Research projects

Project CEMAPRE internal

TitleLévy processes, fractional differential equations and fractional processes: Financial models and Option pricing
ParticipantsJoão Guerra (Principal Investigator), Henrique Guerreiro
SummaryPartial 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 (as the CGMY), the pricing PIDE can be transformed in a Fractional
partial differential equation involving fractional derivatives.

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. The fBm is a
one-parameter family of Gaussian processes derived from the Brownian motion. The main feature of
fBm is that the increments are no longer independent, so the process has long-range memory. Some
applications to finance can be formulated in terms of Stochastic Differential Equations with
to fBm. These kind of equations also involve fractional derivatives and fractional calculus.
Therefore, the techniques used in the study of fractional partial differential equations are also
useful in the study of SDE's driven by fBm.

In this project we will study some particular problems related to Lévy processes, fractional
calculus, fractional partial differential equations, fractional Brownian motion and applications in

I) The option pricing FPDE and numerical methods for this FPDE for market models where the driving
process is a Generalized tempered stable Lévy process and similar processes

II) Volatility modelling using rough fractional volatility models, where the volatility is driven by
a fBm.