Financial modelling and derivatives pricing under alternative Stochastic processes

为风险资产价格过程开发现实的随机模型，并将其应用于一般奇异衍生品的定价，是现代金融数学的核心。大量的研究致力于这一重要领域，这构成了计算和理论上的挑战。目前的模型远远超出了著名的布莱克-斯科尔斯模型，该模型不支持通常观察到的特征，如资产价格回报的波动率聚集和厚尾分布。观察到的杠杆效应和市场隐含的波动率微笑在这个模型中也完全不存在，在该模型的一些数学扩展中也没有很好地复制。这些模型差异的解决是一个根本性的问题，这继续通过使用更现实的模型来激发人们对金融建模和期权定价的兴趣。我的研究目标是在金融数学的这一关键领域和相关关键领域取得进一步进展。这项建议侧重于单一和多资产领域替代随机过程的进一步创新发展，以及在衍生品定价中的几个应用。我的研究的一个基本组成部分涉及使用我们新开发的易于分析的模型家族。到目前为止，我们的研究表明，我们的模型具有丰富的能力，能够真实地描述期权市场的波动性、微笑和偏斜。我最近的工作还为这些新模型开发了首次通过时间密度、势垒选项和经验丰富的回顾选项的解析精确谱展开。我们最近还成功地开发了在我们新的波动率微笑模型的子家族下为奇异期权定价的高效算法。这一提议将继续建立在这种替代随机模型的新发展和应用的基础上。描述这种过程的高效数值算法的发展导致了在金融以及涉及随机过程的数学建模的许多其他领域中的各种应用。这项研究将产生的数学和计算结果预计将对金融数学领域产生重大影响。

The development of realistic stochastic models for risky asset price processes and their implementation to the pricing of generally exotic derivatives is at the heart of modern day financial mathematics. A vast body of research is devoted to this important area which poses computational and theoretical challenges. Current models go well beyond the celebrated Black-Scholes model which does not support commonly observed features such as volatility clustering and fat tail distributions for asset price returns. The observed leverage effect and the market implied volatility smiles are also completely absent in this model and not well replicated in some mathematical extensions of the model. The resolution of these model discrepancies is a fundamental issue and this continues to fuel interest in financial modelling and option pricing with the use of more realistic models. The goal of my research is to make further progress in this and related critical areas of financial mathematics. This proposal focuses on further innovative developments of alternative stochastic processes in the single and multi-asset domain with several applications to derivatives pricing. An underlying component of my research involves the use of our newly developed families of analytically tractable models. Our research to date has shown that our models are rich in their ability to realistically describe option market volatility smiles and skews. My most current work also develops analytically exact spectral expansions for first passage time densities, barrier options and seasoned lookback options for these new models. We have also recently succeeded in developing efficient algorithms for pricing exotic options under subfamilies of our new volatility smile models. This proposal will continue to build on new developments and applications of such alternative stochastic models. The development of efficient numerical algorithms for describing such processes leads to various applications in finance as well as in many other areas of mathematical modelling that involve stochastic processes. The mathematical and computational results that will be generated by this research are expected to significantly impact the field of financial mathematics.