Construct drift-rate model components
DriftRate = drift(A, B)
This constructor specifies the drift-rate component of continuous-time
stochastic differential equations (SDEs). The drift-rate specification
supports the simulation of sample paths of
variables driven by
NBROWNS Brownian motion sources
of risk over
NPERIODS consecutive observation periods,
approximating continuous-time stochastic processes.
The drift-rate specification can be any
function F of the general form:
1 state vector of
1 Brownian motion
A and B are model parameters.
The drift-rate specification is flexible, and provides direct parametric support for static/linear drift models. It is also extensible, and provides indirect support for dynamic/nonlinear models via an interface. This enables you to specify virtually any drift-rate specification.
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.
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
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.
The required input parameters are:
This argument represents the parameter A.
If you specify
This argument represents the parameter B.
If you specify
Object of class
Create a drift-rate function
F = drift(0, 0.1) % Drift rate function F(t,X)
F = Class DRIFT: Drift Rate Specification ------------------------------------- Rate: drift rate function F(t,X(t)) A: 0 B: 0.1
drift object displays like a MATLAB® structure and contains supplemental information, namely, the object's class and a brief description. However, in contrast to the SDE representation, a summary of the dimensionality of the model does not appear, because the
drift class creates a model component rather than a model.
F does not contain enough information to characterize the dimensionality of a problem.
When you specify the input arguments
B as MATLAB arrays,
they are associated with a linear drift parametric form. By contrast,
when you specify either
a function, you can customize virtually any drift-rate specification.
Accessing the output drift-rate parameters
no inputs simply returns the original input specification. Thus, when
you invoke drift-rate 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 drift-rate parameters with inputs, they behave
like functions, giving the impression of dynamic behavior. The parameters
the observation time t and a state vector Xt,
and return an array of appropriate dimension. Specifically, parameters
the corresponding drift-rate component. Even if you originally specified
an input as an array,
drift treats it as a static
function of time and state, by that means guaranteeing that all parameters
are accessible by the same interface.
Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest Rate.” The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.
Ait-Sahalia, 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, Springer-Verlag, 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: Continuous-Time Models. New York: Springer-Verlag, 2004.