Nonlinear diffusion models in financial mathematics

现代金融市场提供和交易越来越复杂的“奇异”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金融变量的价值的时间历史。路径依赖型期权在过去几年里变得越来越受欢迎，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。在量化金融学中，此类衍生品的定价不是一个微不足道的问题。对于非线性扩散模型，许多一般路径依赖期权(如亚式期权)的精确定价公式是未知的，而且精确有效的数值定价算法也相当稀少。这个研究项目的主要目的是开发在精确可解的非线性扩散模型下对金融衍生品定价的有效计算算法。由于路径依赖期权定价的数值密集性，特别是在许多基本维度上，本研究的一个重要部分涉及在高性能计算集群上实现的并行算法的开发。本程序的另一个重点是可解扩散模型。对于一个可解的模型，转移概率密度函数和其他对衍生品定价至关重要的量以封闭的形式表示。扩散模型的可解性使我们可以构造避免近似格式的精确模拟算法。最后，由于现有的模型并不总是能够准确和真实地捕捉到各种基础市场变量的市场期权价格的行为，该程序将致力于开发基于可解扩散模型、切换马尔可夫过程和带跳跃的随机时间变化的新的定价模型。

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.