Imperfect Arbitrage with Wealth Effects

Abstract

This paper presents a continuous-time equilibrium model of arbitrage, based on risk aversion of perfectly competitive arbitrageurs and imperfect capital flows to arbitrage activity. Arbitrageurs make profits from trading against noise trading. Under the assumption of logarithmic utility, arbitrageurs' degree of absolute risk aversion changes with their wealth. When arbitrageurs suffer capital losses due to unfavorable shocks, their willingness to bear risks decreases, and they liquidate some of their risky positions. The liquidation of arbitrageurs' positions tends to amplify the original shocks, and thus generates a wealth effect. The wealth effect can cause arbitrage to be destabilizing in the sense that price volatility can be larger when arbitrageurs are present than when no arbitrageurs exist. The wealth effect offers a mechanism to explain the financial crisis of Long Term Capital Management in 1998. It also produces a motivation for risk managers to take into account the risks created by the trading of other market participants. The model shows that arbitrageurs reduce volatility and provide liquidity on average, but only in extreme circumstances does arbitrage become destabilizing. The wealth effect also provides an explanation for excess volatility and stochastic volatility. In equilibrium, markets are inefficient in the sense that the Sharpe ratio is not driven to zero by arbitrage activity. The long-run mean of the squared Sharpe ratio is primarily determined by the impatience level of arbitrageurs. The equilibrium wealth process for arbitrageurs is endogenously determined. Larger fundamental volatility deters wealth accumulation in arbitrage activity, while larger noise trading volatility induces more.

JEL Classification: G12, G14, D84, D52