Simulate Bates, Heston, and CIR sample paths by quadratic-exponential discretization scheme

## Description

example

[Paths,Times,Z] = simByQuadExp(MDL,NPeriods) simulates NTrials sample paths of a Heston model driven by two Brownian motion sources of risk, or a CIR model driven by one Brownian motion source of risk. Both Heston and Bates models approximate continuous-time stochastic processes by a quadratic-exponential discretization scheme. The simByQuadExp simulation derives directly from the stochastic differential equation of motion; the discrete-time process approaches the true continuous-time process only in the limit as DeltaTimes approaches zero.

example

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

example

[Paths,Times,Z,N] = simByQuadExp(MDL,NPeriods) simulates NTrials sample paths of a Bates model driven by two Brownian motion sources of risk, approximating continuous-time stochastic processes by a quadratic-exponential discretization scheme. The simByQuadExp simulation derives directly from the stochastic differential equation of motion; the discrete-time process approaches the true continuous-time process only in the limit as DeltaTimes approaches zero.

example

[Paths,Times,Z,N] = simByQuadExp(___,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.

## Examples

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Create a bates object.

AssetPrice = 80;
Return = 0.03;
JumpMean = 0.02;
JumpVol = 0.08;
JumpFreq = 0.1;

V0 = 0.04;
Level = 0.05;
Speed = 1.0;
Volatility = 0.2;
Rho = -0.7;
StartState = [AssetPrice;V0];
Correlation = [1 Rho;Rho 1];

batesObj = bates(Return, Speed, Level, Volatility,...
JumpFreq, JumpMean, JumpVol,'startstate',StartState,...
'correlation',Correlation)
batesObj =
Class BATES: Bates 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
JumpFreq: 0.1
JumpMean: 0.02
JumpVol: 0.08

Use simByQuadExp to simulate NTrials sample paths directly from the stochastic differential equation of motion; the discrete-time process approaches the true continuous-time process only in the limit as DeltaTimes approaches zero.

NPeriods = 2;
Paths = 3×2

80.0000    0.0400
64.3377    0.1063
31.5703    0.1009

Times = 3×1

0
1
2

Z = 2×2

0.5377    1.8339
-2.2588    0.8622

N = 2×1

0
0

The output Paths is returned as a (NPeriods + 1)-by-NVars-by-NTrials three-dimensional time-series array.

This example shows how to use simByQuadExp with a Bates 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 bates object.

AssetPrice = 80;
Return = 0.03;
JumpMean = 0.02;
JumpVol = 0.08;
JumpFreq = 0.1;

V0 = 0.04;
Level = 0.05;
Speed = 1.0;
Volatility = 0.2;
Rho = -0.7;
StartState = [AssetPrice;V0];
Correlation = [1 Rho;Rho 1];

Create a bates object.

Bates = bates(Return, Speed, Level, Volatility, ...
JumpFreq, JumpMean, JumpVol,'startstate',StartState, ...
'correlation',Correlation)
Bates =
Class BATES: Bates 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
JumpFreq: 0.1
JumpMean: 0.02
JumpVol: 0.08

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

## Input Arguments

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Stochastic differential equation model, specified as a bates, heston, or cir object. You can create these objects using bates, heston, or cir.

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.

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 a NPeriods-by-1 column vector.

DeltaTimes 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 (specified as DeltaTimes), specified as the comma-separated pair consisting of 'NSteps' and a positive scalar integer.

The simByQuadExp 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 simByQuadExp does not report the output state vector at these intermediate points, the refinement improves accuracy by allowing 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.

Note

If you specify an input noise process (see Z and N), simByQuadExp ignores the value of MonteCarloMethod.

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 one of the following values:

• "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.

Note

If an input noise process is specified using the Z input argument, BrownianMotionMethod is ignored.

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 seen for "brownian-bridge" construction option and the fastest convergence when using the "principal-components" construction option.

Data Types: string | char

Flag to use antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes), specified as the comma-separated pair consisting of 'Antithetic' and a scalar numeric or logical 1 (true) or 0 (false).

When you specify true, simByQuadExp performs sampling such that all primary and antithetic paths are simulated and stored in successive matching pairs:

• Odd trials (1,3,5,...) correspond to the primary Gaussian paths.

• Even trials (2,4,6,...) are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.

Note

If you specify an input noise process (see Z), simByEuler ignores the value of Antithetic.

Data Types: logical

Direct specification of the dependent random noise process for generating the Brownian motion vector (Wiener process) that drives the simulation, specified as the comma-separated pair consisting of 'Z' and a function or an (NPeriods ⨉ NSteps)-by-NBrowns-by-NTrials three-dimensional array of dependent random variates.

If you specify Z as a function, it must return an NBrowns-by-1 column vector, and you must call it with two inputs:

• A real-valued scalar observation time t

• An NVars-by-1 state vector Xt

Data Types: double | function

Dependent random counting process for generating the number of jumps, specified as the comma-separated pair consisting of 'N' and a function or an (NPeriodsNSteps) -by-NJumps-by-NTrials three-dimensional array of dependent random variates. If you specify a function, N must return an NJumps-by-1 column vector, and you must call it with two inputs: a real-valued scalar observation time t followed by an NVars-by-1 state vector Xt.

Note

The N name-value pair argument is supported only when you use a bates object for the MDL input argument.

Data Types: double | function

Flag that indicates how the output array Paths is stored and returned, specified as the comma-separated pair consisting of 'StorePaths' and a scalar numeric or logical 1 (true) or 0 (false).

If StorePaths is true (the default value) or is unspecified, simByQuadExp returns Paths as a three-dimensional time-series array.

If StorePaths is false (logical 0), simByQuadExp returns Paths 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

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

The simByQuadExp function runs processing functions at each interpolation time. The functions must accept the current interpolation time t, and the current state vector Xt and return a state vector that can be an adjustment to the input state.

If you specify more than one processing function, simByQuadExp invokes the functions in the order in which they appear in the cell array. You can use this argument to specify boundary conditions, prevent negative prices, accumulate statistics, plot graphs, and more.

The end-of-period Processes argument allows you to terminate a given trial early. At the end of each time step, simByQuadExp tests the state vector Xt for an all-NaN condition. Thus, to signal an early termination of a given trial, all elements of the state vector Xt must be NaN. This test enables you to define a Processes function to signal early termination of a trial, and offers significant performance benefits in some situations (for example, pricing down-and-out barrier options).

Data Types: cell | function

## Output Arguments

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Simulated paths of correlated state variables for a heston, bates, or cir model, returned as a (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 StorePaths is set to false, simByQuadExp returns Paths as an empty matrix.

Observation times for a heston, bates, or cir model associated with the simulated paths, returned as a (NPeriods + 1)-by-1 column vector. Each element of Times is associated with the corresponding row of Paths.

Dependent random variates for a heston, bates, or cir model for generating the Brownian motion vector (Wiener processes) that drive the simulation, returned as an (NPeriods ⨉ NSteps)-by-NBrowns-by-NTrials three-dimensional time-series array.

Dependent random variates for a bates model for generating the jump counting process vector, returned as a (NPeriods ⨉ NSteps)-by-NJumps-by-NTrials three-dimensional time-series array.

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### Antithetic Sampling

Simulation methods allow you to specify a popular variance reduction technique called antithetic sampling.

This technique attempts to replace one sequence of random observations with another of the same expected value, but smaller variance. In a typical Monte Carlo simulation, each sample path is independent and represents an independent trial. However, antithetic sampling generates sample paths in pairs. The first path of the pair is referred to as the primary path, and the second as the antithetic path. Any given pair is independent of any other pair, but the two paths within each pair are highly correlated. Antithetic sampling literature often recommends averaging the discounted payoffs of each pair, effectively halving the number of Monte Carlo trials.

This technique attempts to reduce variance by inducing negative dependence between paired input samples, ideally resulting in negative dependence between paired output samples. The greater the extent of negative dependence, the more effective antithetic sampling is.

## Algorithms

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### Heston

In the Heston stochastic volatility model, the asset value process and volatility process are defined as

$\begin{array}{l}dS\left(t\right)=\gamma \left(t\right)S\left(t\right)dt+\sqrt{V\left(t\right)}S\left(t\right)d{W}_{S}\left(t\right)\\ dV\left(t\right)=\kappa \left(\theta -V\left(t\right)\right)dt+\sigma \sqrt{V\left(t\right)}d{W}_{V}\left(t\right)\end{array}$

Here:

• γ is the continuous risk-free rate.

• θ is a long-term variance level.

• κ is the mean reversion speed for the variance.

• σ is the volatility of volatility.

### CIR

You can simulate any vector-valued CIR process of the form

$d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)d{W}_{t}$

Here:

• Xt is an NVars-by-1 state vector of process variables.

• S is an NVars-by-NVars matrix of mean reversion speeds (the rate of mean reversion).

• L is an NVars-by-1 vector of mean reversion levels (long-run mean or level).

• D is an NVars-by-NVars diagonal matrix, where each element along the main diagonal is the square root of the corresponding element of the state vector.

• V is an NVars-by-NBrowns instantaneous volatility rate matrix.

• dWt is an NBrowns-by-1 Brownian motion vector.

### Bates

Bates models are bivariate composite models. Each Bates model consists of two coupled univariate models.

• A geometric Brownian motion (gbm) model with a stochastic volatility function and jumps that is expressed as follows.

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

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 that is expressed as follows.

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

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

## References

[1] Andersen, Leif. “Simple and Efficient Simulation of the Heston Stochastic Volatility Model.” The Journal of Computational Finance 11, no. 3 (March 2008): 1–42.

[2] Broadie, M., and O. Kaya. “Exact Simulation of Option Greeks under Stochastic Volatility and Jump Diffusion Models.” In Proceedings of the 2004 Winter Simulation Conference, 2004., 2:535–43. Washington, D.C.: IEEE, 2004.

[3] Broadie, Mark, and Özgür Kaya. “Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes.” Operations Research 54, no. 2 (April 2006): 217–31.

## Version History

Introduced in R2020a

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