SFB 303 Discussion Paper No. B - 41
Author: Christopeit, Norbert
Title: Option Pricing and Optimal Stopping, Part I
Abstract: This is the first part of a report whose intention is to give a systematic treatment of the various approaches
to a theory of rational option pricing. In this first part, only European call options are considered, and the
problem of valuing them is treated within the framework of contigent claim pricing based on arbitrage
considerations in continuous time securities market models. The intuitive ideas behind this approach is the
following. One tires to construct a portfolio in the basic securities that produces the same pattern of cashflow as
the contingent claim and that, in addition, requires no funds to be invested nor allows for withdrawals (in the
terminology to be introduced below: a self-financing trading strategy generating the contingent claim). Then, if
arbitrage opportunities are excluded, there is a unique rational price for the contingent claim, namely the initial
investment in the generating portfolio. Trading is assumed to be done continuously in a frictionless
market.
Since
the economic background of this theory has been widely discussed in the literature, we have put more emphasis
on the mathematical aspects than on economic interpretations. Our presentation uses standard results from
semimartingale theory and stochastic integration. It differs from the approaches in the literature by the way in
which admissible portfolios are specified.In the second part of this report, problems of optimal stopping arising
from the consideration of American options will be treated in the spirit of McKean (1965) and Samuelson
(1965).
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Creation-Date: May 1985
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