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# Hedging by Sequential Regression: An Introduction to the Mathematics of Option Trading

Abstract

It is widely acknowledge that there has been a major breakthrough in the mathematical theory of option trading. This breakthrough, which is usually summarized 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 sophistication; 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 eliminated completely.

Since October 1987, the mood has become 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 intrinsic risk may be much smaller than the a priori risk, but in general one should not expect it to vanish completely. 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. This was initiated by Kreps and Harrison; see, e.g. the excellent survey of Harrison and Pliska (1981,1983). The idea of embedding the Black–Scholes approach into a sequential regression scheme goes back to joint work of the first author with D. Sondermann. In continuous time and under martingale assumptions, this was worked out in Schweizer (1984) and Föllmer and Sondermann (1986). Schweizer (1988) deals with these problems in a general semimartingale model.

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Corresponding author
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D–5300 Bonn
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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

F. Black and M. Scholes The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81 (1973), 637659.

J. C. Cox and S. A. Ross The Valuation of Options for Alternative Stochastic Processes”, Journal of Financial Economics 3 (1976), 145166.

J. M. Harrison and S. R. Pliska Martingales and Stochastic Integrals in the Theory of Continuous Trading”, Stochastic Processes and their Applications 11 (1981), 215260.

J. M. Harrison and S. R. Pliska A Stochastic Calculus Model of Continuous Trading: Complete Markets”, Stochastic Processes and their Applications 15 (1983), 313316.

K. Itô Multiple Wiener Integral”, Journal of the Mathematical Society of Japan 3 (1951), 157169.

R. C. Merton Theory of Rational Option Pricing”, Bell Journal of Economics and Management Science 4 (1973), 141183

N. Wiener Differential-Space”, Journal of Mathematics and Physics 2 (1923), 131174; reprinted in: Wiener, N. “Collected Works”, Volume 1, MIT Press (1976).

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ASTIN Bulletin: The Journal of the IAA
• ISSN: 0515-0361
• EISSN: 1783-1350
• URL: /core/journals/astin-bulletin-journal-of-the-iaa