Hidden markov models in quantitative finance

我的研究项目包括研究市场变量动态建模问题的相关项目，这是财务评估、风险管理和资产配置的基本问题。这些项目可以分为两大类。第一类项目考虑了金融模型的使用，这些模型包含了假定遵循隐马尔可夫模型的经济体制的转变。在此框架下，希望(i)通过求解偏微分方程或常微分方程系统获得金融合约公平价格的数值近似值，（ii）对死亡率和利率都具有随机演化且不一定相互独立的保险和养老金相关产品进行估值，以及（iii）使用Esscher变换或构建其他风险中性措施在马尔可夫市场的不完全设置中进行估值。第二类项目涉及金融模型的使用，其中经济制度的切换由一个隐藏的弱马尔可夫链在离散或连续时间内调制。弱马尔可夫链能够用记忆捕捉金融过程的动态。在该公式下，(i)将采用随机滤波理论来估计模型参数；（ii）估计的参数将用于为金融衍生品定价，计算风险措施和设计最佳资产配置；（三）将讨论几个重要的统计推断问题。解决上述问题将使金融机构能够相应地量化和管理其风险状况，并为其他研究人员分析金融相关问题提供动力。这项研究工作将有助于建立一个有效、运作良好和稳定的金融市场。

My research program comprises of related projects investigating issues on the dynamic modelling of market variables, which is a fundamental concern in financial valuation, risk management and asset allocation. These projects can be grouped into two general categories. The first category of projects considers the use of financial models that incoporate the shift of economic regimes assumed to follow a hidden Markov model. Under this framework, it is desired to (i) obtain numerical approximation for the fair price of financial contracts by solving a system of partial or ordinary differential equations, (ii) value insurance and pension-related products where both mortality and interest rates have stochastic evolution not necessarily independent of each other and (iii) employ Esscher transform or construct other risk-neutral measures to perform valuation in the incomplete setting of a Markov market.  The second category of projects deals with the use of financial models where the switching of economic regimes is modulated by a hidden weak Markov chain in either discrete or continuous time. Weak Markov chains are capable of capturing the dynamics of financial processes with memory. Under this formulation, (i) the theory of stochastic filtering will be adopted to estimate model parameters; (ii) estimated parameters will then be used to price financial derivatives, calculate risk measures and design an optimal allocation of assets; and (iii) several important statistical inference questions will be addressed.  Resolving the aforementioned issues would enable financial institutions to quantify and manage their risk profile accordingly and provide impetus to other researchers in the analysis of related problems in finance. This research work will be instrumental in the attainment of an efficient, well-functioning and stable financial market.