Computer-Aided Stochastic Calculus with
                   Applications in Finance

                    Dr. Colin P. Williams

                      Visiting Scholar
                Knowledge Systems Laboratory
                    Stanford University

                   Variable Systems, Inc.

    Modern financial engineering allows the composition of a portfolio of
investments to be tailored so that it meets the desired risk-return
characteristics of an investor. Usually such portfolios include so-called
"derivative securities". These are financial securities whose value is
dependent upon some other, more fundamental, variable. Common examples
include stock options, stock index options and futures contracts like those
quoted in the Wall Street Journal every day.

    Unfortunately derivative securities are notoriously difficult to intuit.
The recent bankruptcy of Orange County and the demise of Barings PLC indicate
the importance of assessing the true value of a derivative security and the
risks to which an investor in such a security is exposed.

    Fortunately, contrary to the media perception, derivatives can be placed
on a solid theoretical foundation. However, until recently it has only been
feasible to work with fairly small analytic models. With the advent of more
sophisticated computer algebra tools, more complex models can be handled. In
this talk I will show how computer algebra tools can assist in deriving
valuation and risk formulae for derivative securities. I will begin by
showing how a stochastic model for the behavior of stock prices and Ito's
lemma can lead to a partial differential equation whose solution is the value
of a stock option. I will then show how the techniques can be generalized to
derive a partial differential equation for valuing any kind of derivative
security.

Keywords: derivative securities, options, Black-Scholes, risk-neutral
valuation, risk, stochastic, Ito

References

[1] J. Cox & M. Rubinstein "Options Markets", Prentice-Hall, Inc. (1985)

[2] C. W. Gardiner, "Handbook of Stochastic Methods", 2nd Edition,
Springer-Verlag, (1985)

[3] J. Hull "Introduction to Futures & Options Markets", Prentice Hall, Inc.
(1991)