Some Further Results on the Tempered Multistable Approach

Typical approaches incorporating jumps in financial dynamics, such as the Variance Gamma and CGMY models, can be made to depart from the i.i.d. hypothesis by using a stochastic clock. In such a context, the introduction of a dispersion of the clock is equiva- lent to the introduction of a dispersion of the volatility itself. A distinct route that yields
comparable features is that of adding a jump component to a stochastic volatility process, or of considering, in discrete time, leptokurtic innovations within a GARCH process. In this article, we take a third route and we provide a study on tempered multistable pro- cesses, which convey both jumps and autocorrelation from their very construction, and on some of their applications in finance. We obtain the multivariate characteristic function of the asymmetrical field-based tempered multistable process and we study the autocorrela- tions that stem from the use of this model. We concentrate on three types of applications in finance: we study the term structures of Value-at-Risk that can be obtained with this model, we perform a calibration on stock index data, and we also conduct a calibration on derivatives prices.