Finance

Mathematics

Stochastic differential equation

Parametrix

Partial differential equation

Financial economics

Mathematical optimization

Mathematical analysis

Econometrics

Economics

Reinsurance

Parabolic partial differential equation

SABR volatility model

Portfolio

Applied mathematics

Efficient frontier

Stochastic volatility

Volatility (finance)

Scandinavian Actuarial Journal

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.

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