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
isTrue
(the default value) or is unspecified, thensimByTransition
returnsPaths
as a three-dimensional time series array.If
StorePaths
isFalse
(logical0
), thensimByTransition
returns thePaths
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
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 Xt,
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
Xt 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
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.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.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 R2020bR2022b: 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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