simByEuler
Description
[
simulates Paths
,Times
,Z
,N
] = simByEuler(MDL
,NPeriods
)NTrials
sample paths of Bates bivariate models driven
by NBrowns
Brownian motion sources of risk and
NJumps
compound Poisson processes representing the arrivals
of important events over NPeriods
consecutive observation
periods. The simulation approximates continuous-time stochastic processes by the
Euler approach.
[
specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.Paths
,Times
,Z
,N
] = simByEuler(___,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
Simulate Bates Sample Paths by Euler Approximation
Create a bates
object.
AssetPrice = 80; Return = 0.03; JumpMean = 0.02; JumpVol = 0.08; JumpFreq = 0.1; V0 = 0.04; Level = 0.05; Speed = 1.0; Volatility = 0.2; Rho = -0.7; StartState = [AssetPrice;V0]; Correlation = [1 Rho;Rho 1]; batesObj = bates(Return, Speed, Level, Volatility,... JumpFreq, JumpMean, JumpVol,'startstate',StartState,... 'correlation',Correlation)
batesObj = Class BATES: Bates 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 JumpFreq: 0.1 JumpMean: 0.02 JumpVol: 0.08
Use simByEuler
to simulate NTrials
sample paths of this Bates bivariate model driven by NBrowns
Brownian motion sources of risk and NJumps
compound Poisson processes representing the arrivals of important events over NPeriods
consecutive observation periods. This function approximates continuous-time stochastic processes by the Euler approach.
NPeriods = 2; [Paths,Times,Z,N] = simByEuler(batesObj,NPeriods)
Paths = 3×2
80.0000 0.0400
90.8427 0.0873
32.7458 0.1798
Times = 3×1
0
1
2
Z = 2×2
0.5377 0.9333
-2.2588 2.1969
N = 2×1
0
0
The output Paths
is 3
x
2
dimension. Row number 3
is from NPeriods
+
1
. The first row is defined by the bates
name-value pair argument StartState
. The remaining rows are simulated data. Paths
always has two columns because the Bates model is a bivariate model. In this case, the first column is the simulated asset price and the second column is the simulated volatilities.
Quasi-Monte Carlo Simulation with simByEuler
Using a Bates Model
Create a bates
object.
AssetPrice = 80; Return = 0.03; JumpMean = 0.02; JumpVol = 0.08; JumpFreq = 0.1; V0 = 0.04; Level = 0.05; Speed = 1.0; Volatility = 0.2; Rho = -0.7; StartState = [AssetPrice;V0]; Correlation = [1 Rho;Rho 1]; batesObj = bates(Return, Speed, Level, Volatility,... JumpFreq, JumpMean, JumpVol,'startstate',StartState,... 'correlation',Correlation)
batesObj = Class BATES: Bates 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 JumpFreq: 0.1 JumpMean: 0.02 JumpVol: 0.08
Define the quasi-Monte Carlo simulation using the optional name-value arguments for 'MonteCarloMethod'
,'QuasiSequence'
, and 'BrownianMotionMethod'
.
[paths,time,z] = simByEuler(batesObj,10,'ntrials',4096,'montecarlomethod','quasi','quasisequence','sobol','BrownianMotionMethod','principal-components');
Input Arguments
MDL
— Stochastic differential equation model
object
Stochastic differential equation model, specified as a
bates
object. You can create a
bates
object using bates
.
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,Z,N] =
simByEuler(bates,NPeriods,'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 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 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 the comma-separated pair consisting of
'NSteps'
and a positive scalar integer.
The simByEuler
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
simByEuler
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 the comma-separated pair consisting of
'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
), simByEuler
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 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.
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
N
— Dependent random counting process for generating number of jumps
random numbers from Poisson distribution with
parameter JumpFreq
from a bates
object (default) | three-dimensional array | function
Dependent random counting process for generating the number of jumps,
specified as the comma-separated pair consisting of
'N'
and a function or an
(NPeriods
⨉ NSteps
)
-by-NJumps
-by-NTrials
three-dimensional array of dependent random variates.
Note
If you specify a function, N
must return an
NJumps
-by-1
column
vector, and you must call it with two inputs: a real-valued
scalar observation time t followed by 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 the comma-separated pair consisting of
'StorePaths'
and a scalar numeric or logical
1
(true
) or
0
(false
).
If
StorePaths
istrue
(the default value) or is unspecified,simByEuler
returnsPaths
as a three-dimensional time-series array.If
StorePaths
isfalse
(logical0
),simByEuler
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 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
and N
), simByEuler
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 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.
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
simByEuler
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
The simByEuler
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,
simByEuler
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, and plot graphs.
The end-of-period Processes
argument allows you to
terminate a given trial early. At the end of each time step,
simByEuler
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
, simByEuler
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.
N
— Dependent random variates for generating jump counting process vector
array
Dependent random variates used to generate the jump counting process
vector, returned as an (NPeriods ⨉
NSteps)
-by-NJumps
-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 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
Bates models are bivariate composite models. Each Bates model consists of two coupled univariate models.
One model is a geometric Brownian motion (
gbm
) model with a stochastic volatility function and jumps.This model usually corresponds to a price process whose volatility (variance rate) is governed by the second univariate model.
The other model is a Cox-Ingersoll-Ross (
cir
) square root diffusion model.This model describes the evolution of the variance rate of the coupled Bates price process.
This simulation engine provides a discrete-time approximation of the underlying
generalized continuous-time process. The simulation is derived directly from the
stochastic differential equation of motion. Thus, the discrete-time process approaches
the true continuous-time process only as DeltaTime
approaches
zero.
References
[1] Deelstra, Griselda, and Freddy Delbaen. “Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term.” Applied Stochastic Models and Data Analysis, Vol. 14, No. 1, 1998, pp. 77–84.
[2] Higham, Desmond, and Xuerong Mao. “Convergence of Monte Carlo Simulations Involving the Mean-Reverting Square Root Process.” The Journal of Computational Finance, Vol. 8, No. 3, (2005): 35–61.
[3] Lord, Roger, Remmert Koekkoek, and Dick Van Dijk. “A Comparison of Biased Simulation Schemes for Stochastic Volatility Models.” Quantitative Finance, Vol. 10, No. 2 (February 2010): 177–94.
Version History
Introduced in R2020aR2022b: 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
Topics
- Implementing Multidimensional Equity Market Models, Implementation 5: Using the simByEuler Method
- 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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