Nonlinear diffusion models in financial mathematics

点击翻译按钮获取中文摘要

Modern day financial markets offer and trade increasingly more complex, "exotic" products. A path-dependent derivative is a financial contract whose value depends on the time history of values of other, more basic, underlying financial variables. Path-dependent options have become increasingly popular over the last few years because of the greater precision with which they allow investors to choose or avoid exposure to well-defined sources of risk. Pricing of such derivatives is a non-trivial problem in quantitative finance. For nonlinear diffusion models, exact pricing formulas for many general path dependent options such as Asian style options are not known, and, moreover, accurate and efficient numerical pricing algorithms are rather scarce. The primary aim of this research program is the development of efficient computational algorithms for pricing financial derivatives under exactly solvable nonlinear diffusion models. Due to the numerically intensive nature of pricing path-dependent options, particularly in many underlying dimensions, a significant part of this research involves the development of parallel algorithms to be implemented on the high performance computing clusters. The other focus of this program is on the solvable diffusion models. For a solvable model, the transition probability density functions and other quantities that are fundamental to derivatives pricing are represented in closed form. The solvability of a diffusion model allows us to construct exact simulation algorithms avoiding approximate schemes. Finally, since the existing models do not always succeed in capturing the behaviour of market option prices accurately and realistically enough across various underlying market variables, the program will be aimed at the development of new pricing models based on the unification of a solvable diffusion model, a switching Markov process, and stochastic time change with jumps.