heston

Heston model

Description

Creates and displays a heston object, which derives 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

Syntax

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

Heston = heston(___,Name,Value)

Description

Heston = heston(Return,Speed,Level,Volatility) creates a default heston object.

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 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.

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

Return — Return represents the parameter μ

array or deterministic function of time or deterministic function of time and state

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 X_t.

Data Types: double | function_handle

Speed — Return represents the parameter S

array or deterministic function of time or deterministic function of time and state

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 X_t.

Data Types: double | function_handle

Level — Level represents the parameter L

array or deterministic function of time or deterministic function of time and state

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 X_t.

Data Types: double | function_handle

Volatility — Volatility represents the instantaneous volatility of the CIR stochastic variance model

scalar or deterministic function of time or deterministic function of time and state

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 specifyVolatility 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 X_t.

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.

Output Arguments

Heston — Heston model

Heston object

Heston model, returned as a Heston object.

Properties

StartTime — Starting time of first observation, applied to all state variables

0 (default) | scalar

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

Data Types: double

StartState — Initial values of state variables

1 (default) | scalar, column vector, or matrix

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 — Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes)

NBrowns-by-NBrowns identity matrix representing independent Gaussian processes (default) | positive semidefinite matrix | deterministic function

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

Simulation — User-defined simulation function or SDE simulation method

simulation by Euler approximation (simByEuler) (default) | function | SDE simulation method

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

Data Types: function_handle

Drift — Drift rate component of continuous-time stochastic differential equations (SDEs)

Read-only: value stored from drift-rate function (default) | drift object or function accessible by (t, X_t)

This property is read-only.

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

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:

$$F(t,X_t)=A(t)+B(t)X_t$$

where:

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

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

The displayed parameters for a drift object are:

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

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

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

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, X_t) interface. However, you can specify the components A and B as functions that adhere to the common (t, X_t) 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 — Diffusion rate component of continuous-time stochastic differential equations (SDEs)

Read-only: value stored from diffusion-rate function (default) | diffusion object or functions accessible by (t, X_t)

This property is read-only.

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

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:

$$G(t,X_t)=D(t,X_t^{\alpha (t)})V(t)$$

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, X_t) interface.

The displayed parameters for a diffusion object are:

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

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

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

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, X_t) interface. However, you can specify the components A and B as functions that adhere to the common (t, X_t) 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 (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 process sample paths by Milstein approximation
simByMilstein2	Simulate Heston process sample paths by second order Milstein approximation

Examples

Create a heston Object

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:

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

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

Create a heston object to represent the model:

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

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

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

More About

Instance Hierarchy

There are inheritance relationships among the SDE classes.

The following figure illustrates the inheritance relationships.

For more information, see SDE Models, SDE Classes and Associated Simulation Methods, and SDE Class Hierarchy.

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

R2023b: Added simByMilstein2 method

Use the simByMilstein2 method to approximate a numerical solution of a stochastic differential equation.

R2023a: Added simByMilstein method

Use the simByMilstein method to approximate a numerical solution of a stochastic differential equation.