Quantitative approaches to modelling and simulation in some problems of mathematical finance

Abstract

We study two important issues that arise in modern mathematical finance. The first issue is the estimation of a path-dependent derivative. Unlike a vanilla option, path-dependent derivatives have no closed-form solution for its value and Greeks in general. Hence the general approach is to use the Euler scheme but it has the shortcoming that the simulation results have bias. To resolve this, we devise an exact simulation scheme for the boundary hitting time of Brownian motion, and then we use this to estimate the value and the Greeks of a path-dependent derivative efficiently. The second issue is the correlation skew in equity markets that stand out especially when market collapses. Though there have been an evident correlation skew in equity markets, most of models handle the correlation as a fixed constant derived from its historical correlation. To reflect this phenomenon to the modelling and to the estimation of multi-asset derivatives, we derive a local correlation model of which dynamic is determined by the volatilities of underlying assets. And we demonstrate from experiments that our correlation model outperforms the use of a historical correlation in its forecasting ability.