First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits

Meiqiao Ai; Zhimin Zhang; Wenguang Yu

doi:10.3934/jimo.2021039

Article Contents

2022, Volume 18, Issue 3: 1689-1707. 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
^* Corresponding author: Zhimin Zhang

Received: July 31, 2020

Revised: October 31, 2020

Published: May 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. 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

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Table 2. 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

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