Mean-Variance Asset-Liability Management with State-Dependent Risk Aversion

TL;DR

This paper develops a model for asset-liability management using mean-variance optimization with a state-dependent risk aversion, solving complex FBSDEs to find equilibrium strategies that depend on the liability state.

Contribution

It introduces a novel state-dependent risk aversion function and derives equilibrium strategies through solving bivariate FBSDEs in a continuous-time setting.

Findings

Equilibrium strategies are feedback controls based on the liability.

The model accounts for time-inconsistency in risk preferences.

Explicit solutions are obtained for the control problem.

Abstract

In this paper, we consider the asset-liability management under the mean-variance criterion. The financial market consists of a risk-free bond and a stock whose price process is modeled by a geometric Brownian motion. The liability of the investor is uncontrollable and is modeled by another geometric Brownian motion. We consider a specific state-dependent risk aversion which depends on a power function of the liability. By solving a flow of FBSDEs with bivariate state process, we obtain the equilibrium strategy among all the open-loop controls for this time-inconsistent control problem. It shows that the equilibrium strategy is a feedback control of the liability.