Non-zero-sum Stochastic Differential Games for Asset-Liability Management with Stochastic Inflation and Stochastic Volatility

Abstract

This paper investigates the optimal asset-liability management problems for two managers subject to relative performance concerns in the presence of stochastic inflation and stochastic volatility. The objective of the two managers is to maximize the expected utility of their relative terminal surplus with respect to that of their competitor. The problem of finding the optimal investment strategies for both managers is modeled as a non-zero-sum stochastic differential game. Both managers have access to a financial market consisting of a risk-free asset, a risky asset, and an inflation-linked index bond. The risky asset’s price process and uncontrollable random liabilities are not only affected by the inflation risk but also driven by a general class of stochastic volatility models embracing the constant elasticity of variance model, the family of state-of-the-art 4/2 models, and some path-dependent models. By adopting a backward stochastic differential equation (BSDE) approach to overcome the possibly non-Markovian setting, closed-form expressions for the equilibrium investment strategies and the corresponding value functions are derived under power and exponential utility preferences. Moreover, explicit solutions to some special cases of our model are provided. Finally, we perform numerical studies to illustrate the influence of relative performance concerns on the equilibrium strategies and draw some economic interpretations.