Hedging derivatives: from finance to actuarial science

金融和保险产品的定价、套期保值和风险管理是近二十年来金融和保险研究的重点。虽然巨大的研究努力已经解决了重要的方面，仍然有“难题”尚未解决。这个研究项目集中在金融计量经济学和精算学文献的两个主要方面。 该项目的第一部分调查金融衍生工具，特别是期权和波动性衍生工具的定价和对冲。虽然绝大多数数学金融文献在连续时间内研究这些问题，主要是由于这种设置提供的易处理性，但利息的数量通常是在固定日期采样的，因此，离散时间设置可能更合适。在这个项目中，我们开发了新的方法，定价和对冲的期权和方差互换的基础上广泛的一类非线性时间序列模型，我们说明了它们的连续时间限制的相互作用，通过研究相应的数量之间的收敛性的兴趣。我们计划进一步研究如何将这些方法应用于对冲保险产品。 在提案的第二部分，我们探讨了保险公司面临的主要挑战之一，即量化其最低资本要求，以及将其最佳投资于多样化的金融资产组合。出于偿付能力II指令，我们提出了新的优化问题的非寿险公司，不同的偿付能力约束的监管机构。这些优化问题将在静态和动态设置中实现。

The pricing, hedging and risk management of financial and insurance products have been key objects of study in over two decades. Although tremendous research efforts have addressed important aspects, there are still 'puzzles' yet to be solved. This research project focus on two main strands from the financial econometrics and actuarial science literature. The first part of the project investigates the pricing and hedging of financial derivatives, in particular options and volatility derivatives. Although a vast majority of the mathematical finance literature examines these in continuous time, mainly due to the tractability offered by this setup, quantities of interest are in general sampled at fixed dates and therefore, a discrete-time setting might be more appropriate. In this project we develop new methods for pricing and hedging of options and variance swaps based on a broad class of non-linear time series models and we illustrate the interplay with their continuous time limits, by studying the convergence between the corresponding quantities of interest. We further plan to investigate how these methods can be applied to hedging insurance products. In the second part of the proposal, we explore one of the major challenges faced by an insurance company regarding the quantification of its minimum required capital and its optimal investment into a well-diversified portfolio of financial assets. Motivated by the Solvency II Directives, we propose new optimization problems for non-life insurance companies, with different solvency constraints imposed by regulators. These optimization problems will be implemented in both static, and dynamic settings.