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Simulate Heston sample paths with transition density



[Paths,Times] = simByTransition(MDL,NPeriods) simulates NTrials sample paths of Heston bivariate models driven by two Brownian motion sources of risk. simByTransition approximates continuous-time stochastic processes by the transition density.


[Paths,Times] = simByTransition(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.


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Simulate Heston sample paths with transition density.

Define the parameters for the heston object.

Return = 0.03;
Level = 0.05;
Speed = 1.0;
Volatility = 0.2;

AssetPrice = 80;
V0 = 0.04;
Rho = -0.7;
StartState = [AssetPrice;V0];
Correlation = [1 Rho;Rho 1];

Create a heston object.

hestonObj = heston(Return,Speed,Level,Volatility,'startstate',StartState,'correlation',Correlation)
hestonObj = 
   Class HESTON: Heston Bivariate Stochastic Volatility
     Dimensions: State = 2, Brownian = 2
      StartTime: 0
     StartState: 2x1 double array 
    Correlation: 2x2 double array 
          Drift: drift rate function F(t,X(t)) 
      Diffusion: diffusion rate function G(t,X(t)) 
     Simulation: simulation method/function simByEuler
         Return: 0.03
          Speed: 1
          Level: 0.05
     Volatility: 0.2

Define the simulation parameters.

nPeriods = 5;   % Simulate sample paths over the next five years
Paths = simByTransition(hestonObj,nPeriods);
Paths = 6×2

   80.0000    0.0400
   92.9915    0.0343
  108.6211    0.0737
   52.9617    0.1012
   46.9805    0.1243
   54.3704    0.0571

Input Arguments

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Stochastic differential equation model, specified as a heston object. For more information on creating a heston object, see heston.

Data Types: object

Number of simulation periods, specified as a positive scalar integer. The value of NPeriods determines the number of rows of the simulated output series.

Data Types: double

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [Paths,Times] = simByTransition(Heston,NPeriods,'DeltaTimes',dt)

Simulated trials (sample paths) of NPeriods observations each, specified as the comma-separated pair consisting of 'NTrials' and a positive scalar integer.

Data Types: double

Positive time increments between observations, specified as the comma-separated pair consisting of 'DeltaTimes' and a scalar or NPeriods-by-1 column vector.

DeltaTime represents the familiar dt found in stochastic differential equations, and determines the times at which the simulated paths of the output state variables are reported.

Data Types: double

Number of intermediate time steps within each time increment dt (defined as DeltaTimes), specified as the comma-separated pair consisting of 'NSteps' and a positive scalar integer.

The simByTransition function partitions each time increment dt into NSteps subintervals of length dt/NSteps, and refines the simulation by evaluating the simulated state vector at NSteps − 1 intermediate points. Although simByTransition does not report the output state vector at these intermediate points, the refinement improves accuracy by enabling the simulation to more closely approximate the underlying continuous-time process.

Data Types: double

Flag for storage and return method that indicates how the output array Paths is stored and returned, specified as the comma-separated pair consisting of 'StorePaths' and a scalar logical flag with a value of True or False.

  • If StorePaths is True (the default value) or is unspecified, then simByTransition returns Paths as a three-dimensional time series array.

  • If StorePaths is False (logical 0), then simByTransition returns the Paths output array as an empty matrix.

Data Types: logical

Sequence of end-of-period processes or state vector adjustments, specified as the comma-separated pair consisting of 'Processes' and a function or cell array of functions of the form


simByTransition applies processing functions at the end of each observation period. The processing functions accept the current observation time t and the current state vector Xt, and return a state vector that might adjust the input state.

If you specify more than one processing function, simByTransition invokes the functions in the order in which they appear in the cell array.

Data Types: cell | function

Output Arguments

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Simulated paths of correlated state variables, returned as an (NPeriods + 1)-by-NVars-by-NTrials three-dimensional time series array.

For a given trial, each row of Paths is the transpose of the state vector Xt at time t. When the input flag StorePaths = False, simByTransition returns Paths as an empty matrix.

Observation times associated with the simulated paths, returned as an (NPeriods + 1)-by-1 column vector. Each element of Times is associated with the corresponding row of Paths.

More About

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Transition Density Simulation

The CIR SDE has no solution such that r(t) = f(r(0),⋯).

In other words, the equation is not explicitly solvable. However, the transition density for the process is known.

The exact simulation for the distribution of r(t_1 ),⋯,r(t_n) is that of the process at times t_1,⋯,t_n for the same value of r(0). The transition density for this process is known and is expressed as


Heston Model

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.


    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.


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


[1] Glasserman, Paul Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

[2] Van Haastrecht, Alexander, and Antoon Pelsser. "Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model." International Journal of Theoretical and Applied Finance. 13, no. 01 (2010): 1–43.

Introduced in R2020b