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 Xt
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
istrue
(the default value) or is unspecified,simByMilstein2
returnsPaths
as a three-dimensional time-series array.If
StorePaths
isfalse
(logical0
),simByMilstein2
returnsPaths
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
The simByMilstein2
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,
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
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
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
Xt 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.
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].
Further refining of the Euler scheme gives out a metho with a weak order 2:
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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