Pricing and Hedging Equity-Linked Products Using Risk Measures

该项目的主要目标是开发基于风险度量的股票指数年金的定价和对冲技术。股票指数年金（EIA）是一种保险产品，其收益与股票市场的表现挂钩。它在保证最低回报率的同时，对股票指数（如标准普尔500指数）的表现提供有限的参与。在过去的20年里，环保是最具创新性的年金产品。自出道以来，他们变得越来越受欢迎；根据美国人寿保险和市场研究协会（LIMRA） 2012年第四季度美国个人年金销售调查显示，美国个人年金销售额稳步增长，2012年创下339亿美元的历史新高。股票指数型年金通常包含一些死亡、退保、提款和累积担保，以保护投保人。这些担保的成本是根据财务和精算模型进行评估的。对这些产品进行评估的研究使用了各种方法；参见Hardy（“投资保证：股票挂钩人寿保险的建模和风险管理”，2003）及其参考文献。我主要感兴趣的是应用优化技术来降低股票指数年金投资组合的风险。在一个完全的市场中，所有的金融债权都可以被完美地对冲。然而，在不完全金融市场中，证券的数量少于可能结果的数量。在这种情况下，完美的对冲是不可能实现的。当考虑到复杂的潜在资产模型、死亡风险和撤资时，这是合适的框架。在这种情况下，可以使用不同的定价和对冲技术；我们将重点关注风险措施。在使用风险措施定义对冲策略时，可以实现两个不同的目标。一方面，保险公司可以在给定的风险水平下最小化对冲成本。另一方面，发行人可以最大限度地减少与自我融资策略选择的对冲策略相关的风险。Gaillardetz和Moghtadai（使用迭代风险度量的股票关联产品评估，2013）关注后者，其中对冲策略是由潜在的风险资产和无风险资产组成的。利用Rockafeller和Uryasev（一般损失分布的条件风险价值，2002）的结果，将尾部风险价值的评估转化为线性规划问题，我们提出将对冲策略推广到多个风险资产。我们还将限制我们的对冲策略为自筹资金策略，并尽量减少给定的风险措施。这可以用带状态变量的线性规划方法来解决。退保和退保期权的定价通常假定投保人是理性的。它要求投资者充分了解金融市场，从而始终能够做出最优决策。这种强烈的假设导致了价格的上限，因为它考虑了发行者最坏的情况。在提出的风险度量方法下，投降和退出过程将在物理概率度量下建模，这意味着将使用一些内生和外生效应。

The main objective of this project is to develop pricing and hedging techniques for equity-indexed annuities based on risk measures. An equity-indexed annuity (EIA) is an insurance product with benefits linked to the performance of an equity market. It provides limited participation in the performance of an equity index (e.g. S&P 500) while guaranteeing a minimum rate of return. First introduced by Keyport Life Insurance Co in 1995, EIAs have been the most innovative annuity product over the last 20 years. They have become increasingly popular since their debut; the sales of EIAs have steadily increased and hit a record high of $33.9 billion in 2012, as indicated in the fourth quarter 2012 U.S. Individual Annuities Sales survey from Life Insurance and Market Research Association (LIMRA).Equity-indexed annuities are usually wrapped with some death, surrender, withdrawal, and accumulation guarantees that protect policyholders. The costs of these guarantees are evaluated based on financial and actuarial models. Research on evaluating these products has been carried using various approaches; see Hardy ("Investment Guarantees: Modeling and Risk Management for Equity-Linked life Insurance", 2003) and the references therein. I am primarily interested in the application of optimization techniques to reduce the risk of equity-indexed annuity portfolios. In a complete market, all financial claims can be hedged perfectly. However, in an incomplete financial market, the number of securities is less than the number of possible outcomes. In this case, the perfect hedge is unattainable. This is the appropriate framework when complex underlying asset models, mortality risks, and withdrawals are considered. There are different pricing and hedging techniques that could be used in such cases; we shall focus on risk measures. Two different objectives could be achieved while defining the hedging strategy using risk measures. On the one hand, the insurance company could minimize the hedging cost for a given risk level. On the other hand, the issuer could minimize the risk associated to a hedging strategy chosen among self-financing ones. Gaillardetz and Moghtadai (Equity-Linked Product Evaluation Using Iterated Risk Measures, 2013) focus on the latter where the hedging strategy is formed of the underlying risky and risk-free assets. Using the result of Rockafeller and Uryasev (Conditional value-at-risk for general loss distributions, 2002), which converts the evaluation of the tail value-at-risk into a linear programming problem, we propose to generalize the hedging strategies to multiple risky assets. We will also restrict our hedging strategies to self-financing strategies and try to minimize a given risk measure. This could be solved using the linear programming approach with state variables. Surrender and withdrawal options are usually priced assuming that policyholders are rational. It requires that the investors be fully informed of the financial market and thus can always take optimal decisions. This strong assumption leads to an upper bound for prices since it considers the worse case scenario for the issuer. Under the proposed risk measure approach, surrender and withdrawal processes will be modeled under the physical probability measures, which means that some endogenous and exogenous effects will be used.