زمان گسسته به حداقل رساندن خطر محلی فرآیندهای پرداخت و برنامه های کاربردی برای قراردادهای بیمه عمر حقوق صاحبان سهام مرتبط

We develop a theory of local risk minimization for payment processes in discrete time, and apply this theory to the pricing and hedging of equity-linked life-insurance contracts. Thus, we extend the work of Møller (2001a) in several directions: from risk minimization (which is done under a martingale measure) to local risk minimization (which is done under an arbitrary measure), from single claims to payment processes, from complete financial markets to possibly incomplete financial markets, from a single risky asset to several risky assets, and from finite state spaces to general state spaces. Moreover, we show that, when tradable financial assets are independent of mortality, a locally risk-minimizing hedging strategy for most claims in the combined financial and mortality market (such as those arising from equity-indexed annuities) may be expressed as the product of two simpler locally risk-minimizing hedging strategies: one for a purely financial claim, the other for a traditional (i.e. non-equity-linked) life-insurance claim. Finally, we also show, under general assumptions, that the minimal measure for the combined market is the product of the minimal measure for the financial market and the physical measure for the mortality.

The aim of this paper is to study local risk minimization of payment processes in discrete time, particularly in the context of equity-linked life-insurance contracts. For such contracts, the typical (discounted) monthly payment consists of a sum of terms of the form equation(1) HS⋅HT,HS⋅HT, Turn MathJax on where the random variable HSHS is contingent on the price history of stocks, mutual funds, options, and bonds, while the random variable HTHT is contingent on the survival or death of a policyholder. For example, HSHS might be the payoff of an S&P500 option and HTHT, the indicator function of the event “the policyholder is still alive”. More generally, HSHS can be interpreted as a purely financial claim , and HTHT, as a traditional (i.e. non-equity-linked) life-insurance claim. While the no-arbitrage price of a purely financial claim is based on the expected value of its discounted payoff under a risk-neutral probability measure View the MathML sourceQ, the actuarial price of a traditional life-insurance claim is based–according to the equivalence principle–on the expected value of its discounted payoff under the real-world probability measure View the MathML sourceP. Hence, when pricing the product claim (1), which is a mixture of financial and insurance claims, it is not immediately obvious which methodology to adopt. An elegant solution to this problem is provided by local risk minimization, a general approach to price and hedge claims, introduced by Schweizer (1988). Roughly speaking, the idea is to minimize, at every trading period, the mean square hedging error (i.e. the cost increment) of a not necessarily self-financing hedging strategy for a given claim. The initial value of this locally risk-minimizing strategy can then be interpreted as a fair price for the claim. Remarkably, this fair price agrees not only with the no-arbitrage price of a financial claim (in the case of a complete market), but also with the actuarial price of an insurance claim. It is then only natural to use local risk minimization to price and hedge the product claim (1). This idea was pioneered by Møller, 1998 and Møller, 2001a, who considers risk minimization (a particular case of local risk minimization) of equity-linked life-insurance contracts in the context of a complete (Black–Scholes or Cox–Ross–Rubinstein) financial market. Local risk minimization leads to convenient linear pricing and hedging rules: the fair price and the locally risk-minimizing strategy of a sum of claims H1+⋯+HnH1+⋯+Hn are the sum of the fair prices π1+⋯+πnπ1+⋯+πn and the sum of the locally risk-minimizing strategies φ1+⋯+φnφ1+⋯+φn, respectively, of these claims. This linearity property simplifies the pricing and hedging of a payment process (i.e. a series of claims occurring at different times) and of a portfolio of several equity-linked life-insurance contracts. Local risk minimization also leads to an intuitive factorization property: when the financial claim HSHS and the insurance claim HTHT are independent, the fair price and the locally risk-minimizing strategy of the product claim (1) are the product of the fair prices πSπTπSπT and the product of the locally risk-minimizing strategies φSφTφSφT, respectively, of the two claims HSHS and HTHT. This factorization property is observed by Møller, 1998 and Møller, 2001a, for risk minimization, and by Barbarin, 2007a, Barbarin, 2007b, Barbarin, 2008a and Barbarin, 2008b, for continuous-time (local) risk minimization. Here, we establish this factorization property (Theorem 1 and Corollary 2) for discrete-time local risk minimization under a general setting: we consider multi-dimensional asset price processes taking values in a general state space, incomplete financial markets, path-dependent contingent claims, and payment processes. Moreover, the fair price of a claim can be written as the expected discounted value of its payoff under a so-called minimal measure . We show ( Theorem 2) that when the probability space View the MathML source(ΩS,FS,PS) governing the financial market is independent of the probability space View the MathML source(ΩT,FT,PT) governing mortality, the minimal measure on the product of these two spaces is the product of the minimal measure View the MathML sourceQˆS for the financial market and the real-world measure View the MathML sourcePT for mortality. This result makes it possible to price, in the combined financial and mortality market, claims that cannot necessarily be written as the product of a financial claim HSHS and an insurance claim HTHT. This paper is organized as follows. In the next section, we review the literature on (local) risk minimization and its applications to life insurance. Section 3 develops a theory of local risk minimization for payment processes in discrete time. In that section, local risk minimization is considered in a general framework, without any particular application to life insurance. Section 4 introduces mortality in our framework, and applies local risk minimization to equity-linked life-insurance contracts. Section 5 concludes the paper.

The first half of this paper presents a general theory of local risk minimization in discrete time. This theory is applicable to payment processes and to multi-dimensional asset price processes taking values in arbitrary state spaces. Our approach relies on the definition (3) of the discounted value of a hedging strategy introduced by Lamberton et al. (1998). This ensures that a replicating and self-financing hedging strategy is also locally risk-minimizing, which is not the case with the original definition (2) of Schweizer (1988) and Föllmer and Schweizer (1988). In the second half of this paper, the local risk minimization theory developed in the first half is then applied to equity-linked life-insurance contracts. This extends the work of Møller (2001a) to more general settings: path-dependent contingent claims, payment processes, possibly incomplete financial markets, several risky assets, and arbitrary state spaces. Our main results are several factorization theorems relying on the independence between the financial market and the policyholder’s mortality. Under this independence assumption, a locally risk-minimizing strategy for a product claim HS⋅HTHS⋅HT is given by the product of two simpler locally risk-minimizing strategies: one for the purely financial claim HSHS, the other for the traditional life-insurance claim HTHT. Similarly, the fair price of the product claim HS⋅HTHS⋅HT is given by the product of the fair price of the purely financial claim HSHS and the fair price of the traditional life-insurance claim HTHT. Still under the same independence assumption, we also show that the minimal measure for the combined financial and mortality market is the product of the minimal measure for the financial market and the physical measure for the mortality. This last result allows to price, in the combined market, claims that cannot necessarily be expressed as the product of a purely financial claim and a traditional life-insurance claim. We postpone to future research the modeling of the surrender of the policyholder. This problem is of practical importance in jurisdictions with nonforfeiture laws, which automatically embed a surrender option into life-insurance products. The exercise of this option is controlled by the policyholder, whose behavior is potentially influenced by financial markets. Hence, although local risk minimization can still be used (see Barbarin, 2007a and Barbarin, 2008a and Vandaele and Vanmaele (2009) for examples in continuous time), surrender modeling requires relaxing our independence assumption between the financial and insurance markets. Moreover, the policyholder’s behavior is not necessarily rational and thus, challenging to model accurately.