BFS 2002

Contributed Talk

We propose to model share prices with exponentials of time-changed $\alpha$-stable Lévy processes. In particular, if $X(t)=S(T(t))$ are logaithmic returns, we find empirical evidence, using high-frequency data, of heavy tailedness of returns in the natural and changed time scales, and of long memory in the process $T(t)$ modelling the so-called business time scale. These results are quite puzzling, as they cannot be explained by any of the subordinated models recently introduced in the financial and mathematical literature. Furthermore, we partially extend this approach to model the joint distribution of returns with a certain class of operator stable laws with diagonal exponent, thus allowing each return to have a different index of tail thickness. The resulting model features rich dependence structure and is again consistent with prices described by the exponential of a Lévy process. Finally, we discuss estimation and simulation issues related to this type of processes.