simByEuler

Simulate Bates sample paths by Euler approximation

Syntax

[Paths,Times,Z,N] = simByEuler(MDL,NPeriods)

[Paths,Times,Z,N] = simByEuler(___,Name,Value)

Description

[Paths,Times,Z,N] = simByEuler(MDL,NPeriods) simulates NTrials sample paths of Bates bivariate models driven by NBrowns Brownian motion sources of risk and NJumps compound Poisson processes representing the arrivals of important events over NPeriods consecutive observation periods. The simulation approximates continuous-time stochastic processes by the Euler approach.

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

Examples

Simulate Bates Sample Paths by Euler Approximation

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 simByEuler to simulate NTrials sample paths of this Bates bivariate model driven by NBrowns Brownian motion sources of risk and NJumps compound Poisson processes representing the arrivals of important events over NPeriods consecutive observation periods. This function approximates continuous-time stochastic processes by the Euler approach.

NPeriods = 2;
[Paths,Times,Z,N] = simByEuler(batesObj,NPeriods)

Paths = 3×2

   80.0000    0.0400
   90.8427    0.0873
   32.7458    0.1798

Times = 3×1

     0
     1
     2

Z = 2×2

    0.5377    0.9333
   -2.2588    2.1969

N = 2×1

     0
     0

The output Paths is 3 x 2 dimension. Row number 3 is from NPeriods + 1. The first row is defined by the bates name-value pair argument StartState. The remaining rows are simulated data. Paths always has two columns because the Bates model is a bivariate model. In this case, the first column is the simulated asset price and the second column is the simulated volatilities.

Input Arguments

MDL — Stochastic differential equation model

object

Stochastic differential equation model, specified as a bates object. You can create a bates object using bates.

Data Types: object

NPeriods — Number of simulation periods

positive scalar integer

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 Pair 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,N] = simByEuler(bates,NPeriods,'DeltaTimes',dt)

'NTrials' — Simulated trials (sample paths)

1 (single path of correlated state variables) (default) | positive scalar integer

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

'DeltaTimes' — Positive time increments between observations

1 (default) | scalar | column vector

Positive time increments between observations, specified as the comma-separated pair consisting of 'DeltaTimes' and a scalar or an 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

'NSteps' — Number of intermediate time steps within each time increment dt

1 (indicating no intermediate evaluation) (default) | positive scalar integer

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 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

'Antithetic' — Flag to use antithetic sampling to generate the Gaussian random variates

false (no antithetic sampling) (default) | logical with values true or false

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, 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.

Note

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

Data Types: logical

'Z' — Direct specification of the dependent random noise process for generating Brownian motion vector

generates correlated Gaussian variates based on the Correlation member of the SDE object (default) | function | three-dimensional array of dependent random variates

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 as an (NPeriods ⨉ NSteps)-by-NBrowns-by-NTrials three-dimensional array of dependent random variates.

Note

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 X_t

Data Types: double | function

'N' — Dependent random counting process for generating number of jumps

random numbers from Poisson distribution with parameter JumpFreq from a bates object (default) | three-dimensional array | 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 (NPeriods ⨉ NSteps) -by-NJumps-by-NTrials three-dimensional array of dependent random variates.

Note

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 X_t.

Data Types: double | function

'StorePaths' — Flag that indicates how Paths is stored and returned

true (default) | logical with values true or false

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

If StorePaths is false (logical 0), simByEuler returns Paths as an empty matrix.

Data Types: logical

'Processes' — Sequence of end-of-period processes or state vector adjustments

simByEuler makes no adjustments and performs no processing (default) | function | cell array of functions

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

$$X_t=P(t,X_t)$$

The simByEuler function runs processing functions at each interpolation time. The functions must accept the current interpolation time t, and the current state vector X_t and return a state vector that can 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.

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 X_t for an all-NaN condition. Thus, to signal an early termination of a given trial, all elements of the state vector X_t 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

Paths — Simulated paths of correlated state variables

array

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 X_t at time t. When StorePaths is set to false, simByEuler returns Paths as an empty matrix.

Times — Observation times associated with simulated paths

column vector

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.

Z — Dependent random variates for generating Brownian motion vector

array

Dependent random variates 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.

N — Dependent random variates for generating jump counting process vector

array

Dependent random variates used to generate the jump counting process vector, returned as an (NPeriods ⨉ NSteps)-by-NJumps-by-NTrials three-dimensional time series array.

More About

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 that has the same expected value but a 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 other pairs, 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

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

• One model is a geometric Brownian motion (gbm) model with a stochastic volatility function and jumps.

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

  This model usually corresponds to a price process whose volatility (variance rate) is governed by the second univariate model.

• The other model is 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 Bates price process.

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 DeltaTimes approaches zero.

References

[1] Deelstra, Griselda, and Freddy Delbaen. “Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term.” Applied Stochastic Models and Data Analysis. 14, no. 1, 1998, 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 8, no. 3, (2005): 35–61.

[3] Lord, Roger, Remmert Koekkoek, and Dick Van Dijk. “A Comparison of Biased Simulation Schemes for Stochastic Volatility Models.” Quantitative Finance 10, no. 2 (February 2010): 177–94.