# simByMilstein

Simulate `Heston` process sample paths by Milstein approximation

Since R2023a

## Syntax

``[Paths,Times,Z] = simByMilstein(MDL,NPeriods)``
``[Paths,Times,Z] = simByMilstein(___,Name=Value)``

## Description

example

````[Paths,Times,Z] = simByMilstein(MDL,NPeriods)` simulates `NTrials` sample paths of Heston bivariate models driven by two `NBrowns` Brownian motion sources of risk approximating continuous-time stochastic processes by the Milstein approximation.`simByMilstein` provides a discrete-time approximation of the underlying generalized continuous-time process. The simulation is derived directly from the stochastic differential equation of motion; the discrete-time process approaches the true continuous-time process only in the limit as `DeltaTime` approaches zero.```

example

````[Paths,Times,Z] = simByMilstein(___,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

collapse all

This example shows how to use `simByMilstein` 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 `simByMilstein` with the optional name-value arguments for `MonteCarloMethod`, `QuasiSequence`, and `BrownianMotionMethod`.

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

## Input Arguments

collapse all

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

Example: ```[Paths,Times,Z] = simByMilstein(Heston_obj,NPeriods,NTrials=10,DeltaTime=dt)```

Simulated trials (sample paths) of `NPeriods` observations each, specified as `NTrials` and a positive scalar integer.

Data Types: `double`

Positive time increments between observations, specified as `DeltaTime` and a scalar or an `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 `NSteps` and a positive scalar integer.

The `simByMilstein` 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 `simByMilstein` 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 to use antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes), specified as `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`), `simByMilstein` 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 `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 Xt

Data Types: `double` | `function`

Flag that indicates how the output array `Paths` is stored and returned, specified as `StorePaths` and a scalar numeric or logical `1` (`true`) or `0` (`false`).

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

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

Data Types: `logical`

Monte Carlo method to simulate stochastic processes, specified as `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`), `simByMilstein` ignores the value of `MonteCarloMethod`.

Data Types: `string` | `char`

Low discrepancy sequence to drive the stochastic processes, specified as `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.

If you specify an input noise process (see `Z`), `simByMilstein` ignores the value of `QuasiSequence`.

Data Types: `string` | `char`

Brownian motion construction method, specified as `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`

Sequence of end-of-period processes or state vector adjustments, specified as `Processes` and a function or cell array of functions of the form

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

The `simByMilstein` 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, `simByMilstein` 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, `simByMilstein` 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

collapse all

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 `StorePaths` is set to `false`, `simByMilstein` 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`.

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.

collapse all

### Milstein Method

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

The Milstein method is an extension of the Euler-Maruyama method, which is a first-order numerical method for SDEs. The Milstein method adds a correction term to the Euler-Maruyama method that takes into account the second-order derivative of the SDE. This correction term improves the accuracy of the approximation, especially for SDEs with non-linearities.

### 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

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

`$d{X}_{t}=\mu \left({X}_{t}\right)dt+\sigma \left({X}_{t}\right)d{W}_{t}$`

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].

`${X}_{t+1}={X}_{t}+\mu \left({X}_{t}\right)dt+\sigma \left({X}_{t}\right)d{W}_{t}+\frac{1}{2}\sigma \left({X}_{t}\right){\sigma }^{/}\left({X}_{t}\right)\left(d{W}_{t}^{2}-dt\right)$`

## 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 R2023a