# simByTransition

## Description

`[`

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

,`Times`

] = simByTransition(___,`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

### Use simByTransition with `heston`

Object

Simulate Heston sample paths with transition density.

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.

hestonObj = heston(Return,Speed,Level,Volatility,'startstate',StartState,'correlation',Correlation)

hestonObj = 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

Define the simulation parameters.

```
nPeriods = 5; % Simulate sample paths over the next five years
Paths = simByTransition(hestonObj,nPeriods);
Paths
```

`Paths = `*6×2*
80.0000 0.0400
99.7718 0.0201
124.7044 0.0176
52.7914 0.1806
75.3173 0.1732
76.7572 0.1169

### Quasi-Monte Carlo Simulation Using Heston Model

This example shows how to use `simByTransition`

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 `simByTransition`

with the optional name-value argument for `'MonteCarloMethod'`

, `'QuasiSequence'`

, and `'BrownianMotionMethod'`

.

[paths,time] = simByTransition(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. For more information on creating a
`heston`

object, see `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.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **```
[Paths,Times] =
simByTransition(Heston,NPeriods,'DeltaTime',dt)
```

`NTrials`

— Simulated trials (sample paths)

`1`

(single path of correlated state
variables) (default) | positive 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 `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

`1`

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

Number of intermediate time steps within each time increment
*dt* (defined as `DeltaTime`

),
specified as the comma-separated pair consisting of
`'NSteps'`

and a positive scalar integer.

The `simByTransition`

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
`simByTransition`

does not report the output state
vector at these intermediate points, the refinement improves accuracy by
enabling the simulation to more closely approximate the underlying
continuous-time process.

**Data Types: **`double`

`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

**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.

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`

`StorePaths`

— Flag for storage and return method

`True`

(default) | logical with `True`

or
`False`

Flag for storage and return method 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, then`simByTransition`

returns`Paths`

as a three-dimensional time series array.If

`StorePaths`

is`False`

(logical`0`

), then`simByTransition`

returns the`Paths`

output array as an empty matrix.

**Data Types: **`logical`

`Processes`

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

`simByTransition`

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

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

`simByTransition`

applies processing functions at the
end of each observation period. The processing functions accept the
current observation time *t* and the current state
vector *X*_{t},
and return a state vector that might adjust the input state.

If you specify more than one processing function,
`simByTransition`

invokes the functions in the
order in which they appear in the cell array.

**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 the input flag
`StorePaths`

= `False`

,
`simByTransition`

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`

.

## More About

### Transition Density Simulation

The CIR SDE has no solution such that
*r*(*t*) =
*f*(*r*(0),⋯).

In other words, the equation is not explicitly solvable. However, the transition density for the process is known.

The exact simulation for the distribution of
*r*(*t*_1
),⋯,*r*(*t*_*n*) is that of
the process at times
*t*_1,⋯,*t*_*n* for the same
value of *r*(0). The transition density for this process is known
and is expressed as

$$\begin{array}{l}r(t)=\frac{{\sigma}^{2}(1-{e}^{-\alpha (t-u)}}{4\alpha}{x}_{d}^{2}\left(\frac{4\alpha {e}^{-\alpha (t-u)}}{{\sigma}^{2}(1-{e}^{-\alpha (t-u)})}r(u)\right),t>u\\ \text{where}\\ d\equiv \frac{4b\alpha}{{\sigma}^{2}}\end{array}$$

### Heston Model

Heston models are bivariate composite models.

Each Heston model consists of two coupled univariate models:

A geometric Brownian motion (

`gbm`

) model with a stochastic volatility function.$$d{X}_{1t}=B(t){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}$$

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

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 GBM price process.

## References

[1] Glasserman, Paul *Monte Carlo Methods in Financial
Engineering*, New York: Springer-Verlag, 2004.

[2] Van Haastrecht, Alexander, and
Antoon Pelsser. "Efficient, Almost Exact Simulation of the Heston Stochastic Volatility
Model." *International Journal of Theoretical and Applied Finance*,
Vol. 13, No. 01 (2010): 1–43.

## Version History

**Introduced in R2020b**

### R2022b: Perform Brownian bridge and principal components construction

Perform Brownian bridge and principal components construction using the name-value
argument `BrownianMotionMethod`

.

### R2022a: Perform Quasi-Monte Carlo simulation

Perform Quasi-Monte Carlo simulation using the name-value arguments
`MonteCarloMethod`

and
`QuasiSequence`

.

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