The Financial Modeling package supports a wide range of stochastic processes used in Financial Engineering. This includes processes for modeling equity prices, mean-reverting processes, pure jump processes, jump diffusions as well as multi-variate Ito processes. In addition, the package provides tools for building more complicated processes from simple ones. Below is the list of supported processes:

Here are some related commands.

For the first example consider the standard Wiener process.

The previous command created a new Maple variable representing the standard Wiener process. This is an ordinary Maple variable with additional attributes containing some information about the underlying stochastic process. This variable is typically accessed in the form , where  is the time of interest. For example, the following command can be used to generate some replications of the sample path for the underlying stochastic process.

You can now generate a larger sample and use tools from the Statistics package to analyze the generated data.

You can also simulate any expression involving our stochastic variable.

Simple expressions involving multiple stochastic parameters can be handled the same way.

Alternatively, you can implement any kind of path function using Maple procedures. This function will be called on every replication of the sample path. In the case of a single stochastic factor the path function will be passed a one-dimensional array containing the sample path. In the case of multiple stochastic factors, the path function will be passed a two-dimensional array where each row contains a sample path for the corresponding stochastic factor.

Consider, for example, a path function that returns  In the case when the path function is given explicitly, you must know the number of time steps in advance. You will use  time steps in this example, which means that the sample path will contain  points.

Now you can compute the expected value of this path function.

Compare this to:

Consider an example involving two stochastic factors (the same as above).

Here is a procedure that computes the expected value of the absorbing Wiener process.

An Ito process is a stochastic process  governed by the stochastic differential equation (SDE)

where  is the drift parameter and  is the volatility parameter. The coefficients of the above SDE can be extracted using the Drift and Diffusion commands.

Here is a slightly more complicated example.

Here is an example involving a multi-dimensional process.

Here is an Ito process defined explicitly.

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