Minimal description risky asset modeling

The geometric Brownian motion (Black-Scholes) model for the price of a risky asset which provides the theoretical underpinning of the huge financial derivatives industry stipulates that the log returns are independent and identically distributed and Gaussian. However, the empirical evidence shows the distribution to be leptokurtic (much higher peak and heavier tail than the Gaussian) and it also reveals strong dependence. The talk will be concerned with (1) the stylized features which one finds in real data and how these can be incorporated into models which are still simple enough to be practically useful and (2) some of the challenging mathematical questions which ensue.