استراتژی فرابورس به حداقل رساندن ریسک محلی برای قراردادهای بیمه زندگی واحد مرتبط در یک بازار مالی فرایند لوی

In [Riesner, M., 2006. Hedging life insurance contracts in a Lévy process financial market. Insurance Math. Econom. 38, 599–608] the (locally) risk-minimizing hedging strategy for unit-linked life insurance contracts is determined in an incomplete financial market driven by a Lévy process. The considered risky asset is not a martingale under the original measure and therefore, a change of measure to the minimal martingale measure is performed. The goal of this paper is to show that the risk-minimizing hedging strategy under the new martingale measure which is found in the paper cited above is not the locally risk-minimizing strategy under the original measure. Finally, the real locally risk-minimizing strategy is derived and a relationship between the number of risky assets held in the proposed portfolio cited in the above-mentioned paper and the one proposed here is given.

In Föllmer and Schweizer (1991) and in Schweizer, 1990, Schweizer, 1993 and Schweizer, 2001 the concept of locally risk-minimizing hedging strategies to hedge claims was introduced for the case that the discounted risky asset is a semimartingale. In Møller (1998) the hedging portfolio is constructed for unit-linked life insurance contracts with a pure endowment and a term insurance with single premium in the complete Black–Scholes market. The risky asset under consideration is a continuous semimartingale. In this Black–Scholes setting the locally risk-minimizing hedging strategy under the original measure is equivalent with the risk-minimizing hedging strategy under the minimal martingale measure. We emphasize that this equivalence is only proved for continuous semimartingales as it is the case in the Black–Scholes market. In Riesner (2006) this theory of locally risk-minimizing hedging strategies is combined with the results of Møller (1998) but for a risky asset of which the price process is discontinuous as it follows a geometric Lévy process. However, determining the risk-minimizing hedging strategy under the minimal martingale measure does not provide the locally risk-minimizing hedging strategy in the Lévy case, as we will show in this paper. Moreover, we will derive the locally risk-minimizing hedging strategy by a direct construction of the Föllmer–Schweizer decomposition. Hereto, we explicitly check whether the risky asset satisfies all the necessary conditions in order to have the equivalence between the locally risk-minimizing hedging strategy and the Föllmer–Schweizer decomposition. In the literature the equivalence is often applied in a wrong way. Therefore, we find it important to give an overview concerning ((pseudo) locally) risk-minimizing hedging strategies in Section 2. After repeating the setting of Riesner (2006) in Section 3, we then show in Section 4 that in Riesner (2006) the risk-minimizing hedging strategy under the new measure was found, but that this strategy is not the locally risk-minimizing one under the original measure. In Section 5 we show how to determine the real locally risk-minimizing hedging strategy under the original measure and we calculate the associated risk process. Finally in Section 6 we adapt the results of Riesner (2006) for unit-linked life insurance contracts with a pure endowment and a term insurance.

This paper highlights the problems when trying to determine the locally risk-minimizing hedging strategy in case the price process of the underlying risky asset is discontinuous. In a lot of cases, the equivalence between the Föllmer–Schweizer decomposition and the locally risk-minimizing hedging strategy will still hold, but there is no longer an equivalence between the Galtchouk–Kunita–Watanabe decomposition and the Föllmer–Schweizer decomposition. As shown here, the easiest way to determine in the latter case the Föllmer–Schweizer decomposition is by explicitly imposing all the necessary conditions.