Stochastic Differential Equation (SDE) model
Creates and displays general stochastic differential equation
(SDE
) models from user-defined drift and diffusion rate
functions.
Use sde
objects to simulate sample paths of
NVars
state variables driven by NBROWNS
Brownian motion sources of risk over NPeriods
consecutive observation
periods, approximating continuous-time stochastic processes.
An sde
object enables you to simulate any vector-valued SDE of the form:
where:
Xt is an
NVars
-by-1
state vector of process
variables.
dWt is an
NBROWNS
-by-1
Brownian motion
vector.
F is an
NVars
-by-1
vector-valued
drift-rate function.
G is an
NVars
-by-NBROWNS
matrix-valued
diffusion-rate function.
creates a default SDE
= sde(DriftRate
,DiffusionRate
)SDE
object.
creates a SDE
= sde(___,Name,Value
)SDE
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 SDE
object has the following Properties:
StartTime
— Initial observation
time
StartState
— Initial state at time
StartTime
Correlation
— Access function for the
Correlation
input argument, 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
interpolate | Brownian interpolation of stochastic differential equations |
simulate | Simulate multivariate stochastic differential equations (SDEs) |
simByEuler | Euler simulation of stochastic differential equations (SDEs) |
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, sde
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.