Abstract
This paper addresses the risk-minimization problem, with and without mortality securitization, à la Föllmer–Sondermann for a large class of equity-linked mortality contracts when no model for the death time is specified. This framework includes situations in which the correlation between the market model and the time of death is arbitrary general, and hence leads to the case of a market model where there are two levels of information—the public information, which is generated by the financial assets, and a larger flow of information that contains additional knowledge about the death time of an insured. By enlarging the filtration, the death uncertainty and its entailed risk are fully considered without any mathematical restriction. Our key tool lies in our optional martingale representation, which states that any martingale in the large filtration stopped at the death time can be decomposed into precise orthogonal local martingales. This allows us to derive the dynamics of the value processes of the mortality/longevity securities used for the securitization, and to decompose any mortality/longevity liability into the sum of orthogonal risks by means of a risk basis. The first main contribution of this paper resides in quantifying, as explicitly as possible, the effect of mortality on the risk-minimizing strategy by determining the optimal strategy in the enlarged filtration in terms of strategies in the smaller filtration. Our second main contribution consists of finding risk-minimizing strategies with insurance securitization by investing in stocks and one (or more) mortality/longevity derivatives such as longevity bonds. This generalizes the existing literature on risk-minimization using mortality securitization in many directions.
Highlights
In this paper, we manage the risk of a life insurance portfolio that faces two main types of risk, financial risk and mortality or longevity risk, by designing quadratic hedging strategies à la Föllmer–Sondermann, introduced in [1], with and without mortality securization
We present our mathematical model, which is constituted by the initial market model and a death time, and recalls our optional martingale representation result that we use throughout the paper
We addressed the risk-minimization problem for mortality liabilities by designing quadratic hedging strategies à la Föllmer–Sondermann with and without insurance securization when no model for the death time is specified
Summary
We manage the risk of a life insurance portfolio that faces two main types of risk, financial risk and mortality or longevity risk, by designing quadratic hedging strategies à la Föllmer–Sondermann, introduced in [1], with and without mortality securization. Even though our results can be extended to more general quadratic hedging approaches, we opted to focus on the Föllmer–Sondermann method to well illustrate our main ideas The literature addressing this objective has become quite rich in the last decade, while the existing literature makes assumptions on the triplet (F, S, τ ) that can be translated, in one way or another, to a sort of independence and/or no correlation between the financial market—represented by the pair (F, S)—and the mortality represented by the death time τ. For the sake of easy exposition, the proof of some intermediate technical lemmas are relegated to the appendix
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