# simByTransition

Simulate Heston sample paths with transition density

Since R2020b

## Description

[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.

example

[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.

You can perform quasi-Monte Carlo simulations using the name-value arguments for MonteCarloMethod, QuasiSequence, and BrownianMotionMethod. For more information, see Quasi-Monte Carlo Simulation.

example

## Examples

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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
Paths = 6×2

80.0000    0.0400
99.7718    0.0201
124.7044    0.0176
52.7914    0.1806
75.3173    0.1732
76.7572    0.1169

This example shows how to use simByTransition with a Heston model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead pseudo random numbers.

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.

Heston = heston(Return,Speed,Level,Volatility,'startstate',StartState,'correlation',Correlation)
Heston =
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

Perform a quasi-Monte Carlo simulation by using simByTransition with the optional name-value argument for 'MonteCarloMethod', 'QuasiSequence', and 'BrownianMotionMethod'.

[paths,time] = simByTransition(Heston,10,'ntrials',4096,'MonteCarloMethod','quasi','QuasiSequence','sobol','BrownianMotionMethod','principal-components');

## 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 pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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

Monte Carlo method to simulate stochastic processes, specified as the comma-separated pair consisting of 'MonteCarloMethod' and a string or character vector with one of the following values:

• "standard" — Monte Carlo using pseudo random numbers

• "quasi" — Quasi-Monte Carlo using low-discrepancy sequences

• "randomized-quasi" — Randomized quasi-Monte Carlo

Data Types: string | char

Low discrepancy sequence to drive the stochastic processes, specified as the comma-separated pair consisting of 'QuasiSequence' and a string or character vector with the following value:

• "sobol" — Quasi-random low-discrepancy sequences that use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension

Note

If MonteCarloMethod option is not specified or specified as"standard", QuasiSequence is ignored.

Data Types: string | char

Brownian motion construction method, specified as the comma-separated pair consisting of 'BrownianMotionMethod' and a string or character vector with one of the following values:

• "standard" — The Brownian motion path is found by taking the cumulative sum of the Gaussian variates.

• "brownian-bridge" — The last step of the Brownian motion path is calculated first, followed by any order between steps until all steps have been determined.

• "principal-components" — The Brownian motion path is calculated by minimizing the approximation error.

The starting point for a Monte Carlo simulation is the construction of a Brownian motion sample path (or Wiener path). Such paths are built from a set of independent Gaussian variates, using either standard discretization, Brownian-bridge construction, or principal components construction.

Both standard discretization and Brownian-bridge construction share the same variance and, therefore, the same resulting convergence when used with the MonteCarloMethod using pseudo random numbers. However, the performance differs between the two when the MonteCarloMethod option "quasi" is introduced, with faster convergence for the "brownian-bridge" construction option and the fastest convergence for the "principal-components" construction option.

Data Types: string | char

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

${X}_{t}=P\left(t,{X}_{t}\right)$

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.

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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

$\begin{array}{l}r\left(t\right)=\frac{{\sigma }^{2}\left(1-{e}^{-\alpha \left(t-u\right)}}{4\alpha }{x}_{d}^{2}\left(\frac{4\alpha {e}^{-\alpha \left(t-u\right)}}{{\sigma }^{2}\left(1-{e}^{-\alpha \left(t-u\right)}\right)}r\left(u\right)\right),t>u\\ \text{where}\\ d\equiv \frac{4b\alpha }{{\sigma }^{2}}\end{array}$

### 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.

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

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.

## References

[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, Vol. 13, No. 01 (2010): 1–43.

## Version History

Introduced in R2020b

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