Mathematical Modelling and Problem Solving in Finance and Insurance

该研究计划旨在开发金融和保险问题的先进数学模型，以帮助科普未来的金融危机，减轻它们对社会的负面影响。我的研究计划有四个主要的研究方向：1）建模，定价，证券化和管理与寿命增加相关的金融风险; 2）在流动性成本存在的情况下定价和对冲金融工具; 3）研究不同金融/精算制度（如牛市和熊市制度）的金融和保险问题。我的研究计划的长期目标是整合这三个研究方向。长寿风险是指由于寿命增加，养老基金没有足够的钱支付退休人员的情况。长寿风险的管理对社会至关重要，特别是在维持金融和养老金体系稳定方面。长寿风险转移金融工具的设计，如长寿挂钩衍生品是至关重要的，因为保险业无法单独处理这种风险。因此，需要数学模型来为这些金融工具定价。经典资产定价理论假设交易者作为价格接受者，也就是说，他们对支付或接收的价格没有影响。这一假设的放松导致价格对已实现收益的影响，这被称为流动性成本。在流动性成本的存在下，期权的定价变得更加复杂，因为人们必须找出流动性对期权价格的影响。制度转换是指数学金融/精算模型的特征受到几种制度影响的情况（例如非危机制度与危机制度）。这种制度转换与金融和精算数据中观察到的某些周期性是一致的。 长寿风险和流动性成本处理不当可能引发下一次金融危机。 该研究计划的主要创新方面是将适当的数学技术巧妙地应用于社会问题，例如长寿风险，这是一个当前和紧迫的问题。本研究提案的影响将是减轻未来金融危机对社会的负面影响。这是通过以下方式实现的：1）设计新的金融工具并为其定价，以减轻风险（例如，长寿债券将减少长寿风险），2）提供金融衍生工具的准确估值/定价（通过纳入流动性成本效应），以及3）考虑市场转换机制模型（例如危机与非危机），这反过来又会导致更准确地估计金融危机的影响。 总的来说，以前的模型似乎忽略了在流动性成本和市场转换制度的背景下设计和定价减轻风险的金融工具。HQP将被纳入本研究计划。

This proposed research program aims to develop advanced mathematical models of financial and insurance problems to help cope with future financial crises by mitigating the negative effects they will have on society. My research program has four major research directions: 1) the modelling, pricing, securitization, and management of the financial risks associated with increasing longevity; 2) pricing and hedging financial instruments in the presence of liquidity costs; 3) studying financial and insurance problems with different financial/actuarial regimes such as bull and bear market regimes. The long term objective of my research program is integrating the three research directions. Longevity risk refers to situations when pension funds do not have enough money to pay the retiree due to increased longevity. The management of longevity risk is crucial for society, especially in maintaining the stability of the financial and pension system. The design of longevity risk transfer financial instruments such as longevity linked derivatives is of paramount importance because the insurance industry alone can not handle this risk. Therefore, there is a need for mathematical models to price such financial instruments. Classical asset pricing theory assumes that traders act as price takers, that is, they have no impact on the prices paid or received. The relaxation of this assumption leads to price impacts on realized returns, which are refer to as liquidity costs. In the presence of liquidity costs the pricing of an option becomes more involved since one has to find out the liquidity effects on the option price. Regime switching refers to situations when the characteristics of mathematical financial/actuarial models are affected by several regimes (e.g. non crisis versus crisis regimes). The regime switching is consistent with some cyclicality observed in financial and actuarial data. The longevity risk and the mishandling of liquidity costs may trigger the next financial crisis. The main innovative aspect of this research program is the skillful application of appropriate mathematical techniques to societal problems such as the longevity risk, a current and pressing issue. The impact of this research proposal will be in mitigating the negative effects on society of future financial crises. This is done by: 1) designing and pricing of new financial instruments to mitigate risk (e.g. longevity bonds that will lessen longevity risk), 2) providing accurate valuation/pricing of financial derivatives (through incorporating the liquidity costs effects), and 3) considering market switching regimes modelling (such as crisis versus non crisis) which in turn would lead to a more accurate estimation of financial crises effects.      In general previous models seemed to have overlooked designing and pricing risk mitigating financial instruments in the context of liquidity costs and market switching regimes. HQP will be included in this research program.