基于相对熵方法的长寿债券定价研究

doi:10.16381/j.cnki.issn1003-207x.2019.05.004

摘要/Abstract

摘要： 本文在风险中性测度理论的框架下对经典的长寿风险衍生品——EIB/BNP型长寿债券进行了定价。文章首先使用Cairns-Blake-Dowd两因子模型，模拟预测了我国高龄人口未来的死亡率路径，然后通过引入三种相对熵，即Kullback-Leibler相对熵、Tsallis相对熵以及Rényi相对熵，并依据最小相对熵准则确定最优风险中性测度，进而完成对长寿债券的定价。在定价的过程中，本文融入已在我国市场发行的不同期限金融债的风险溢价信息，使定价结果更合理地反映我国金融市场规律。从计算方法上看，不同相对熵模型的定价保证了计算结果的稳健性和可信性；本文还发现对冲女性长寿风险所需的溢价整体上要高于对冲男性长寿风险所需的溢价，表明女性长寿风险面临的不确定性程度更大。本文还分别比较了仅融合单一市场风险价格以及考虑多个市场风险价格作为先验信息时定价结果的差异，发现定价的结果对市场信息依赖程度较高。这也表明了相对熵方法的可塑性较强，随着市场完全度的提高和我国金融市场的发展，该方法的定价结果将更趋于合理。文章最后一部分是总结和启示。

关键词: 长寿债券, 风险中性测度, Kullback-Leibler相对熵, Tsallis相对熵, Ré, nyi相对熵

Abstract: Along with the raise of living standard and improvement of medical technology, average life expectancy in China is rising substantially. The unexpected improvement of average life expectancy renders the insurers to face the challenge of excessive annuity payout, and even brings about to bankrupt risk to them. This type of risk is called longevity risk. The studies on how to manage longevity risk is very popular all over the world, since China is not the only country suffering from longevity and aging. Many scholars point out that we can employ the special financial instruments, which are called longevity derivatives, into hedging the longevity risk. Compared with the mechanism design, previous and present studies are more interested on how to price the longevity derivatives appropriately. In this article, the typical derivative, the longevity bond is priced which is issued by European Investment Bank and BNP Paribas in year 2004. The future mortality of male and female aged people is simulated with Cairns-Blake-Dowd two-factor model. Then EIB/BNP bond is priced by using the risk-neutral measure theory. Since there are infinitely risk-neutral measures in an incomplete market, the relative entropy methods are introduced, to select the optimal risk-neutral measure. Three different types of relative entropy, namely Kullback-Leibler relative entropy, Tsallis relative entropy and Rényi,are applied in order to ensure the robustness. The pricing result also incorporating the market prices of risk as the useful information, in order to make the longevity bond rationally priced. The difference between pricing male mortality linked longevity bonds and that of female morality linked longevity bonds are compared, finding that high risk premia is required when hedging the longevity risk arising from female aged people, which means that improvement of female mortality reveals more uncertainty than that of male mortality, especially when the maturity is long. The case when incorporating unique market price of risk and the case when incorporating total 6 market prices of risk are also compared. It can be concluded that the pricing result varies greatly according to the number of market prices provided by the market. The simulation outcomes when using Kullback-Leibler relative entropy, Tsallis relative entropy and Rényi relative entropy are highly consistent. Thus it is also concluded that the when pricing longevity bonds with relative entropy approach,the outcomes rely much on the completeness of market. This indicates that relative entropy method is very flexible, and the more complete the financial market is, the more reasonable the pricing result will reveal. If the longevity risk is to be managed effectively by means of financial derivatives, the most important issue is to perfect the financial market. And since the relative entropy approach shows great superiority in pricing longevity derivatives, this method can be further developed, and wide application can be made in the field of longevity risk management.

Key words: longevity bond, risk-neutral measure, Kullback-Leibler relative entropy, Tsallis entropy, Rényi relative entropy

中图分类号:

F830.91

宋平凡, 谭常春, 祁毓. 基于相对熵方法的长寿债券定价研究[J]. 中国管理科学, 2019, 27(5): 32-41.

SONG Ping-fan, TAN Chang-chun, QI Yu. Pricing Longevity Bond with Relative Entropy Approach[J]. Chinese Journal of Management Science, 2019, 27(5): 32-41.

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