Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer

Pengxu Xie; Lihua Bai; Huayue Zhang

doi:10.3934/jimo.2022020

Article Contents

2023, Volume 19, Issue 3: 1827-1845. Doi: 10.3934/jimo.2022020

This issue Previous Article An inertial triple-projection algorithm for solving the split feasibility problem Next Article Product line extension with a green added product: Impacts of segmented consumer preference on supply chain improvement and consumer surplus

Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer

1.
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
2.
School of Finance, Nankai University, Tianjin 300071, China

^*Corresponding author: Huayue Zhang
^*Corresponding author: Huayue Zhang

Received: September 2021

Revised: December 2021

Early access: February 2022

Published: March 2023

Abstract / Introduction Full Text(HTML) Figure(3) Related Papers Cited by

Abstract

This paper studies the optimal proportional reinsurance and investment strategy for an insurer who invests one paired assets, where their price spread is described by Ornstein-Uhlenbeck (O-U) processes. The insurer's objective is to maximize the expected exponential utility of the terminal wealth in a finite time horizon under two risk models: a classical risk model and a diffusion model. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we characterize the optimal strategies and provide a verification result for the value function via the exponential integrability of the square of an O-U process. Finally, numerical examples are performed to obtain sensitivity analysis.

Keywords:

Mathematics Subject Classification: Primary: 91G10; Secondary: 93E20.

Citation:

Full Text(HTML)

Figure 1. The sample path of $ u^* $

Download: Full-size image PowerPoint slide

Figure 2. The influence of $ a $ on $ u^* $

Download: Full-size image PowerPoint slide

Figure 3. The influence of $ z $ on $ u^* $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	L. H. Bai and J. Y. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975. doi: 10.1016/j.insmatheco.2007.11.002.
[2]	F. E. Benth and K. H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stoch. Anal. Appl., 23 (2005), 687-704. doi: 10.1081/SAP-200064457.
[3]	W. K. Bertram, Analytic solutions for optimal statistical arbitrage trading, Physica A: Statistical Mechanics and its Applications, Issue 11, 389 (2010), 2234-2243.
[4]	M. Boguslavsky and E. Boguslavskaya, Arbitrage under power, Risk, 17 (2004), 69-73.
[5]	M. Brachetta and H. Schmidli, Optimal reinsurance and investment in a diffusion model, Decis. Econ. Finance, 43 (2020), 341-361. doi: 10.1007/s10203-019-00265-8.
[6]	S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958. doi: 10.1287/moor.20.4.937.
[7]	R. J. Elliott, J. van der Hoek and W. P. Malcolm, Pairs trading, Quant. Finance, 5 (2005), 271-276. doi: 10.1080/14697680500149370.
[8]	E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative value arbitrage rule, Social Science Electronic Publishing, 19 (2006), 797-827.
[9]	H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph Series, 8., Distributed by Richard D. Irwin, Inc., Homewood, Ill., 1979.
[10]	J. Grandell, Aspects of Risk Theory, Springer, New York, 1991. doi: 10.1007/978-1-4613-9058-9.
[11]	B. H$\phi$jgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quant. Finance, 4 (2004), 315-327. doi: 10.1088/1469-7688/4/3/007.
[12]	Z. B. Liang, L. H. Bai and J. Y. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Appl. Methods, 32 (2011), 587-608. doi: 10.1002/oca.965.
[13]	Y. X. Lin, M. McCrae and C. Gulati, Loss protection in pairs trading through minimum profit bounds: A cointegration approach, J. Appl. Math. Decis. Sci., 2006 (2006), 1-14. doi: 10.1155/JAMDS/2006/73803.
[14]	R. C. Merton, Option pricing when the underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.
[15]	S. Mudchanatongsuk, J. A. Primbs and W. Wong, Optimal Pairs Trading: A Stochastic Control Approach, American Control Conference IEEE, 2008. doi: 10.1109/ACC.2008.4586628.
[16]	D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128. doi: 10.1080/10920277.2005.10596214.
[17]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 1 (2001), 55-68. doi: 10.1080/034612301750077338.
[18]	G. Vidyamurthy, Pairs Trading: Quantitative Methods and Analysis, John Wiley Sons, 2004.
[19]	K. Vladislav, Optimal Convergence Trading, Technical report, EconWPA, 2004.
[20]	H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009.