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In 1973, Black and Scholes [1] developed the mathematical theory of perfectly hedged options in an arbitrage-free lognormal random walk asset price model. Fundamental for the mathematical formulation and deeply rooted in the mind of both theoreticians and practitioners, evolved the notion that a general strategy of pricing and coverage of derivatives involves the hedging away of all risks. However, we do not live in a Black-Scholes world:
* asset prices do not follow a continuous time lognormal processes, but may exhibit large jumps and be distributed with fat tails, in addition to present subtle correlations between increments at different times (the markets are ‘incomplete’),
* the replicating portfolio cannot be implemented exactly, since it involves continuous rebalancing: several ‘imperfections’ such as transaction costs, delays and lack of liquidity necessarily introduce a finite time scale for transactions.
Confronted with these practical problems, many authors have attempted to generalize the Black and Scholes strategy. Here, we present an intuitively appealing formalism which is flexible enough to address these problems efficiently and show how it can be implemented practically. Our method is based on a global (integral) representation of wealth balance, in contrast to Black-Scholes local (dfferential) approach. The mathematical tool-box is that of functional integration and derivation, in contrast to the standard Ito calculus and partial differential equation formalism. We propose two main new concepts:
* The minimization of the non zero residual risk as a criterion to fix an optimal strategy [2]. One should note in this respect that somewhat related ideas have previously been proposed in the mathematical literature in a rather formal way [3]. Furthermore, our hypothesis are more general and the resulting formulae are easily implemented numerically.
* For strongly fluctuating assets, option pricing and its hedging is obtained by a ‘tail chiseling’ of wealth distributions as a treatment of large risks [4].
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In 1973, Black and Scholes [1] developed the mathematical theory of perfectly hedged options in an arbitrage-free lognormal random walk asset price model. Fundamental for the mathematical formulation and deeply rooted in the mind of both theoreticians and practitioners, evolved the notion that a general strategy of pricing and coverage of derivatives involves the hedging away of all risks. However, we do not live in a Black-Scholes world:
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