Stochastic modelling in mathematical and computational finance

数学和计算金融中的随机建模

• 批准号: RGPIN-2015-04125
• 负责人: Hyndman, Cody
• 金额: $ 1.02万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=654177
• 关键词: Stochastic; modelling; mathematical; computational; finance

Stochastic modelling in mathematical and computational finance is the focus of the proposed research program.  Mathematical and computational finance considers the uncertain future behaviour of financial or economic variables and systems; presents a theory for the valuation and risk management of derivative securities; and provides a quantitative framework for examining various investment, managerial, and regulatory decisions. Various probabilistic models, statistical estimation methods, and computational algorithms which are motivated by financial applications shall be considered.  Addressing applied problems in a realistic framework also drives new theoretical research and is the impetus for novel theoretical advances in probability and statistics.  The research integrates various aspects of mathematical and computational finance starting with the development of new stochastic models for fundamental financial and economic quantities such as interest rates and asset volatility. We shall develop new pricing and risk management theories methodologies that allow market participants to value and hedge financial derivatives exposed to credit risk.  We also plan to study the stochastic equations which characterize these new valuation and risk management methods, deriving explicit solutions where possible but  focusing on realistic modelling which requires the creation of new efficient computational algorithms for solving these equations.***The first objective of the proposed research program is the study of forward -backward stochastic differential equations (FBSDEs) and applications in mathematical finance. An FBSDE is a coupled system of stochastic equations with components that evolve forward in time from a specified initial condition and components that evolve backward in time from a random terminal condition.  We shall use FBSDEs to characterize a new pricing methodology for credit risk derivatives such as defaultable bonds and extend this method.  The second objective is the development of numerical methods for the solution of FBSDEs since the class of FBSDEs with explicit solutions is limited.  We shall further develop a new numerical method we created, based on the fast Fourier transform, to higher dimensions.  The third objective involves the study of problems in mathematical finance that can be characterized as producing or depending on large amounts of high- dimensional data. Our goal is to extend to financial contexts certain modelling and statistical techniques for high- dimensional data that effectively reduce the dimension to the point that an accurate lower dimensional model can be implemented.  Infinite dimensional models of forward interest rate processes shall be the first example considered so that, by reducing the dimension,  we can create new parsimonious financial models that preserve the key features of the theoretical model and the observed data.**

数学和计算融资中的随机建模是拟议研究计划的重点。数学和计算融资认为财务或经济变量和系统的未来行为不确定；提出了衍生证券价值和风险管理的理论；并提供了一个定量框架，用于检查各种投资，管理和监管决策。应考虑各种概率模型，统计估计方法和计算算法。解决现实框架中的应用问题也推动了新的理论研究，这是概率和统计学新理论进步的动力。该研究纳入了数学和计算融资的各个方面，从开发新的随机模型开始，用于基本财务和经济量，例如利率和资产波动。我们将开发新的定价和风险管理理论方法，使市场参与者能够重视和对冲金融衍生品接触信用风险。我们还计划研究特征这些新价值和风险管理方法的随机方程，并在可能的情况下得出明确的解决方案，但专注于现实的建模，这需要创建用于求解这些方程式的新有效的计算算法。 FBSDE是一个随机方程的耦合系统，其组件是从指定的初始条件和成分从随机终端条件中向后发展的。我们将使用FBSDE来表征用于信用风险衍生品（例如默认债券）的新定价方法，并扩展了此方法。第二个目标是开发用于FBSDE的溶液的数值方法，因为具有显式溶液的FBSDES类是有限的。我们将进一步开发一种我们基于快速傅立叶变换创建的新数值方法。第三个目标涉及研究数学金融问题的研究，这些问题可以根据大量的高维数据而定为产生或定义。我们的目标是将某些建模和统计技术扩展到高维数据的某些建模和统计技术，从而有效地将维度降低到可以实现准确的较低维模型的程度。远期利率过程的无限维度模型应为第一个考虑的示例，以便通过降低维度，我们可以创建新的简约财务模型，以保留理论模型的关键特征和观察到的数据。** **