مصون سازی Quantile برای قراردادهای بیمه عمر مرتبط با ارزش سهام با نرخ بهره تصادفی

This paper studies the problem of pricing equity-linked life insurance contracts, and also focuses on the valuation of insurance contracts with stochastic guarantee. The contracts under consideration are based on two risky assets which satisfy a two-factor jump-diffusion model: one asset is responsible for future gains, and the other one is a stochastic guarantee. As most life insurance products are long-term contracts, it is more practical to consider the problem in a stochastic interest rate environment. In our setting, the stochastic interest rate behaviour is also described by a jump-diffusion model. In addition, quantile hedging technique is developed and exploited to price such finance/insurance contracts with initial capital constraints. Explicit formulas for both the price of the contracts and the survival probability are obtained. Our results are illustrated by numerical example based on financial indexes Russell 2000 and S&P 500.

Equity-linked life insurance contracts have been studied since the middle of the 1970s. This type of contracts links the benefit payable at the maturity time with the market value of some reference portfolio, such as stocks, foreign currencies etc. Thus, the benefit of such contracts is uncertain while it is fixed for the traditional contracts. Compared with traditional ones, these innovative products can bring the insurance companies as well as the clients more benefit and improve the insurance companies’ competitiveness in the modern financial system. In North America and the UK, equity-linked life insurance contracts are typically provided with guarantee. Therefore, the topic of pricing equity-linked life insurance contracts with guarantee has attracted most scholars’ attention. Brennan and Schwartz (1976), Boyle and Schwartz (1977) are the first papers appeared in this area. The authors decomposed the benefit of the contracts into a guaranteedamount and a call (put) option on the reference portfolio, then they used Black-Scholes model to evaluate the contracts. Moreover, Moeller (1998, 2001) applied the mean-variance hedging method to calculate the price of the contracts. The guarantee of the contracts in all those papers is deterministic or fixed. Ekern and Persson (1996) priced the contracts with different guarantees, fixed and stochastic using fair pricing valuation. Kirch and Melnikov (2005), Melnikov and Romanyuk (2008) also applied efficient hedging method to price the equity-linked life insurance contracts with stochastic guarantee. Quantile hedging technique, as an imperfect hedging technique, was developed in several publications by Foellmer and Leukert (1999), and we exploit further the most important paper on this topic. It can successfully hedge the option with maximal probability in the class of self-financing strategies with restricted initial capital. This technique has been proposed by Melnikov (2004) as pricing and hedging methodology for equity-linked life insurance contracts in the Black-Scholes framework. Later it was extended by Melnikov and Skornyakova (2005) to a two factor jump-diffusion model with constant interest rate, where the second risky asset could be considered as a stochastic guarantee for the contracts. Up till now, many research papers in the area work with a constant interest rate r . However, as insurance products are usually long-term contract, they could be more sensitive to the changes in the interest rates. Therefore, it is more practical to consider a stochastic interest rate in the financial market. Gao, et al. (2010) considered the problem of pricing equity-linked life insurance contracts by means of quantile hedging and stochastic interest rate. They studied this topic in the framework of the Black- Scholes market model driven by two independent Wiener processes and a stochastic interest rate via HJM model (See [13]). The guarantee of the contracts in their study depends on a constant rate of return g and timet . It is well-known that discontinuous models for both the stochastic interest rate and the value of risky assets are more realistic. Extending the paper of Gao et al. (2010), we consider two risky assets S1 and S2 satisfying a two-factor jump-diffusion model, where the asset S2 is less risky than S1 , and it can be seen as a stochastic guarantee of the equity-linked life insurance contract. We study the problem in the framework of Melnikov and Skornyakova (2005). But in contrast with that paper, we use a generalised HJM jump-diffusion model for the term structure of interest rate r (t ) , which is similar to the framework of Shirakawa (1991), and Chiarella & Sklibosios (2003). Assuming independence of financial and insurance (mortality) risks, we apply quantile hedging to price equity-linked life insurance contracts with initial capital constraints. The paper is structured as follows. In Section 2, we review jump-diffusion models and introduce the HJM term structure framework. Then we describe finance/insurance contracts under consideration. In Section3, we briefly describe quantile hedging technique and present our main pricing results. Section 4 illustrates our results with a numerical example. In Section 5 some future work is discussed. Appendix A, B and C contain technical details of proofs.

In this paper, we generalized the results by Melnikov and Skornyakova (2005) and Gao, et al. (2010) on pricing equity-linked life insurance. We choose the two-factor jump-diffusion model and generalized HJM model in our study in order to better describe the real financial market. The presence of mortality risk usually causes budget constraint on a hedge and makes it impossible for insurance companies to exactly replicate the payoff of a contract. Thus, we apply the quantile hedging technique to price the contracts when the perfect hedging is impossible. A natural extension of this work is to consider non-constant parameters in assets’ models, for example, a stochastic volatilityσ (t ) . Besides stochastic interest rate r (t ) , stochastic volatility σ (t ) is another important factor in pricing long-term equity-linked life insurance contracts. It is possible to incorporate both factors into the jump-diffusion model. In this paper, we only consider a single premium contract. Further, we could work with a periodic premium contract. Such contract could be an equity-linked endowment insurance polity with asset value guarantee with periodic premiums, where the buyer is committed to pay regularly a predetermined premium to the insurance company.