Optimal investment and reinsurance strategies under 4/2 stochastic volatility model

Abstract

This paper studies a mean-variance investment-reinsurance problem under a new stochastic volatility model, namely the 4/2 stochastic volatility model. Solving this problem requires a deep understanding of a class of parabolic partial differential equations (PPDEs). By the parametrix method and the integral transform method, we derive explicit solutions to the PPDEs in several special cases. Through the Lie symmetry analysis, we obtain a four-parameter family of the 4/2 stochastic volatility models such that the corresponding PPDEs have closed-form solutions. The efficient strategy and the efficient frontier of the mean-variance problem are represented by using the closed-form solutions to PPDEs. Numerical examples for the obtained efficient frontier are provided by Monto Carlo method.

Keywords:

• Reinsurance
• parabolic partial differential equation
• 4/2 stochastic volatility model
• parametrix method
• Lie symmetries
• mean-variance optimization

Acknowledgements

We thank two anonymous referees for helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Wenyuan Wang acknowledges support in part by the National Natural Science Foundation of China [grant numbers 12171405 and 11661074]. Dmitry Muravey acknowledges support by the Russian Science Foundation [grant number 20-68-47030]. Yang Shen acknowledges support by the Australian Research Council [grant number DE200101266]. Yan Zeng acknowledges support by the National Natural Science Foundation of China [grant number 71771220] and the Nature Science Foundation of Guangdong Province of China [grant number 2022A1515011472].