Hedging By Sequential Regression - Introduction To The Mathematics Of Option Trading(1).pdf

HEDGING BY SEQUENTIAL REGRESSION: AN INTRODUCTION TO
THE MATHEMATICS OF OPTION TRADING
BY H. FOLLMER and M. SCHWEIZER
ETH Zfirich
1. INTRODUCTION
It is widely acknowledge that there has been a major breakthrough in the
mathematical theory of option trading. This breakthrough, which is usually sum-
marized by the Black-Scholes formula, has generated a lot of excitement and a
certain mystique. On the mathematical side, it involves advanced probabilistic
techniques from martingale theory and stochastic calculus which are accessible
only to a small group of experts with a high degree of mathematical sophisti-
cation; hence the mystique. In its practical implications it offers exciting prospects.
Its promise is that, by a suitable choice of a trading strategy, the risk involved
in handling an option can be pletely.
Since October 1987, the mood has e more sober. But there are also
mathematical reasons which suggest that expectations should be lowered. This
will be the main point of the present expository'account. We argue that, typically,
the risk involved in handling an option has an irreducible intrinsic part. This intrin-
sic risk may be much smaller than the apriori risk, but in general one should not
expect it to pletely. In this more sober perspective, the mathematical
technique behind the Black-Scholes formula does not lose any of its importance.
In fact, it should be seen as a sequential regression scheme whose purpose is to
reduce the a priori risk to its intrinsic core.
We begin with a short introduction to the Black-Scholes formula in terms of
currency options. Then we develop a general regression scheme in discrete time,
first in an elementary two-period model, and then in a multiperiod model which
involves martingale considerations and sets the stage for extensions to continuous
time. Our method is based on the interpretation and extension of the
Black-Scholes formula in terms of martingale theory.