Geometric Brownian motion model
Creates and displays geometric Brownian motion (GBM) models, which derive from
the cev
(constant elasticity of variance)
class.
Geometric Brownian motion (GBM) models allow you to simulate sample paths of
NVars
state variables driven by NBrowns
Brownian motion sources of risk over NPeriods
consecutive observation
periods, approximating continuous-time GBM stochastic processes. Specifically, this
model allows the simulation of vector-valued GBM processes of the form
where:
Xt is an
NVars
-by-1
state vector of process
variables.
μ is an
NVars
-by-NVars
generalized
expected instantaneous rate of return matrix.
D is an
NVars
-by-NVars
diagonal matrix,
where each element along the main diagonal is the corresponding element of
the state vector Xt.
V is an
NVars
-by-NBrowns
instantaneous
volatility rate matrix.
dWt is an
NBrowns
-by-1
Brownian motion
vector.
creates a default GBM
= gbm(Return
,Sigma
)GBM
object.
Specify required input parameters as one of the following types:
A MATLAB® array. Specifying an array indicates a static (non-time-varying) 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 t
as its only input argument. Otherwise, a parameter is assumed to be
a function of time t and state
X(t) and is invoked with both input
arguments.
creates a GBM
= gbm(___,Name,Value
)GBM
object with additional options specified by
one or more Name,Value
pair arguments.
Name
is a property name and Value
is
its corresponding value. Name
must appear inside single
quotes (''
). You can specify several name-value pair
arguments in any order as
Name1,Value1,…,NameN,ValueN
The GBM
object has the following Properties:
StartTime
— Initial observation time
StartState
— Initial state at
StartTime
Correlation
— Access function for the
Correlation
input, callable as a function
of time
Drift
— Composite drift-rate function,
callable as a function of time and state
Diffusion
— Composite diffusion-rate
function, callable as a function of time and state
Simulation
— A simulation function or
method
Return
— Access function for the input
argument Return
, callable as a function of
time and state
Sigma
— Access function for the input
argument Sigma
, callable as a function of
time and state
interpolate | Brownian interpolation of stochastic differential equations |
simulate | Simulate multivariate stochastic differential equations (SDEs) |
simByEuler | Euler simulation of stochastic differential equations (SDEs) |
simBySolution | Simulate approximate solution of diagonal-drift GBM processes |
When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.
Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these parameters with no inputs, they behave like simple properties and allow you to test the data type (double vs. function, or equivalently, static vs. dynamic) of the original input specification. This is useful for validating and designing methods.
When you invoke these parameters with inputs, they behave like functions, giving the
impression of dynamic behavior. The parameters accept the observation time
t and a state vector
Xt, and return an array of appropriate
dimension. Even if you originally specified an input as an array, gbm
treats it as a static function of time and state, by that means guaranteeing that all
parameters are accessible by the same interface.
[1] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies, vol. 9, no. 2, Apr. 1996, pp. 385–426.
[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, vol. 54, no. 4, Aug. 1999, pp. 1361–95.
[3] Glasserman, Paul. Monte Carlo Methods in Financial Engineering. Springer, 2004.
[4] Hull, John. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.
[5] Johnson, Norman Lloyd, et al. Continuous Univariate Distributions. 2nd ed, Wiley, 1994.
[6] Shreve, Steven E. Stochastic Calculus for Finance. Springer, 2004.
bm
| cev
| diffusion
| drift
| interpolate
| nearcorr
| simByEuler
| simulate