sde
Stochastic Differential Equation (SDE
) model
Description
Creates and displays a general stochastic differential equation
(SDE
) model 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.
Creation
Description
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 timeStartState
— Initial state at timeStartTime
Correlation
— Access function for theCorrelation
input argument, callable as a function of timeDrift
— Composite drift-rate function, callable as a function of time and stateDiffusion
— Composite diffusion-rate function, callable as a function of time and stateSimulation
— A simulation function or method
Input Arguments
Properties
Object Functions
interpolate | Brownian interpolation of stochastic differential equations (SDEs) for
SDE , BM , GBM ,
CEV , CIR , HWV ,
Heston , SDEDDO , SDELD , or
SDEMRD models |
simulate | Simulate multivariate stochastic differential equations (SDEs) for
SDE , BM , GBM ,
CEV , CIR , HWV ,
Heston , SDEDDO , SDELD ,
SDEMRD , Merton , or Bates
models |
simByEuler | Euler simulation of stochastic differential equations (SDEs) for
SDE , BM , GBM ,
CEV , CIR , HWV ,
Heston , SDEDDO , SDELD , or
SDEMRD models |
Examples
More About
Algorithms
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.
References
[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.
Version History
Introduced in R2008a
See Also
Topics
- Base SDE Models
- Representing Market Models Using SDE Objects
- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Price American Basket Options Using Standard Monte Carlo and Quasi-Monte Carlo Simulation
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Quasi-Monte Carlo Simulation
- Performance Considerations