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Euler simulation of stochastic differential equations (SDEs)



[Paths,Times,Z] = simByEuler(MDL,NPeriods) simulates NTrials sample paths of NVars correlated state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods. simByEuler uses the Euler approach to approximate continuous-time stochastic processes.


[Paths,Times,Z] = simByEuler(___,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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Load the Data and Specify the SDE Model

load Data_GlobalIdx2
prices  = [Dataset.TSX Dataset.CAC Dataset.DAX ...
    Dataset.NIK Dataset.FTSE Dataset.SP];

returns =  tick2ret(prices);

nVariables  = size(returns,2);
expReturn   = mean(returns);
sigma       = std(returns);
correlation = corrcoef(returns);
t           = 0;
X           = 100;
X           = X(ones(nVariables,1));

F = @(t,X) diag(expReturn)* X;
G = @(t,X) diag(X) * diag(sigma);

SDE = sde(F, G, 'Correlation', ...
    correlation, 'StartState', X);

Simulate a Single Path Over a Year

nPeriods = 249;      % # of simulated observations
dt       =   1;      % time increment = 1 day
[S,T] = simByEuler(SDE, nPeriods, 'DeltaTime', dt);

Simulate 10 trials and examine the SDE model

[S,T] = simulate(SDE, nPeriods, 'DeltaTime', dt, 'nTrials', 10);

whos S
  Name        Size               Bytes  Class     Attributes

  S         250x6x10            120000  double              

Plot the paths

plot(T, S(:,:,1)), xlabel('Trading Day'), ylabel('Price')
title('First Path of Multi-Dimensional Market Model')
legend({'Canada' 'France' 'Germany' 'Japan' 'UK' 'US'},...
    'Location', 'Best')

Figure contains an axes object. The axes object with title First Path of Multi-Dimensional Market Model contains 6 objects of type line. These objects represent Canada, France, Germany, Japan, UK, US.

The Cox-Ingersoll-Ross (CIR) short rate class derives directly from SDE with mean-reverting drift (SDEMRD): dXt=S(t)[L(t)-Xt]dt+D(t,Xt12)V(t)dW

where D is a diagonal matrix whose elements are the square root of the corresponding element of the state vector.

Create a cir object to represent the model: dXt=0.2(0.1-Xt)dt+0.05Xt12dW.

CIR = cir(0.2, 0.1, 0.05)  % (Speed, Level, Sigma)
CIR = 
   Class CIR: Cox-Ingersoll-Ross
     Dimensions: State = 1, Brownian = 1
      StartTime: 0
     StartState: 1
    Correlation: 1
          Drift: drift rate function F(t,X(t)) 
      Diffusion: diffusion rate function G(t,X(t)) 
     Simulation: simulation method/function simByEuler
          Sigma: 0.05
          Level: 0.1
          Speed: 0.2

Simulate a single path over a year using simByEuler.

nPeriods = 249;      % # of simulated observations
dt       =   1;      % time increment = 1 day
[Paths,Times] = simByEuler(CIR,nPeriods,'Method','higham-mao','DeltaTime', dt)
Paths = 250×1


Times = 250×1


Input Arguments

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Stochastic differential equation model, specified as an sde, bm, gbm, cev, cir, hwv, heston, sdeddo, sdeld, or sdemrd object.

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,Z] = simByEuler(SDE,NPeriods,'DeltaTime',dt)

Method to handle negative values, specified as the comma-separated pair consisting of 'Method' and a character vector or string with a supported value.


The Method argument is only supported when using a CIR object. For more information on creating a CIR object, see cir.

Data Types: char | string

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.

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

The simByEuler 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 simByEuler 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

Flag indicates whether simByEuler uses antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes). This argument is specified as the comma-separated pair consisting of 'Antithetic' and a scalar logical flag with a value of True or False.

When you specify True, simByEuler 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.


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 used to generate the Brownian motion vector (Wiener process) that drives the simulation. This argument is specified as the comma-separated pair consisting of 'Z' and a function or as 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

Flag 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, simByEuler returns Paths as a three-dimensional time series array.

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

Data Types: logical

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


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

If you specify more than one processing function, simByEuler 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.

Data Types: cell | function

Output Arguments

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Simulated paths of correlated state variables, 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 the input flag StorePaths = False, simByEuler returns Paths as an empty matrix.

Observation times 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 used to generate the Brownian motion vector (Wiener processes) that drive the simulation, returned as a (NPeriods ⨉ NSteps)-by-NBrowns-by-NTrials three-dimensional time series array.

More About

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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.


This function simulates any vector-valued SDE of the form



  • X is an NVars-by-1 state vector of process variables (for example, short rates or equity prices) to simulate.

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

  • F is an NVars-by-1 vector-valued drift-rate function.

  • G is an NVars-by-NBrowns matrix-valued diffusion-rate function.

simByEuler simulates NTrials sample paths of NVars correlated state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, using the Euler approach to approximate continuous-time stochastic processes.

  • This simulation engine provides a discrete-time approximation of the underlying generalized continuous-time process. The simulation is derived directly from the stochastic differential equation of motion. Thus, the discrete-time process approaches the true continuous-time process only as DeltaTime approaches zero.

  • The input argument Z allows you to directly specify the noise-generation process. This process takes precedence over the Correlation parameter of the sde object and the value of the Antithetic input flag. If you do not specify a value for Z, simByEuler generates correlated Gaussian variates, with or without antithetic sampling as requested.

  • The end-of-period Processes argument allows you to terminate a given trial early. At the end of each time step, simByEuler 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 a user-defined 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).


[1] Deelstra, G. and F. Delbaen. “Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term.” Applied Stochastic Models and Data Analysis., 1998, vol. 14, no. 1, pp. 77–84.

[2] Higham, Desmond, and Xuerong Mao. “Convergence of Monte Carlo Simulations Involving the Mean-Reverting Square Root Process.” The Journal of Computational Finance, vol. 8, no. 3, 2005, pp. 35–61.

[3] Lord, Roger, et al. “A Comparison of Biased Simulation Schemes for Stochastic Volatility Models.” Quantitative Finance, vol. 10, no. 2, Feb. 2010, pp. 177–94

Introduced in R2008a