Abstracts

Among the typical approaches incorporating jumps in financial dynamics, we can cite the Variance Gamma and the CGMY models. In such models, the departure from the i.i.d. hypothesis can be achieved by using a stochastic clock. Indeed, introducing a dispersion of the clock can be seen as equivalent to introducing a dispersion of the volatility itself. In the CGMY model, this amounts to introducing a stochastic C, where C is the parameter driving the size of fluctuations. In this article, we take a different route by directly modeling the fluctuations of the fourth moment of asset return distributions. We do so because the kurtosis is the key driver of financial crises and we believe it is important to directly describe the nature of this indicator. In the example of the CGMY model, our approach amounts to modeling a time-varying Y, where Y is the parameter that is related to the size of tails and to kurtosis. To tackle this problem, we propose a tempered multistable setting and derive its main characteristics. We apply this setting to both risk management and option pricing.