From Brownian motion to operational risk: Statistical physics and financial markets

High-frequency returns of the DAX German blue chip stock index are used to test geometric Brownian motion, the standard model for financial time series. Even on a 15-s time scale, the linear correlations of DAX returns have a zero-time delta function which carries 90% of the weight, while the remaining 10% are positively correlated with a decay time of 53 s and negatively correlated on a 9.4-min scale. The probability density of the returns possesses fat tails with power laws whose exponents continuously increase with time scales. It is suggested that hydrodynamic turbulence may provide a phenomenological framework for the description of these data, and at the same time, open a way to use them for risk-management purposes, e.g. option pricing and hedging. Option pricing also is the cornerstone of credit valuation, an area of much practical importance not considered explicitly in most other physics-inspired papers on finance. Finally, operational risk is introduced as a new risk category currently emphasized by regulators, which will become important in many banks in the near future.