Nonlinear diffusion models in financial mathematics

现代金融市场提供和交易越来越复杂的“异国情调”产品。路径依赖衍生品是一种金融合约，其价值取决于其他更基本的潜在金融变量的时间历史。路径依赖期权在过去几年中越来越受欢迎，因为它们允许投资者更精确地选择或避免暴露于明确定义的风险来源。这类衍生品的定价是定量金融学中一个重要的问题。对于非线性扩散模型，许多一般的路径相关期权如亚式期权的精确定价公式还不清楚，而且精确有效的数值定价算法也相当缺乏.本研究计划的主要目的是开发有效的计算算法，为金融衍生品定价下精确可解的非线性扩散模型。由于数值密集的性质定价路径依赖的选项，特别是在许多基本的尺寸，本研究的一个重要组成部分涉及到开发的并行算法上实现的高性能计算集群。这个程序的另一个重点是可解扩散模型。对于可解模型，转移概率密度函数和其他对衍生品定价至关重要的量都以封闭形式表示。扩散模型的可解性使我们能够构造精确的模拟算法，避免近似方案。最后，由于现有的模型并不总是成功地捕捉市场期权价格的行为准确和现实的各种潜在的市场变量，该计划将旨在开发新的定价模型的基础上统一的可解扩散模型，切换马尔可夫过程，和随机时间变化跳跃。

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.