Simulate approximate solution of diagonaldrift HWV
and GBM
processes
[Paths, Times, Z] = simBySolution(MDL, NPERIODS)
[Paths, Times, Z] = simBySolution(MDL, NPERIODS, 'Name1',
Value1,'Name2', Value2, ...)
The simBySolution
method simulates NTRIALS
sample
paths of NVARS
correlated state variables, driven
by NBROWNS
Brownian motion sources of risk over NPERIODS
consecutive
observation periods, approximating continuoustime HullWhite/Vasicek
(HWV
), and geometric Brownian motion (GBM
)
shortrate models by an approximation of the closedform solution.
Consider a separable, vectorvalued HWV
model
of the form:
$$d{X}_{t}=S(t)[L(t){X}_{t}]dt+V(t)d{W}_{t}$$  (1812) 
where:
X is an NVARSby1 state vector of process variables.
S is an NVARSbyNVARS matrix of mean reversion speeds (the rate of mean reversion).
L is an NVARSby1 vector of mean reversion levels (longrun mean or level).
V is an NVARSbyNBROWNS instantaneous volatility rate matrix.
W is an NBROWNSby1 Brownian motion vector.
or a separable, vectorvalued GBM
model of
the form:
$$d{X}_{t}=\mu (t){X}_{t}dt+D(t,{X}_{t})V(t)d{W}_{t}$$  (1813) 
where:
X_{t} is an NVARS
by1
state vector of process variables.
μ is an NVARS
byNVARS
generalized expected instantaneous rate of return matrix.
V is an NVARS
byNBROWNS
instantaneous
volatility rate matrix.
dW_{t} is an NBROWNS
by1
Brownian motion vector.
The simBySolution
method simulates
the state vector X_{t} using
an approximation of the closedform solution of diagonaldrift models.
When evaluating the expressions, simBySolution
assumes
that all model parameters are piecewiseconstant over each simulation
period.
In general, this is not the exact solution to the models, because the probability distributions of the simulated and true state vectors are identical only for piecewiseconstant parameters.
When parameters are piecewiseconstant over each observation period, the simulated process is exact for the observation times at which X_{t} is sampled.
MDL  HullWhite/Vasicek (HWV ) or geometric Brownian
motion (GBM ) model. 
NPERIODS  Positive scalar integer number of simulation periods. The value of this argument determines the number of rows of the simulated output series. 
Specify optional input arguments as variablelength lists of
matching parameter name/value pairs: 'Name1'
, Value1
, 'Name2'
, Value2
,
... and so on. The following rules apply when specifying parametername
pairs:
Specify the parameter name as a character string, followed by its corresponding parameter value.
You can specify parameter name/value pairs in any order.
Parameter names are case insensitive.
You can specify unambiguous partial string matches.
Valid parameter names are:
NTRIALS  Positive scalar integer number of simulated trials (sample
paths) of NPERIODS observations each. If you do
not specify a value for this argument, the default is 1 ,
indicating a single path of correlated state variables. 
DeltaTime  Scalar or NPERIODS by1 column vector of
positive time increments between observations. DeltaTime represents
the familiar dt found in stochastic differential
equations, and determines the times at which simBySolution reports
the simulated paths of the output state variables. If you do not specify
a value for this argument, the default is 1 .

NSTEPS  Positive scalar integer number of intermediate time steps within
each time increment dt (specified as DeltaTime ). simBySolution 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 simBySolution 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 continuoustime process. If you do not
specify a value for NSTEPS , the default is 1 ,
indicating no intermediate evaluation. 
Antithetic  Scalar logical flag that indicates whether antithetic sampling
is used to generate the Gaussian random variates that drive the Brownian
motion vector (Wiener processes). When Antithetic is TRUE (logical 1 ), simBySolution performs
sampling such that all primary and antithetic paths are simulated
and stored in successive matching pairs:
Antithetic to be
any value other than TRUE ,simBySolution assumes
that it is FALSE (logical 0 )
by default, and does not perform antithetic sampling. When you specify
an input noise process (see Z ), simBySolution ignores
the value of Antithetic . 
Z  Direct specification of the dependent random noise process
used to generate the Brownian motion vector (Wiener process) that
drives the simulation. Specify this argument as a function, or as
an (NPERIODS * NSTEPS) byNBROWNS byNTRIALS array
of dependent random variates. If you specify Z as
a function, it must return an NBROWNS by1 column
vector, and you must call it with two inputs:
Z , simBySolution generates
correlated Gaussian variates based on the Correlation member
of the SDE object. 
StorePaths  Scalar logical flag that indicates how simBySolution stores
the output array Paths and returns it to the caller.
If StorePaths is TRUE (the default
value) or is unspecified, simBySolution returns Paths as
a threedimensional time series array. If StorePaths is FALSE (logical 0 ), simBySolution returns
the Paths output array as an empty matrix. 
Processes  Function or cell array of functions that indicates a sequence
of endofperiod processes or state vector adjustments of the form $${X}_{t}=P(t,{X}_{t})$$ simBySolution applies
processing functions at the end of each observation period. These
functions must accept the current observation time t and
the current state vector X_{t},
and return a state vector that may be an adjustment to the input state. If
you specify more than one processing function, If
you do not specify a processing function, 
Paths  (NPERIODS + 1) byNVARS byNTRIALS threedimensional
time series array, consisting of simulated paths of correlated state
variables. For a given trial, each row of Paths is
the transpose of the state vector X_{t} at
time t. When the input flag StorePaths = FALSE , simBySolution returns Paths as
an empty matrix. 
Times  (NPERIODS + 1) by1 column vector of observation
times associated with the simulated paths. Each element of Times is
associated with the corresponding row of Paths .

Z  (NPERIODS * NSTEPS) byNBROWNS byNTRIALS threedimensional
time series array of dependent random variates used to generate the
Brownian motion vector (Wiener processes) that drive the simulation. 
Implementing Multidimensional Equity Market Models, Implementation 6: Using GBM Simulation Methods
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