Hedging by Sequential Regression: an Introduction to the Mathematics of Option Trading
Hedging by Sequential Regression: an Introduction to the Mathematics of Option Trading
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通过序贯回归进行对冲：期权交易数学简介
DOI：
10.1017/s0515036100008606
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发表时间：
1988
期刊：
影响因子：
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通讯作者：
Martin Schweizer
中科院分区：
文献类型：
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作者：
Hans Föllmer;Martin Schweizer
人们普遍认为，期权交易的数学理论已经有了重大突破。这一突破通常用布莱克-斯科尔斯公式来总结，产生了很多兴奋和一定的神秘感。在数学方面，它涉及到来自鞅理论和随机微积分的高级概率技术，只有具有高度数学复杂性的一小群专家才能接触到这些技术；因此才有神秘性。在其实际意义上，它提供了令人兴奋的前景。它的承诺是，通过选择合适的交易策略，可以完全消除处理期权所涉及的风险。自1987年10月以来，市场情绪变得更加清醒。但也有数学上的原因表明，人们的预期应该降低。这将是本说明性陈述的主要观点。我们认为，通常情况下，处理期权所涉及的风险具有不可减少的内在部分。这种内在风险可能比先验风险小得多，但总的来说，人们不应指望它会完全消失。在这种更冷静的视角下，布莱克-斯科尔斯公式背后的数学技巧并没有失去它的任何重要性。事实上，它应该被视为一种序贯回归方案，其目的是降低其内在核心的先验风险。我们首先简要介绍一下布莱克-斯科尔斯公式在货币期权方面的情况。然后，我们发展了一个离散时间的一般回归方案，首先是在一个基本的两周期模型中，然后是在一个多周期模型中，它涉及到了鞅的考虑，并为扩展到连续时间奠定了基础。我们的方法是建立在对Black-Scholes公式的解释和推广基础上的。这是由克雷普斯和哈里森首创的，例如，哈里森和普利斯卡的出色调查(1981,1983)。将布莱克-斯科尔斯方法嵌入到序贯回归方案中的想法可以追溯到第一作者与D·桑德曼的合作工作。这是Schweizer(1984)和Föllmer和Sondermann(1986)在连续时间和鞅假设下得到的结果。Schweizer(1988)在一个一般的半鞅模型中讨论了这些问题。
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.