|
Calling Sequence
|
|
ItoProcess(, mu, sigma)
ItoProcess(, mu, sigma, x, t)
ItoProcess(X, Sigma)
|
|
Parameters
|
|
|
-
|
the initial value
|
mu
|
-
|
the drift parameter
|
sigma
|
-
|
volatility parameter
|
X
|
-
|
Vector of one-dimensional Ito processes
|
Sigma
|
-
|
matrix
|
|
|
|
|
Description
|
|
•
|
The ItoProcess command creates a new one- or multi-dimensional Ito process, which is a stochastic process governed by the stochastic differential equation (SDE)
|
where
–
|
is the drift parameter
|
–
|
is the diffusion parameter
|
and
–
|
is the standard Wiener process.
|
•
|
The parameter defines the initial value of the underlying stochastic process. It must be a real constant.
|
•
|
The parameter mu is the drift. In the simplest case of a constant drift mu is real number (that is, any expression of type realcons). Time-dependent drift can be given either as an algebraic expression or as a Maple procedure. If mu is given as an algebraic expression, then the parameter t must be passed to specify which variable in mu should be used as a time variable. A Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift.
|
•
|
The parameter sigma is the diffusion. Similar to the drift parameter, the volatility can be constant or time-dependent.
|
•
|
One can use the ItoProcess command to construct a multi-dimensional Ito process with the given correlation structure. To be more precise, assume that is an -dimensional vector whose components , ..., are one-dimensional Ito processes. Let ,...,, and ,..., be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an -dimensional Ito process such that
|
where is an -dimensional Wiener process whose covariance matrix is Sigma. Note that the matrix Sigma must have numeric coefficients.
|
|
Examples
|
|
>
|
|
| (4) |
| (5) |
You can generate sample paths for this stochastic process (in order to do this, we must assign numeric values to mu and sigma).
>
|
|
Here is an example of a multi-dimensional Ito process.
>
|
|
>
|
|
| (15) |
>
|
|
>
|
|
>
|
|
>
|
|
| (20) |
In this example, construct a two-dimensional Ito process using two one-dimensional projections and a given covariance matrix.
>
|
|
>
|
|
>
|
|
| (25) |
| (26) |
>
|
|
| (27) |
>
|
|
| (28) |
|
|
References
|
|
|
Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.
|
|
Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.
|
|
|
Compatibility
|
|
•
|
The Finance[ItoProcess] command was introduced in Maple 15.
|
|
|
|