simByMilstein2

Simulate diagonal diffusion Merton sample paths by second order Milstein approximation

Since R2023b

Syntax

[Paths,Times,Z,N] = simByMilstein2(MDL,NPeriods)

[Paths,Times,Z,N] = simByMilstein2(___,Name=Value)

Description

[Paths,Times,Z,N] = simByMilstein2(MDL,NPeriods) simulates NTrials sample paths of NVars correlated state variables 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 the continuous-time Merton jump diffusion process by the second order Milstein approximation.

simByMilstein2 is only valid for diagonal diffusion SDE models.

[Paths,Times,Z,N] = simByMilstein2(___,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

Quasi-Monte Carlo Simulation with Milstein2 Scheme Using Merton Model

This example shows how to use simByMilstein2 with a Merton 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 merton object.

AssetPrice = 80;
Return = 0.03;
Sigma = 0.16;
JumpMean = 0.02;
JumpVol = 0.08;
JumpFreq = 2;

Create a merton object.

mertonObj = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol,StartState=AssetPrice)

mertonObj = 
   Class MERTON: Merton Jump Diffusion
   ----------------------------------------
     Dimensions: State = 1, Brownian = 1
   ----------------------------------------
      StartTime: 0
     StartState: 80
    Correlation: 1
          Drift: drift rate function F(t,X(t)) 
      Diffusion: diffusion rate function G(t,X(t)) 
     Simulation: simulation method/function simByEuler
          Sigma: 0.16
         Return: 0.03
       JumpFreq: 2
       JumpMean: 0.02
        JumpVol: 0.08

Perform a quasi-Monte Carlo simulation by using simByMilstein2 with the optional name-value arguments for MonteCarloMethod, QuasiSequence, and BrownianMotionMethod.

[paths,times] = simByMilstein2(mertonObj,10,Ntrials=4096,MonteCarloMethod="quasi",QuasiSequence="sobol",BrownianMotionMethod="principal-components");

Input Arguments

MDL — Stochastic differential equation model

Merton object

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

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 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.

Example: [Paths,Times,Z,N] = simByMilstein2(Merton_obj,NPeriods,NTrials=1000,DeltaTime=dt,NSteps=10)

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

DeltaTime — Positive time increments between observations

1 (default) | scalar | column vector

Positive time increments between observations, specified as the comma-separated pair consisting of 'DeltaTime' 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

NSteps — Number of intermediate time steps within each time increment

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

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

Antithetic — Flag to use antithetic sampling to generate 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 merton 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.

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

MonteCarloMethod — Monte Carlo method to simulate stochastic processes

"standard" (default) | string with values "standard", "quasi", or "randomized-quasi" | character vector with values 'standard', 'quasi', or 'randomized-quasi'

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), simByEuler ignores the value of MonteCarloMethod.

Data Types: string | char

QuasiSequence — Low discrepancy sequence to drive stochastic processes

"sobol" (default) | string with value "sobol" | character vector with value 'sobol'

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

BrownianMotionMethod — Brownian motion construction method

"standard" (default) | string with value "brownian-bridge" or "principal-components" | character vector with value 'brownian-bridge' or 'principal-components'

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

Data Types: string | char

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 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 X_t at time t. When StorePaths is set to false, simByMilstein2 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 used to generate 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 used for generating jump counting process vector

array

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

More About

Second-Order Milstein Method

The second-order Milstein method is a numerical method for approximating solutions to stochastic differential equations (SDEs).

The second order Milstein method is an extension of the Euler-Maruyama method, which is a first-order numerical method for SDEs. The second-order Milstein method is more accurate than the Euler-Maruyama method and is often used to solve SDEs with strong or weak convergence rates.

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

This function simulates any vector-valued SDE of the following form:

$$d{X}_t=B(t,X_t)X_{t}dt+D(t,X_t)V(t,x_t)d{W}_t+Y(t,X_t,N_t)X_{t}d{N}_t$$

Here:

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

• B(t,X_t) is an NVars-by-NVars matrix of generalized expected instantaneous rates of return.

• D(t,Xt) is an NVars-by-NVars diagonal matrix in which each element along the main diagonal is the corresponding element of the state vector.

• V(t,Xt) is an NVars-by-NVars matrix of instantaneous volatility rates.

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

• Y(t,Xt,Nt) is an NVars-by-NJumps matrix-valued jump size function.

• dN_t is an NJumps-by-1 counting process vector.

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.

Consider the process X satisfying a stochastic differential equation of the form.

$$d{X}_t=\mu (X_t)dt+\sigma (X_t)d{W}_t$$

The attempt of including a term of O(dt) in the drift refines the Euler scheme and results in the algorithm derived by Milstein [1].

$$X_{t+1}=X_t+\mu (X_t)dt+\sigma (X_t)d{W}_t+\frac{1}{2}\sigma (X_t){\sigma }^{/}(X_t)(d{W}_t^2-dt)$$

Further refining of the Euler scheme gives out a metho with a weak order 2:

$$\begin{array}{l}{X}_{t+1}=X_t+\mu (X_t)dt+\sigma (X_t)d{W}_t+\frac{1}{2}\sigma (X_t){\sigma }^{/}(X_t)(d{W}_t^2-dt)+{\mu }^{/}(X_t)\sigma (X_t)dI\\ +\frac{1}{2}\left(\frac{1}{2}{\sigma }^2(X_t){\mu }^{//}(X_t)+\mu (X_t){\mu }^{/}(X_t)\right)d{t}^2+\left(\frac{1}{2}{\sigma }^2(X_t){\sigma }^{//}(X_t)+\mu (X_t){\sigma }^{/}(X_t)\right)(d{W}_{t}dt-dI)\end{array}$$

where dI is given by the area of the triangle with base dt and height dW.

References

[1] Milstein, G.N. "A Method of Second-Order Accuracy Integration of Stochastic Differential Equations."Theory of Probability and Its Applications, 23, 1978, pp. 396–401.

Version History

Introduced in R2023b