# simBySolution

Simulate approximate solution of diagonal-drift `GBM`

processes

## Description

`[`

adds optional name-value pair arguments. `Paths`

,`Times`

,`Z`

] = simBySolution(___,`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

### Simulating Equity Markets Using GBM Simulation Functions

Use GBM simulation functions. Separable GBM models have two specific simulation functions:

An overloaded Euler simulation function (

`simulate`

), designed for optimal performance.A

`simBySolution`

function that provides an approximate solution of the underlying stochastic differential equation, designed for accuracy.

Load the `Data_GlobalIdx2`

data set and specify the SDE model as in Representing Market Models Using SDE Objects, and the GBM model as in Representing Market Models Using SDELD, CEV, and GBM Objects.

load Data_GlobalIdx2 prices = [Dataset.TSX Dataset.CAC Dataset.DAX ... Dataset.NIK Dataset.FTSE Dataset.SP]; returns = tick2ret(prices); nVariables = size(returns,2); expReturn = mean(returns); sigma = std(returns); correlation = corrcoef(returns); t = 0; X = 100; X = X(ones(nVariables,1)); F = @(t,X) diag(expReturn)* X; G = @(t,X) diag(X) * diag(sigma); SDE = sde(F, G, 'Correlation', ... correlation, 'StartState', X); GBM = gbm(diag(expReturn),diag(sigma), 'Correlation', ... correlation, 'StartState', X);

To illustrate the performance benefit of the overloaded Euler approximation function (`simulate`

), increase the number of trials to `10000`

.

nPeriods = 249; % # of simulated observations dt = 1; % time increment = 1 day rng(142857,'twister') [X,T] = simulate(GBM, nPeriods, 'DeltaTime', dt, ... 'nTrials', 10000); whos X

Name Size Bytes Class Attributes X 250x6x10000 120000000 double

Using this sample size, examine the terminal distribution of Canada's TSX Composite to verify qualitatively the lognormal character of the data.

histogram(squeeze(X(end,1,:)), 30), xlabel('Price'), ylabel('Frequency') title('Histogram of Prices after One Year: Canada (TSX Composite)')

Simulate 10 trials of the solution and plot the first trial:

rng('default') [S,T] = simulate(SDE, nPeriods, 'DeltaTime', dt, 'nTrials', 10); rng('default') [X,T] = simBySolution(GBM, nPeriods,... 'DeltaTime', dt, 'nTrials', 10); subplot(2,1,1) plot(T, S(:,:,1)), xlabel('Trading Day'),ylabel('Price') title('1st Path of Multi-Dim Market Model:Euler Approximation') subplot(2,1,2) plot(T, X(:,:,1)), xlabel('Trading Day'),ylabel('Price') title('1st Path of Multi-Dim Market Model:Analytic Solution')

In this example, all parameters are constants, and `simBySolution`

does indeed sample the exact solution. The details of a single index for any given trial show that the price paths of the Euler approximation and the exact solution are close, but not identical.

The following plot illustrates the difference between the two functions:

subplot(1,1,1) plot(T, S(:,1,1) - X(:,1,1), 'blue'), grid('on') xlabel('Trading Day'), ylabel('Price Difference') title('Euler Approx Minus Exact Solution:Canada(TSX Composite)')

The `simByEuler`

Euler approximation literally evaluates the stochastic differential equation directly from the equation of motion, for some suitable value of the `dt`

time increment. This simple approximation suffers from discretization error. This error can be attributed to the discrepancy between the choice of the *dt* time increment and what in theory is a continuous-time parameter.

The discrete-time approximation improves as `DeltaTime`

approaches zero. The Euler function is often the least accurate and most general method available. All models shipped in the simulation suite have the `simByEuler`

function.

In contrast, the `simBySolution`

function provides a more accurate description of the underlying model. This function simulates the price paths by an approximation of the closed-form solution of separable models. Specifically, it applies a Euler approach to a transformed process, which in general is not the exact solution to this `GBM`

model. This is because the probability distributions of the simulated and true state vectors are identical only for piecewise constant parameters.

When all model parameters are piecewise constant over each observation period, the simulated process is exact for the observation times at which the state vector is sampled. Since all parameters are constants in this example, `simBySolution`

does indeed sample the exact solution.

For an example of how to use `simBySolution`

to optimize the accuracy of solutions, see Optimizing Accuracy: About Solution Precision and Error.

### Simulating Equity Markets Using GBM Model with Quasi-Monte Carlo Simulation

This example shows how to use `simBySolution`

with a GBM 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.

Load the `Data_GlobalIdx2`

data set and specify the GBM model as in Representing Market Models Using SDELD, CEV, and GBM Objects.

load Data_GlobalIdx2 prices = [Dataset.TSX Dataset.CAC Dataset.DAX ... Dataset.NIK Dataset.FTSE Dataset.SP]; returns = tick2ret(prices); nVariables = size(returns,2); expReturn = mean(returns); sigma = std(returns); correlation = corrcoef(returns); X = 100; X = X(ones(nVariables,1)); GBM = gbm(diag(expReturn),diag(sigma), 'Correlation', ... correlation, 'StartState', X);

Perform a quasi-Monte Carlo simulation by using `simBySolution`

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

,`'QuasiSequence'`

, and `'BrownianMotionMethod'`

.

[paths,time,z] = simBySolution(GBM, 10,'ntrials',4096,'MonteCarloMethod','quasi','QuasiSequence','sobol','BrownianMotionMethod','brownian-bridge');

## Input Arguments

`MDL`

— Geometric Brownian motion (GBM) model

`gbm`

object

Geometric Brownian motion (GBM) model, specified as a
`gbm`

object that is created using `gbm`

.

**Data Types: **`object`

`NPeriods`

— Number of simulation periods

positive 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,Z] =
simBySolution(GBM,NPeriods,'DeltaTime',dt,'NTrials',10)
```

`NTrials`

— Simulated trials (sample paths) of `NPERIODS`

observations each

`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`

`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 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 *dt* (specified as
`DeltaTime`

)

`1`

(indicating no intermediate
evaluation) (default) | positive 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 `simBySolution`

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
`simBySolution`

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 indicate whether `simBySolution`

uses antithetic sampling to generate the Gaussian random variates

`False`

(no antithetic
sampling) (default) | logical with values `True`

or
`False`

Flag to indicate whether `simBySolution`

uses
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 logical flag with a value of `True`

or
`False`

.

When you specify `True`

,
`simBySolution`

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`

), `simBySolution`

ignores
the value of `Antithetic`

.

**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`

), `simBySolution`

ignores
the value of `MonteCarloMethod`

.

**Data Types: **`string`

| `char`

`QuasiSequence`

— Low discrepancy sequence to drive the 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 one of the following values:

`"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`

),`simBySolution`

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 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
seen for `"brownian-bridge"`

construction option and
the fastest convergence when using the
`"principal-components"`

construction
option.

**Data Types: **`string`

| `char`

`Z`

— Direct specification of the dependent random noise process used to generate the 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 used to
generate 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.

The input argument `Z`

allows you to directly specify
the noise generation process. This process takes precedence over the
`Correlation`

parameter of the input `gbm`

object and the value of
the `Antithetic`

input flag.

**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 the output array `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 logical flag with a
value of `True`

or `False`

.

If `StorePaths`

is `True`

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

returns `Paths`

as a three-dimensional time series
array.

If `StorePaths`

is `False`

(logical
`0`

), `simBySolution`

returns the
`Paths`

output array as an empty matrix.

**Data Types: **`logical`

`Processes`

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

`simBySolution`

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

Sequence of end-of-period processes or state vector adjustments of the
form, 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 `simBySolution`

function runs processing
functions at each interpolation time. They must accept the current
interpolation time *t*, and the current state vector
*X _{t}*, and return a state
vector that may be an adjustment to the input state.

`simBySolution`

applies processing functions at the
end of each observation period. These functions must accept the current
observation time *t* and the current state vector
*X*_{t}, and
return a state vector that may be an adjustment to the input
state.

The end-of-period `Processes`

argument allows you to
terminate a given trial early. At the end of each time step,
`simBySolution`

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 a user-defined
`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).If you specify more than one processing function,
`simBySolution`

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.

**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 the input flag
`StorePaths`

= `False`

,
`simBySolution`

returns `Paths`

as an
empty matrix.

`Times`

— Observation times associated with the simulated paths

column vector

Observation times associated with the simulated paths, returned as a
`(NPERIODS + 1)`

-by-`1`

column vector.
Each element of `Times`

is associated with the
corresponding row of `Paths`

.

`Z`

— Dependent random variates used to generate the Brownian motion vector

array

Dependent random variates used to generate the Brownian motion vector
(Wiener processes) that drive the simulation, returned as a
```
(NPERIODS *
NSTEPS)
```

-by-`NBROWNS`

-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 of the same expected value, but 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 of any other
pair, 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

The `simBySolution`

function simulates `NTRIALS`

sample paths of `NVARS`

correlated state variables, driven by
`NBROWNS`

Brownian motion sources of risk over
`NPERIODS`

consecutive observation periods, approximating
continuous-time GBM short-rate models by an approximation of the closed-form
solution.

Consider a separable, vector-valued GBM model of the form:

$$d{X}_{t}=\mu (t){X}_{t}dt+D(t,{X}_{t})V(t)d{W}_{t}$$

where:

*X*is an_{t}`NVARS`

-by-`1`

state vector of process variables.*μ*is an`NVARS`

-by-`NVARS`

generalized expected instantaneous rate of return matrix.*V*is an`NVARS`

-by-`NBROWNS`

instantaneous volatility rate matrix.*dW*is an_{t}`NBROWNS`

-by-`1`

Brownian motion vector.

The `simBySolution`

function simulates the state vector
*X _{t}* using an approximation of the
closed-form solution of diagonal-drift models.

When evaluating the expressions, `simBySolution`

assumes that all
model parameters are piecewise-constant over each simulation period.

In general, this is *not* the exact solution to the models, because
the probability distributions of the simulated and true state vectors are identical
*only* for piecewise-constant parameters.

When parameters are piecewise-constant over each observation period, the simulated
process is exact for the observation times at which
*X _{t}* is sampled.

Gaussian diffusion models, such as `hwv`

, allow negative states. By default, `simBySolution`

does nothing to prevent negative states, nor does it guarantee that the model be
strictly mean-reverting. Thus, the model may exhibit erratic or explosive growth.

## References

[1] Aït-Sahalia, Yacine. “Testing
Continuous-Time Models of the Spot Interest Rate.” *Review of Financial
Studies* 9, no. 2 ( Apr. 1996): 385–426.

[2] Aït-Sahalia, Yacine.
“Transition Densities for Interest Rate and Other Nonlinear Diffusions.” *The
Journal of Finance* 54, no. 4 (Aug. 1999): 1361–95.

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

[4] Hull, John C.
*Options, Futures and Other Derivatives*. 7th ed, Prentice
Hall, 2009.

[5] Johnson, Norman Lloyd, Samuel
Kotz, and Narayanaswamy Balakrishnan. *Continuous Univariate
Distributions*. 2nd ed. Wiley Series in Probability and Mathematical
Statistics. New York: Wiley, 1995.

[6] Shreve, Steven E.
*Stochastic Calculus for Finance*. New York: Springer-Verlag,
2004.

## Version History

**Introduced in R2008a**

### 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`

.

## See Also

`simByEuler`

| `simulate`

| `gbm`

| `simBySolution`

### Topics

- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Price American Basket Options Using Standard Monte Carlo and Quasi-Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Quasi-Monte Carlo Simulation
- Performance Considerations

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