Mathematical models of Financial Markets

The Black & Scholes theory applies to more general contingent claims, including the Asian options. The relevant Kolmogorov equation is strongly degenerate and requires the development of an “ad hoc” regularity theory for its treatment and for the development of its numerical analysis. A challenge of the research team in this field is the regularity theory for the obstacle problem, which arises in the study of part-dependent American options.

A related goal is to provide solutions to the modelling of the asymmetry of risk in financial markets. There is increasing empirical evidence that classical financial models are not fully adequate and hould be pushed beyond their current limitations. A specific challenge is to capture skewness (directional asymmetry) in return distributions.

The Black & Scholes model relies on the stochastic theory to describe the financial markets in order to deal with the problem of hedging and pricing of European options. The so called Black & Scholes formula provides us with the fair price of an European option, and is widely used to compute the implied volatility of the Stock market.

Image: Numerical pricing of Geometric Asian Options in the Black & Scholes framework. Approximation of the option value using a Boundary Element Method.

 

[Ultimo aggiornamento: 03/02/2021 15:37:00]