Solvable models in option pricing and credit risk

数学金融学研究金融市场中的问题，如投资、对冲、定价和风险管理。在过去的30年里，数学金融学的思想和方法已被全球领先的银行、保险公司和养老基金采用，并已成为金融市场有效运行的关键。数学金融学的核心是金融数据的建模。一个好的模型应该同时具有现实性和计算效率。那些对重要数量(如欧式期权价格)有明确公式的模型被称为“可解”模型，是业内最受欢迎的模型。问题是，只有几个可解的模型，而且大多数模型都太简单了，无法捕捉到市场数据中观察到的现象。我提议做的工作是构建新的可解随机模型(并对现有模型进行分类)，这些模型可能更适合于数学、金融以及应用科学中的其他领域的特定问题。这项研究涉及应用数学、随机过程理论和特殊函数理论等多个学科的交叉。这些新的可解模型在数学金融学中的可能应用包括美式期权的定价、奇异期权、信用风险建模、利率和随机波动率建模。

The Science of Mathematical Finance deals with problems in financial markets, such as investing, hedging, pricing and risk management. Over the last 30 years ideas and methods of Math Finance have been adopted by the leading banks, insurance companies and pension funds around the globe and have become essential for the efficient operation of financial markets.At the heart of Mathematical Finance lies the modelling of financial data. A good model should be realistic and computationally efficient at the same time. Those models which have explicit formulas for important quantities (such as the prices of European options) are called "solvable" and are the most popular in the industry. The problem is that there are just a few solvable models and most of them are just too simple to capture the phenomena observed in market data. The work I propose to do is to construct new solvable stochastic models (and classify existing ones), which might be better suited for the particular problems in math finance, as well as other areas in applied science. This research lies at the intersection of several branches of science, such as applied mathematics, theory of stochastic processes and theory of special functions. The possible applications of these new solvable models in Mathematical Finance include pricing of American type options, exotic options, credit risk modelling, interest rate and stochastic volatility modelling.