Pricing And Hedging Equity-Linked Insurance In An Incomplete Market

Europe and the United States were in high inflation and high interest rates in the lat-ter part of last century. But the insured who bought the traditional type of life insurance products with the fixed interest rates can not share the market appreciation of the benefits to offset the impact of inflation. Many policyholders got back the funds by surrendering and policy loans to switch to other high-interest rate financial instruments in order to obtain a higher return. From 70s of last century, with financial liberalization and inter-nationalization of the tide, Countries had financial deregulation, the boundaries became increasingly blurred between insurance and other financial services industry. Compared to other new financial instruments, the traditional life insurance products with the fixed interest rates had the less flexibility. This had forced the United States and Europe to adjust life traditional insurance industry design direction. The investment-oriented insur-ance came into being. Investment-type insurance products have participating insurance, investment linked insurance, universal life insurance, variable annuities, equity-indexed annuities. The participating insurance and the investment linked insurance were designed in China at the end of the last century. At present investment-type insurance prod-ucts are ushered in another wave of development opportunities. We call the products, investment linked insurance and variable annuities and equity-indexed annuities etc., as equity-linked insurance. This dissertation will study the problems for pricing and hedging of equity-linked insurance.One part of this thesis is valuation of equity-indexed annuities in the incomplete market from Chapter II to Chapter IV. An equity-indexed annuity (EIA) earns a minimum rate of interest and offers a potential gain that is tied to the performance of a stock index, typically S&P 500. EIAs have received considerable attention in the actuarial literature; see, for example, Tiong (2000), Gerber and Shiu (2003), Lin and Tan (2003), Hardy (2003 and 2004), Lee (2003), Jaimungal (2004), Kijima and Wong (2007), Boyle and Tian (2008), Moore (2009) etc. This thesis extend the model to regime Switching, stochastic interest rate, stochastic mortality and jump diffusion. Under these models the market is incomplete, contingent claims cannot be perfectly replicated by self-financing strategies. In other words every contingent claim in such a market will have an intrinsic risk. Since in the incomplete market there are many equivalent martingale measures under which the discounted price process is a martingale, we need some additional criteria to select an appropriate martingale measure to price a contingent claim, such as utility maximization etc. This can lead minimal martingale measure, minimal entropy martingale measure, Esscher measure.Since some rare events (release of an unexpected economic figure, major political changes or even a natural disaster in a major economy) can lead to brusque variations in prices, thus we need consider a more general economic model, assuming that equity index follows a jump diffusion process. Moreover, Lin and Tan (2003), Kijima and Wong (2007)argued that the effects of stochastic interest rates were crucial in pricing EIAs by simulation results, we also need consider the stochastic interest rate model. In Chapterâ…¡we consider valuation of equity-indexed annuities with stochastic interest rate and jump diffusion. We use the change of measure technique and Bayes' rule to derive the close-solution of pricing of point-to-point and annual reset EIAs.The expected life length has increased considerably in many countries during the past decades with the advances made in the health sciences and medicine, life insurance and annuities are exposed to unanticipated changes over time in the mortality rates of the appropriate reference population, which has forced life insurers to use a stochastic model to describe mortality laws. Chapterâ…¢incorporate the stochastic mortality model and the stochastic interest rate model into the valuation of two types of design of equity-indexed annuity: the point-to-point and the annual reset. We construct a general model, in which the stochastic mortality and the stochastic interest rate are dependent of each other. In fact, the model incorporates three specific cases:(1) the stochastic interest rate model, when some parameters of the stochastic mortality model equal to zero. (2) the stochastic mortality model, when some parameters of the stochastic interest rate model equal to zero. (3) the model that the stochastic mortality and the stochastic interest rate are not dependent, when the correlation coefficient between the mortality and the interest rate equals to zero. We derive the pricing formulas in closed form for the two EIAs products, and give the results for above the three special cases. In Case 1, the pricing of EIAs is similar to that of Lin et al.(2003) and Kijima et al. (2006). In order to make the paper be applicable, we introduce some methods for estimating the parameters of the mortality and interest rate models, and conduct several numerical experiments, in which we analyze the relationship between some parameters and the pricing of EIAs.In Chapterâ…£we assume that the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, de-pend on the states of the economy which are modeled by a continuous-time Markov process in jump-diffusion financial market. We employ two approaches to price point-to-point EIA and annual reset EIA in the incomplete market setting:one is the gen-eralized regime-switching Esscher transform under Merton's assumption, and another is risk-minimization. Merton's assumption is that "jump risk" is diversifiable, i.e., the jump risk is not priced. Under the assumption we can obtain the unique equivalent martingale measure by the conditional Esscher transform. Risk-minimization is to select an unique equivalent martingale measure by minimizing the quadratic utility of the losses. Further-more, the difference is given for the two methods of the pricing by numerical simulation results, and the effects of the model parameters on the EIAs pricing are illustrated through numerical experiment.The other part of this thesis is hedging of equity-linked insurances in the incomplete market from Chapter V to Chapterâ…¦. Since some equity-linked insurances have guar-antee, the main problem for the products is how to manage the risk. The method of risk-minimization is one of hedging methods, to obtain optimal strategies in the sense of minimization of intrinsic risk. The theory of risk-minimization for incomplete markets was introduced in Follmer and Sondermann(1986), and developed further in Follmer and Schweizer(1991) and Schweizer(1991,1994 and 2001). Moreover, Chan (1999) addressed the issue of minimal martingale measure for contingent claims driven by a general geo-metric Levy model. M(?)ller(1998,2001) constructed risk-minimizing hedging strategies for unit-linked life insurance contracts with a pure endowment and a single premium term insurance in the Black-Scholes market. Riesner (2006), Vandaele and Vanmaele (2008) derived hedging strategies of unit-linked life insurance contracts in a Levy process financial market. Chapterâ…¤obtains the risk-minimizing hedging strategy for point-to-point EIA under regime switching model. In Riesner (2006) the locally risk-minimizing hedging strategies were found under the minimal martingale measure, but the strate-gies were not the locally risk-minimizing ones under the original measure. Vandaele and Vanmaele (2008) proved the locally risk-minimizing hedging strategies under the origi-nal measure are equivalent with the locally risk-minimizing hedging strategies under the minimal martingale measure only for continuous semimartingales, as it is the case in Black-Scholes market. Chapterâ…¥extends the model and analysis in that of Vandaele and Vanmaele(2008), We assume that parameters of the Levy process which models the dynamic of risky asset in the financial market depend on a finite state Markov chain. The state of the Markov chain can be interpreted as the state of the economy. Under the regime switching Levy model, we obtain the locally risk-minimizing hedging strategies for some unit-linked life insurance products, including both the pure endowment policy and the term insurance contract. In Chapter VII we extend Riesner's (2007) model and get round of its shortcoming in the sense that the strategies are the locally risk-minimizing hedging strategy under the original measure.