Non-Probabilistic Jump Modelling for Financial Derivatives

Abstract

This paper applies the uncertain nonlinear parameter approach, originally by Avellaneda et al. and Lyons, to model non-local changes in financial variables and the resulting impact on portfolios of derivatives and their underlying assets. It formulates the non-probabilistic uncertainty assumptions as a governing system of nonlinear PDEs about both the spatial and the time dimensions of the variables. The solution technique can be decomposed as a control problem for the former and a free-boundary problem for the latter. It is shown that, modelled in a non-probabilistic way any jump in a variable can be treated in the same manner as a dividend on equity.