Create GBM model
GBM = gbm(Return, Sigma)
GBM = gbm(Return, Sigma, 'Name1', Value1, 'Name2',
Value2, ...)
This function creates and displays geometric Brownian motion
(GBM
) models, which derive from the CEV
(constant
elasticity of variance) class. Use GBM
models to
simulate sample paths of NVARS
state variables
driven by NBROWNS
Brownian motion sources of risk
over NPERIODS
consecutive observation periods,
approximating continuoustime GBM
stochastic processes.
This function allows simulation of vectorvalued GBM
processes
of the form:
$$d{X}_{t}=\mu (t){X}_{t}dt+D(t,{X}_{t})V(t)d{W}_{t}$$  (186) 
where:
X_{t} is an NVARS
by1
state vector of process variables.
μ is an NVARS
byNVARS
generalized expected instantaneous rate of return matrix.
D is an NVARS
byNVARS
diagonal
matrix, where each element along the main diagonal is the corresponding
element of the state vector X_{t}.
V is an NVARS
byNBROWNS
instantaneous
volatility rate matrix.
dW_{t} is an NBROWNS
by1
Brownian motion vector.
Specify required input parameters as one of the following types:
A MATLAB^{®} array. Specifying an array indicates a static (nontimevarying) parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.
A MATLAB function. Specifying a function provides indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported via an interface, because all implementation details are hidden and fully encapsulated by the function.
Note: You can specify combinations of array and function input parameters as needed. Moreover, a parameter is identified as a deterministic function
of time if the function accepts a scalar time 
The required input parameters are:
Return  Return represents the parameter μ.
If you specify Return as an array and it must be
an NVARS byNVARS matrix representing
the expected (mean) instantaneous rate of return. As a deterministic
function of time, when Return is called with a
realvalued scalar time t as its only input, Return must
produce an NVARS byNVARS matrix.
If you specify Return as a function of time and
state, it must return an NVARS byNVARS matrix
when invoked with two inputs:

Sigma  Sigma represents the parameter V.
If you specify Sigma as an array, it must be an NVARS byNBROWNS matrix
of instantaneous volatility rates or as a deterministic function of
time. In this case, each row of Sigma corresponds
to a particular state variable. Each column corresponds to a particular
Brownian source of uncertainty, and associates the magnitude of the
exposure of state variables with sources of uncertainty. As a deterministic
function of time, when Sigma is called with a realvalued
scalar time t as its only input, Sigma must
produce an NVARS byNBROWNS matrix.
If you specify Sigma as a function of time and
state, it must return an NVARS byNBROWNS matrix
of volatility rates when invoked with two inputs:
Although the 
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:
StartTime  Scalar starting time of the first observation, applied to all
state variables. If you do not specify a value for StartTime ,
the default is 0 . 
StartState  Scalar, NVARS by1 column vector, or NVARS byNTRIALS matrix
of initial values of the state variables. If If If If you do not specify
a value for 
Correlation  Correlation between Gaussian random variates drawn to generate
the Brownian motion vector (Wiener processes). Specify Correlation as
an NBROWNS byNBROWNS positive
semidefinite matrix, or as a deterministic function C(t) that
accepts the current time t and returns an NBROWNS byNBROWNS positive
semidefinite correlation matrix.A As a deterministic function
of time, If you do not specify a value for 
Simulation  A userdefined simulation function or SDE simulation method.
If you do not specify a value for Simulation , the
default method is simulation by Euler approximation (simByEuler ). 
GBM  Geometric Brownian motion model with the following displayed
parameters:

AitSahalia, Y., "Testing ContinuousTime Models of the Spot Interest Rate," The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.
AitSahalia, Y., "Transition Densities for Interest Rate and Other Nonlinear Diffusions," The Journal of Finance, Vol. 54, No. 4, August 1999.
Glasserman, P., Monte Carlo Methods in Financial Engineering, New York: SpringerVerlag, 2004.
Hull, J. C., Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.
Johnson, N. L., S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: John Wiley & Sons, 1995.
Shreve, S. E., Stochastic Calculus for Finance II: ContinuousTime Models, New York: SpringerVerlag, 2004.