Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

Wei Wang; Yang Shen; Linyi Qian; Zhixin Yang

doi:10.3934/jimo.2021072

Article Contents

2022, Volume 18, Issue 4: 2369-2399. Doi: 10.3934/jimo.2021072

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Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

1.
School of Mathematics and Statistics, Ningbo University, 818 Fenghua Road, Ningbo 315211, China
2.
School of Risk and Actuarial Studies and CEPAR, University of New South Wales, Sydney, NSW 2052, Australia
3.
Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai 200241, China
4.
Department of Mathematical Sciences, Ball State University, Muncie 47304, USA

^* Corresponding author: Linyi Qian
^* Corresponding author: Linyi Qian

Received: June 30, 2020

Revised: January 31, 2021

Published: July 2022

This work was supported by the Humanity and Social Science Youth Foundation of the Ministry of Education of China (18YJC910012), the National Natural Science Foundation of China (11771147, 12071147), "Shuguang Program" supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission(18SG25), the State Key Program of National Natural Science Foundation of China (71931004), the Discovery Early Career Researcher Award (DE200101266) of the Australian Research Council, the Zhejiang Provincial Natural Science Foundation of China (LY17G010003), the 111 Project(B14019), Ningbo City Natural Science Foundation (202003N4144) and the Humanity and Social Science Foundation of Ningbo University (XPYB19002)

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Abstract

This paper studies the hedging problem of unit-linked life insurance contracts in an incomplete market presence of self-exciting (clustering) effect, which is described by a Hawkes process. Applying the local risk-minimization method, we manage to obtain closed-form expressions of the locally risk-minimizing hedging strategies for both pure endowment and term insurance contracts. Besides, we demonstrate the existence of the minimal martingale measure and perform numerical analyses. Our numerical results indicate that jump clustering has a significant impact on the optimal hedging strategies.

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Mathematics Subject Classification: Primary: 91B05, 91B16; Secondary: 60H30.

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Figure 1. Paths of intensity process $ \lambda_t $ and stock price process $ S_t $

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Figure 2. Number of stock $ \xi_{t}^{PE\ast} $ with different strike prices $ K $

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Figure 3. Effects of the jump size $ Z $ on $ \xi_{0}^{PE\ast} $ and $ \Delta_{0} $. We assume the jump size $ Z_j\in U(-0.1, 0.1) $ and $ Z_j\in U(-0.5, 0.5) $ in the left panel and the right panel of Figure 3, respectively

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Figure 4. Effects of the intensity $ \lambda $ on $ \xi_{0}^{PE\ast} $

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