Dynamic Asset Allocation with Relative Wealth Concerns in Incomplete Markets

In dynamic portfolio choice problems, stochastic state variables such as stochastic volatility lead to adjustments of the optimal stock demand referred to as hedge terms or Merton-Breeden terms. By deriving an explicit solution in a multi-agent framework with a stochastic opportunity set, we show that relative wealth concerns give rise to new hedge terms beyond the ordinary ones. This is because the agents hedge against both exogenous changes in the state variable and endogenous decisions of the other agent. Depending on the parametrization of the model, these new terms can significantly change the investors' hedging demands. We also show that both heterogeneity in risk aversion or relative wealth concerns can have similar effects on the heterogeneity in portfolio decisions. Formally, we study a non-cooperative, non-zero sum stochastic differential game for which we prove a verification theorem in a setting with an unspanned state variable