## Project CEMAPRE internal

Title | Lévy processes and fractional processes: Financial models and Option pricing |

Participants | Pedro Febrer, João Guerra (Principal Investigator), Henrique Guerreiro |

Summary | Partial integro-differential equations (PIDE's) and fractional partial differential equations (FPDE's) appear in option pricing models with jumps (modeled by Lévy processes). These equations generalize the Black-Scholes PDE when the continuous diffusion dynamics for the underlying price is replaced by a Lévy process dynamics (including jumps). For certain Lévy processes, the pricing PIDE can be transformed in a Fractional partial differential equation involving fractional derivatives. Moreover, the analysis of a space-time fractional diffusion allows to express the price of a call option the price of an European call option as a Mellin Barnes integral and applying a multidimensional residue theorem, this price can be expressed as a numerical series. A particular fractional process, the fractional Brownian motion (fBm) has been extensively studied as it finds applications in many different situations, e.g., modelling turbulence or the log returns of a stock. This process is also very useful to model stochastic volatility, in particular it is used in the so-called rough volatility models. These models are highly consistent with empirical data. However, some open problems remain. In particular, they seem unable to explain the smile effect associated to the options on the VIX volatility index. In this project we will study some particular problems related to Lévy processes, fractional calculus, fractional Brownian motion and applications in Finance: I) Using the Mellin-Barnes integral and applying a multidimensional residue theorem, we want to obtain a series formula for the price of an European call option driven by the Variance Gamma process and generalizations of this process and compare the numerical efficiency of this formula with other numerical methods. II) Volatility modelling using rough fractional volatility models, where the volatility is driven by a fBm. |