simByQuadExp
Simulate Bates
, Heston
, and
CIR
sample paths by quadratic-exponential discretization
scheme
Since R2020a
Syntax
Description
[
simulates Paths
,Times
,Z
] = simByQuadExp(MDL
,NPeriods
)NTrials
sample paths of a Heston model driven by
two Brownian motion sources of risk, or a CIR model driven by one Brownian
motion source of risk. Both Heston and Bates models approximate continuous-time
stochastic processes by a quadratic-exponential discretization scheme. The
simByQuadExp
simulation derives 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.
[
specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.Paths
,Times
,Z
] = simByQuadExp(___,Name,Value
)
[
simulates Paths
,Times
,Z
,N
] = simByQuadExp(MDL
,NPeriods
)NTrials
sample paths of a Bates model driven by
two Brownian motion sources of risk, approximating continuous-time stochastic
processes by a quadratic-exponential discretization scheme. The
simByQuadExp
simulation derives 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.
[
specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.Paths
,Times
,Z
,N
] = simByQuadExp(___,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 Quadratic-Exponential Discretization Scheme
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 simByQuadExp
to simulate NTrials
sample paths directly from the stochastic differential equation of motion; the discrete-time process approaches the true continuous-time process only in the limit as DeltaTimes
approaches zero.
NPeriods = 2; [Paths,Times,Z,N] = simByQuadExp(batesObj,NPeriods)
Paths = 3×2
80.0000 0.0400
64.3377 0.1063
31.5703 0.1009
Times = 3×1
0
1
2
Z = 2×2
0.5377 1.8339
-2.2588 0.8622
N = 2×1
0
0
The output Paths
is returned as a (NPeriods
+ 1
)-by-NVars
-by-NTrials
three-dimensional time-series array.
Quasi-Monte Carlo Simulation Using Bates Model
This example shows how to use simByQuadExp
with a Bates 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 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];
Create a bates
object.
Bates = bates(Return, Speed, Level, Volatility, ... JumpFreq, JumpMean, JumpVol,'startstate',StartState, ... 'correlation',Correlation)
Bates = 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
Perform a quasi-Monte Carlo simulation by using simByQuadExp
with the optional name-value arguments for 'MonteCarloMethod'
,'QuasiSequence'
, and 'BrownianMotionMethod'
.
[paths,time,z] = simByQuadExp(Bates,10,'ntrials',4096,'montecarlomethod','quasi','quasisequence','sobol','BrownianMotionMethod','brownian-bridge');
Input Arguments
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] =
simByQuadExp(bates_obj,NPeriods,'DeltaTime',dt)
NTrials
— Simulated trials (sample paths) of NPeriods
observations each
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 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 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 simByQuadExp
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
simByQuadExp
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
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
), simByQuadExp
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 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.
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
Antithetic
— Flag to use antithetic sampling to generate the 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
,
simByQuadExp
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 dependent random noise process for generating Brownian motion vector
generates correlated Gaussian variates based on the
Correlation
member of the
heston
, bates
, or
cir
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 an (NPeriods ⨉
NSteps)
-by-NBrowns
-by-NTrials
three-dimensional array of dependent random variates.
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. 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.
Note
The N
name-value pair argument is supported
only when you use a bates
object
for the MDL
input argument.
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
is true
(the
default value) or is unspecified, simByQuadExp
returns Paths
as a three-dimensional time-series
array.
If StorePaths
is false
(logical
0
), simByQuadExp
returns
Paths
as an empty matrix.
Data Types: logical
Processes
— Sequence of end-of-period processes or state vector adjustments
simByQuadExp
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 simByQuadExp
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,
simByQuadExp
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,
simByQuadExp
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 for a heston, bates, or cir
model, 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
Xt at time
t. When StorePaths
is set to
false
, simByQuadExp
returns
Paths
as an empty matrix.
Times
— Observation times associated with simulated paths
column vector
Observation times for a heston, bates, or cir model 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 for generating Brownian motion vector
array
Dependent random variates for a heston, bates, or cir model 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 for a bates model for generating the jump
counting process vector, returned as a (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 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
Heston
In the Heston stochastic volatility model, the asset value process and volatility process are defined as
Here:
γ is the continuous risk-free rate.
θ is a long-term variance level.
κ is the mean reversion speed for the variance.
σ is the volatility of volatility.
CIR
You can simulate any vector-valued CIR process of the form
Here:
Xt is an
NVars
-by-1
state vector of process variables.S is an
NVars
-by-NVars
matrix of mean reversion speeds (the rate of mean reversion).L is an
NVars
-by-1
vector of mean reversion levels (long-run mean or level).D is an
NVars
-by-NVars
diagonal matrix, where each element along the main diagonal is the square root of the corresponding element of the state vector.V is an
NVars
-by-NBrowns
instantaneous volatility rate matrix.dWt is an
NBrowns
-by-1
Brownian motion vector.
Bates
Bates models are bivariate composite models. Each Bates model consists of two coupled univariate models.
A geometric Brownian motion (
gbm
) model with a stochastic volatility function and jumps that is expressed as follows.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 that is expressed as follows.This model describes the evolution of the variance rate of the coupled Bates price process.
References
[1] Andersen, Leif. “Simple and Efficient Simulation of the Heston Stochastic Volatility Model.” The Journal of Computational Finance 11, no. 3 (March 2008): 1–42.
[2] Broadie, M., and O. Kaya. “Exact Simulation of Option Greeks under Stochastic Volatility and Jump Diffusion Models.” In Proceedings of the 2004 Winter Simulation Conference, 2004., 2:535–43. Washington, D.C.: IEEE, 2004.
[3] Broadie, Mark, and Özgür Kaya. “Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes.” Operations Research 54, no. 2 (April 2006): 217–31.
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
bates
| heston
| cir
| simByEuler
| simByTransition
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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