Solvable and Other Stochastic Models for Risk Modeling and Asset Pricing in Quantitative Finance

随机建模是金融数学的基础，是金融学和数学的核心学科。随机过程为量化所有经济部门的金融风险提供了一个自然的环境。金融衍生品是在所有经济部门都大量交易的或有债权。金融数学的一个重要领域仍然是开发和应用现实和易于处理的随机模型来描述风险因素，并在衍生品合约的估值和对冲中实施这些模型。许多金融衍生品也具有内在的路径依赖性。这种具有内在路径依赖关系的复杂金融工具，对于减轻特定风险非常有用。此外，资产的某些类型的路径依赖可以在将期权定价与特定金融公司相关的信用风险联系起来方面发挥关键作用。例如，路径相关的数量，如资产首次到达临界水平的时间或低于给定临界水平的资产占用时间，以及这些数量的变化，对于评估受违约风险影响的金融合同是有用的。我的研究将集中于可处理随机模型的进一步新发展，并将模型应用于定量金融中的各种重要问题，包括风险建模、衍生品定价和对冲，以及对历史市场数据的模型校准。我研究的一个主要部分将继续开发和建立我之前在所谓的可解模型上的进展，这些模型具有当前和新的应用领域，也将把期权定价和信用风险联系起来。这一建议将导致使用这些模型的金融应用的新模型和有效的数值算法。我研究的另一个组成部分将利用蒙特卡罗模拟算法。将使用不同的方法来开发可处理的现实模型。一种方法将尝试将我们已经成功地为单资产可解模型开发的数学方法扩展到多资产领域。另一种方法将利用copula来构建可处理的多资产模型，该模型将考虑到一些真实的市场观察效应。这些模型的实际实施将导致有效的算法模型校准和定价和对冲多资产金融产品。一些算法还将利用高性能计算软件。

Stochastic modelling is fundamental to financial mathematics, which is a core discipline in both finance and mathematics. Stochastic processes provide a natural setting for quantifying financial risk in all sectors of the economy. Financial derivatives are contingent claims that are heavily traded in all sectors of the economy. An important area of financial mathematics remains the development and application of realistic and tractable stochastic models for describing risk factors and the implementation of these models in the valuation and hedging of derivative contracts. Many financial derivatives are also inherently path-dependent. Such complex financial instruments, with built-in path dependencies, are useful for mitigating specific risks. Moreover, certain types of path dependencies of an asset can play a key role in linking option pricing with the credit risk associated to a given financial firm. For example, path-dependent quantities such as the first passage time of an asset to a critical level or the occupation time of the asset below a given critical level, as well as variations of such quantities, are useful for valuing financial contracts subject to default risk. My research will focus on further new developments of tractable stochastic models and implementing the models to various important problems in quantitative finance involving risk modelling, derivative pricing and hedging, and model calibration to historical market data. A main part of my research will continue to exploit and build upon my previous advancements on so-called solvable models with current and new areas of application that will also link option pricing and credit risk. This proposal will lead to new models and efficient numerical algorithms for financial applications employing such models. Another component of my research will exploit Monte Carlo simulation algorithms. Different approaches will be used to develop tractable realistic models. One approach will attempt to extend the mathematical methodology that we have already successfully developed for single-asset solvable models into the multi-asset domain. The other methodology will exploit the use of copulas in constructing tractable multi-asset models that will take into account some real market observed effects. The actual implementation of these models will lead to efficient algorithms for model calibration and pricing and hedging multi-asset financial products. Some of the algorithms will also exploit the use of high-performance computing software.