Fat-Tails and the Physics of Financial Markets
Lisa Borland, Evnine & Associates, Inc.
The dynamics of financial markets and the price formation process is an example of a high-dimensional complex system at work. In great volumes around the globe, individuals and institutions alike are almost continuously placing orders to buy or sell a particular stock at a particluar price. There is clearly a need to understand and model the fluctuations that drive these processes, for purposes that range from pure speculation to the fair pricing of derivative instruments such as options, or for the important task of hedging financial risk. At the same time one would like a model that is somewhat intuitive and analytically tractable. The most popular model, made famous by Black, Scholes and Merton in their Nobel prize winning work which allowed for elegant solutions to the option pricing problem, is essentially a Brownian motion, resulting in Gaussian statistics for the price changes. However, real financial time series exhibit a slew of anomalous statistics - or stylized facts - such as persistent fat tails, long-range memory and time-reversal assymmetry. We present here a non-Gaussian model that generalizes the standard one in a way that reproduces many of the stylized facts while still allowing for closed-form solutions which allow efficient pricing of options and other derivatives such as credit default swaps. The model is based on a statistical feedback process, which aims to incorporate - in a phenomenological fashion - the impact of market participants reacting over one or multiple time-scales.