Kelly Trading And Market Equilibrium

The Kelly framework is the natural multi-period extension of the one-period mean-variance model of Markowitz in the sense that the efficient frontier is characterized by trading strategies having maximal instantaneous Sharpe ratio. We show that Kelly traders naturally trade in such a way as to induce an equilibrium for the instantaneous covariance matrix. This equilibrium, arising from trading alone, has the property that the equilibrium correlation can be described as the saddle point of a stochastic differential game. However, because the game is not necessarily a zero-sum game the equilibrium volatility is shown to be lower than what is predicted from the game. The covariance equilibrium is fully specified by the rate of logarithmic return, the interest rate and the aggregate willingness to leverage seen in the market.