Project CEMAPRE internal
|Title||Modelling short term memory effects in financial derivatives|
|Participants||João Guerra, João Janela (Principal Investigator), Gilson Silva|
|Summary||The pricing of financial derivatives has been a central topic in modern finance for many years,|
since it was initially introduced by Merton, Black and Scholes back in 1973, and the famous
Black-Scholes formula was first proposed. Black-Scholes formula relies on a number of assumptions
that are not satisfied in many situations of practical interest. Successful generalizations of the
Black-Scholes equation requires a balance between two usually conflicting aspects: On one hand, a
sufficiently elaborated stochastic differential equation to simulate the value of the underlying
assets; on the other hand a stochastic equation that provides the desired mathematical properties to
carry out a proper analysis of the problem.
This project concerns with a generalization of the Jump telegraph drift diffusion model, where the
jumps can have stochastic amplitude. The selection of this model is motivated by the fact that it
can be reformulated as a system of PDEs, coupled by the reaction terms, that exhibits memory effects
in both the price and the volatility. Using this example we can compare the performance and accuracy
of the numerical solution of PIDEs with classical methods for PDEs.