# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021039
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## First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China

* Corresponding author: Zhimin Zhang

Received  August 2020 Revised  November 2020 Early access March 2021

Fund Project: Zhimin Zhang is supported by the National Natural Science Foundation of China [grant number 11871121], Natural Science Foundation Project of CQ CSTC [grant number cstc2019jcyj-msxmX0004] and the Fundamental Research Funds for the Central Universities (project number 2020CDJSK02ZH03). Wenguang Yu is supported by the National Social Science Foundation of China (No. 15BJY007), the Taishan Scholars Program of Shandong Province (No. tsqn20161041), the Humanities and Social Sciences Project of the Ministry Education of China (No. 19YJA910002), the Natural Science Foundation of Shandong Province (No. ZR2018MG002), the Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions (No. 1716009), Shandong Provincial Social Science Project Planning Research Project (No. 19CQXJ08), the Risk Management and Insurance Research Team of Shandong University of Finance and Economics, Excellent Talents Project of Shandong University of Finance and Economics, the Collaborative Innovation Center Project of the Transformation of New and old Kinetic Energy and Government Financial Allocation

This paper studies some first passage time problems in a refracted jump diffusion process with hyper-exponential jumps. Closed-form expressions for four functions associated with the first passage time are obtained by solving some ordinary integro-differential equations. In addition, the obtained results are used to value equity-linked death benefit products with state-dependent fees. Specifically, we obtain the closed-form Laplace transform of the fair value of barrier option, which is further recovered by the bilateral Abate-Whitt algorithm. Numerical results confirm that the proposed approach is efficient.

Citation: Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021039
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##### References:
Pricing up-and-in call options when $K$ varies, $B = 120$ and $L = 115$.
 $\sigma = 0.1$ $\sigma = 0.2$ $\lambda$ $K$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ 1 100 64.56585 64.55930 64.54628 68.75548 68.75253 68.74360 105 64.36554 64.35829 64.34406 68.60582 68.60241 68.59308 110 64.16854 64.16063 64.14525 68.45948 68.45562 68.44584 3 100 65.08980 65.08372 65.07131 69.01900 69.01623 69.00753 105 64.89483 64.88807 64.87452 68.87247 68.86926 68.86019 110 64.70320 64.69579 64.68116 68.72922 68.72559 68.71609
 $\sigma = 0.1$ $\sigma = 0.2$ $\lambda$ $K$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ 1 100 64.56585 64.55930 64.54628 68.75548 68.75253 68.74360 105 64.36554 64.35829 64.34406 68.60582 68.60241 68.59308 110 64.16854 64.16063 64.14525 68.45948 68.45562 68.44584 3 100 65.08980 65.08372 65.07131 69.01900 69.01623 69.00753 105 64.89483 64.88807 64.87452 68.87247 68.86926 68.86019 110 64.70320 64.69579 64.68116 68.72922 68.72559 68.71609
Pricing up-and-in call options when $L$ varies, $B = 120$ and $K = 100$.
 $\sigma=0.1$ $\sigma=0.2$ $\lambda$ $L$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ 1 105 64.58681 64.57969 64.56342 68.76237 68.75897 68.74997 125 64.53840 64.53193 64.52085 68.74569 68.74308 68.73608 145 64.48201 64.47513 64.46268 68.71998 68.71769 68.71446 3 105 65.10796 65.10130 65.08595 69.02559 69.02240 69.01370 125 65.06588 65.05994 65.04979 69.00962 69.00720 69.00034 145 65.01405 65.00780 64.99722 68.98490 68.98282 68.97977
 $\sigma=0.1$ $\sigma=0.2$ $\lambda$ $L$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ $\mathcal{M}_{3}$ $\mathcal{M}_{5}$ $\mathcal{M}_{10}$ 1 105 64.58681 64.57969 64.56342 68.76237 68.75897 68.74997 125 64.53840 64.53193 64.52085 68.74569 68.74308 68.73608 145 64.48201 64.47513 64.46268 68.71998 68.71769 68.71446 3 105 65.10796 65.10130 65.08595 69.02559 69.02240 69.01370 125 65.06588 65.05994 65.04979 69.00962 69.00720 69.00034 145 65.01405 65.00780 64.99722 68.98490 68.98282 68.97977
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