Financial markets as adaptive systems

– We show, by studying in detail the market prices of options on liquid markets, that the market has empirically corrected the simple, but inadequate Black-Scholes formula to account for two important statistical features of asset fluctuations: “fat tails” and correlations in the scale of fluctuations. These aspects, although not included in the pricing models, are very precisely reflected in the price fixed by the market as a whole. Financial markets thus behave as rather efficient adaptive systems. Motivations. – Options markets offer an interesting example of the adaptation of a population (the traders) to a complex environment, through trial and errors and natural selection (inefficient traders disappear quickly). The problem is the following: an “option” is an insurance contract protecting its owner against the rise (or fall) of financial assets, such as stocks, currencies, etc. The problem of knowing the value of such contracts has become extremely acute ever since organized option markets opened twenty-five years ago, allowing one to buy or sell options much like stocks. Almost simultaneously, Black and Scholes (bs) proposed their famous option pricing theory, based on a simplified model for stock fluctuations, namely the (geometrical) continuous-time Brownian-motion model [1], [2]. The most important parameter of the model is the “volatility” σ, which is the standard deviation of the market price’s fluctuations. The Black-Scholes model is known to be based on unrealistic assumptions but is nevertheless used as a benchmark by all market participants. Guided by the Black-Scholes theory, but constrained by the fact that “bad” prices lead to “arbitrage opportunities” (that is, an easy way to make money), the option market fixes prices which are close, but significantly and systematically different from the bs formula. The aim of this paper is to exploit a recent reformulation of the Black-Scholes problem [3], [4] which allows one to incorporate non-Gaussian effects, and to compare directly option prices (as determined by the market) to their theoretical value. In order to compare theoretical prices