simByTransition
Simulate CIR
sample paths with transition
density
Description
[
simulates Paths
,Times
] = simByTransition(MDL
,NPeriods
)NTrials
sample paths of NVars
independent state variables driven by the Cox-Ingersoll-Ross (CIR) process sources
of risk over NPeriods
consecutive observation periods.
simByTransition
approximates a continuous-time CIR model
using an approximation of the transition density function.
[
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
Simulate Future Term Structures Using a CIR Model
Using the short rate, simulate the rate dynamics and term structures in the future using a CIR model. The CIR model is expressed as
The exponential affine form of the bond price is
where
and
Define the parameters for the cir
object.
alpha = .1; b = .05; sigma = .05; r0 = .04;
Define the function for bond prices.
gamma = sqrt(alpha^2 + 2*sigma^2); A_func = @(t, T) ... 2*(exp(gamma*(T-t))-1)/((alpha+gamma)*(exp(gamma*(T-t))-1)+2*gamma); C_func = @(t, T) ... (2*alpha*b/sigma^2)*log(2*gamma*exp((alpha+gamma)*(T-t)/2)/((alpha+gamma)*(exp(gamma*(T-t))-1)+2*gamma)); P_func = @(t,T,r_t) exp(-A_func(t,T)*r_t+C_func(t,T));
Create a cir
object.
obj = cir(alpha,b,sigma,'StartState',r0)
obj = Class CIR: Cox-Ingersoll-Ross ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 0.04 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Sigma: 0.05 Level: 0.05 Speed: 0.1
Define the simulation parameters.
nTrials = 100; nPeriods = 5; % Simulate future short over the next five years nSteps = 12; % Set intermediate steps to improve the accuracy
Simulate the short rates. The returning path is a (NPeriods + 1)-by-NVars
-by-NTrials
three-dimensional time-series array. For this example, the size of the output is 6
-by-1
-by-100
.
rng('default'); % Reproduce the same result rPaths = simByTransition(obj,nPeriods,'nTrials',nTrials,'nSteps',nSteps); size(rPaths)
ans = 1×3
6 1 100
rPathsExp = mean(rPaths,3);
Determine the term structure over the next 30 years.
maturity = 30; T = 1:maturity; futuresTimes = 1:nPeriods+1; % Preallocate simTermStruc simTermStructure = zeros(nPeriods+1,30); for i = futuresTimes for t = T bondPrice = P_func(i,i+t,rPathsExp(i)); simTermStructure(i,t) = -log(bondPrice)/t; end end plot(simTermStructure') legend('Current','1-year','2-year','3-year','4-year','5-year') title('Projected Term Structure for Next 5 Years') ylabel('Long Rate Maturity R(t,T)') xlabel('Time')
Quasi-Monte Carlo Simulation Using a CIR Model
The Cox-Ingersoll-Ross (CIR) short rate class derives directly from SDE with mean-reverting drift (SDEMRD
):
where is a diagonal matrix whose elements are the square root of the corresponding element of the state vector.
Create a cir
object to represent the model: .
cir_obj = cir(0.2, 0.1, 0.05) % (Speed, Level, Sigma)
cir_obj = Class CIR: Cox-Ingersoll-Ross ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Sigma: 0.05 Level: 0.1 Speed: 0.2
Define the quasi-Monte Carlo simulation using the optional name-value arguments for 'MonteCarloMethod'
,'QuasiSequence'
, and 'BrownianMotionMethod'
.
[paths,time] = simByTransition(cir_obj,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
cir
object. For more information on creating a
CIR
object, see cir
.
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(CIR,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 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
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
StorePaths
— Flag for storage and return method
True
(default) | logical with values 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
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 that has a convergence rate of O(N-½)."quasi"
— Quasi-Monte Carlo rate of convergence is faster than standard Monte Carlo with errors approaching size of O(N-1)."randomized-quasi"
— Quasi-random sequences, also called low-discrepancy sequences, are deterministic uniformly distributed sequences which are specifically designed to place sample points as uniformly as possible.
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
seen for "brownian-bridge"
construction option and
the fastest convergence when using 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
.
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 may 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 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
Algorithms
Use the simByTransition
function to simulate any vector-valued CIR
process of the form
where
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.
References
[1] Glasserman, P. Monte Carlo Methods in Financial Engineering, New York: Springer-Verlag, 2004.
Version History
Introduced in R2018bR2022b: 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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