Financial markets as adaptative ecosystems

The option markets offer a very interesting example of the adaptation of a population (the traders) to a complex environment, through trial and errors and natural selection (unefficent 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 when organised 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) Brownian motion model. The most important parameter of the model is the ‘volatility’ σ, which is the standard deviation of the market price’s relative fluctuations. Guided by the Black-Scholes theory, but constrained by the fact that ‘bad’ prices lead to arbitrage opportunities, option markets agree on prices which are close, but significantly and systematically different from the BS formula. Surprisingly, a detailed study of the observed market prices clearly shows that, despite the lack of an appropriate model, traders have empirically adapted to incorporate some subtle information on the real statistics of price changes.