}_{2t}=0.2\left[0.1-{X}_{2t}\right]dt+0.05\sqrt{{X}_{2t}}d{W}_{2t}$`

`obj = heston (0.1, 0.2, 0.1, 0.05) % (Return, Speed, Level, Volatility)`
```obj = Class HESTON: Heston Bivariate Stochastic Volatility ---------------------------------------------------- Dimensions: State = 2, Brownian = 2 ---------------------------------------------------- StartTime: 0 StartState: 1 (2x1 double array) Correlation: 2x2 diagonal double array Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Return: 0.1 Speed: 0.2 Level: 0.1 Volatility: 0.05 ```

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## 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 Xt, 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.

 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.

 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.

 Glasserman, Paul. Monte Carlo Methods in Financial Engineering. Springer, 2004.

 Hull, John. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.

 Johnson, Norman Lloyd, et al. Continuous Univariate Distributions. 2nd ed, Wiley, 1994.

 Shreve, Steven E. Stochastic Calculus for Finance. Springer, 2004.