# simByMilstein2

Simulate `Heston`

process sample paths by second order Milstein
approximation

*Since R2023b*

## Syntax

## Description

`[`

simulates `Paths`

,`Times`

,`Z`

] = simByMilstein2(`MDL`

,`NPeriods`

)`NTrials`

sample paths of a Heston bivariate model
driven by two `NBrowns`

Brownian motion sources of risk
approximating continuous-time stochastic processes by the second order Milstein
approximation.

`simByMilstein2`

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.

`simByMilstein2`

is only valid for diagonal diffusion SDE
models.

`[`

specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.`Paths`

,`Times`

,`Z`

] = simByMilstein2(___,`Name=Value`

)

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 Heston Model

This example shows how to use `simByMilstein2`

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 `simByMilstein2`

with the optional name-value arguments for `MonteCarloMethod`

, `QuasiSequence`

, and `BrownianMotionMethod`

.

[paths,time,z] = simByMilstein2(Heston,10,ntrials=4096,MonteCarloMethod="quasi",QuasiSequence="sobol",BrownianMotionMethod="principal-components");

## Input Arguments

`MDL`

— Stochastic differential equation model

`Heston`

object

Stochastic differential equation model, specified as a
`heston`

object. You can create a
`heston`

object using `heston`

.

**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] =
simByMilstein2(Heston_obj,NPeriods,NTrials=10,DeltaTime=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 `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
`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`

`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 `DeltaTime`

),
specified as `NSteps`

and a positive scalar
integer.

The `simByMilstein2`

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
`simByMilstein2`

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 `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`

), `simByMilstein2`

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 `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`

`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 `StorePaths`

and a
scalar numeric or logical `1`

(`true`

)
or `0`

(`false`

).

If

`StorePaths`

is`true`

(the default value) or is unspecified,`simByMilstein2`

returns`Paths`

as a three-dimensional time-series array.If

`StorePaths`

is`false`

(logical`0`

),`simByMilstein2`

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
`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`

), `simByMilstein2`

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 `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`

), `simByMilstein2`

ignores
the value of `QuasiSequence`

.

**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
`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

`simByMilstein2`

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 `Processes`

and a function or cell array
of functions of the form

$${X}_{t}=P(t,{X}_{t})$$

The `simByMilstein2`

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,
`simByMilstein2`

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,
`simByMilstein2`

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*must be

_{t}`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 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.

## 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

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**

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