Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T07:57:50.345Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  29 August 2014

H. Föllmer  and
M. Schweizer
H. Föllmer*
Affiliation:
ETH Zürich
M. Schweizer*
Affiliation:
ETH Zürich
*
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D–5300 Bonn
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D–5300 Bonn
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Invited Papers
Information
ASTIN Bulletin: The Journal of the IAA , Volume 18 , Issue 2 , November 1988 , pp. 147 - 160
Copyright
Copyright © International Actuarial Association 1988

References

Bachelier, L. “Théorie de la Spéculation”, Ann. Sci. Ec. Norm. Sup. III–17 (1900), 21–86.CrossRefGoogle Scholar
Bernoulli, J. “Ars coniectandi” (1713).Google Scholar
Black, F. and Scholes, M. “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81 (1973), 637–659.CrossRefGoogle Scholar
Cox, J. C. and Ross, S. A. “The Valuation of Options for Alternative Stochastic Processes”, Journal of Financial Economics 3 (1976), 145–166.CrossRefGoogle Scholar
Föllmer, H. and Sondermann, D. “Hedging of Non-Redundant Contingent Claims”, in: Hildenbrand, W. and Mas-Colell, A. (eds.), Contributions to Mathematical Economics North-Holland (1986), 205–223.Google Scholar
Harrison, J. M. and Pliska, S. R. “Martingales and Stochastic Integrals in the Theory of Continuous Trading”, Stochastic Processes and their Applications 11 (1981), 215–260.CrossRefGoogle Scholar
Harrison, J. M. and Pliska, S. R. “A Stochastic Calculus Model of Continuous Trading: Complete Markets”, Stochastic Processes and their Applications 15 (1983), 313–316.CrossRefGoogle Scholar
Huygens, Chr. “De ratiociniis in ludo aleae” (1657).Google Scholar
Itô, K. “Multiple Wiener Integral”, Journal of the Mathematical Society of Japan 3 (1951), 157–169.CrossRefGoogle Scholar
Merton, R. C. “Theory of Rational Option Pricing”, Bell Journal of Economics and Management Science 4 (1973), 141–183Google Scholar
Samuelson, P. A. “Rational Theory of Warrant Pricing”, in: Cootner, P. H. (ed.), “The Random Character of Stock Market Prices”, MIT Press, Cambridge, Massachusetts (1964), 506–525.Google Scholar
Schweizer, M. “Varianten der Black-Scholes-Formel”, Diplomarbeit ETHZ, Zürich (1984).Google Scholar
Schweizer, M. “Hedging of Options in a General Semimartingale Model”, Diss. ETHZ no. 8615, Zürich (1988).Google Scholar
Sondermann, D. “Reinsurance in Arbitrage-Free Markets”, discussion paper no. B-82, University of Bonn (1988).Google Scholar
Wiener, N. “Differential-Space”, Journal of Mathematics and Physics 2 (1923), 131–174; reprinted in: Wiener, N. “Collected Works”, Volume 1, MIT Press (1976).CrossRefGoogle Scholar