Long Run Feedback in the Broker Call Money Market

I unravel the basic long run dynamics of the broker call money market, which is the pile of cash that funds margin loans to retail clients (read: continuous time Kelly gamblers). Call money is assumed to supply itself perfectly inelastically, and to continuously reinvest all principal and interest. I show that the relative size of the money market (that is, relative to the Kelly bankroll) is a martingale that nonetheless converges in probability to zero. The margin loan interest rate is a submartingale that converges in mean square to the choke price $r_\infty:=\nu-\sigma^2/2$, where $\nu$ is the asymptotic compound growth rate of the stock market and $\sigma$ is its annual volatility. In this environment, the gambler no longer beats the market asymptotically a.s. by an exponential factor (as he would under perfectly elastic supply). Rather, he beats the market asymptotically with very high probability (think 98%) by a factor (say 1.87, or 87% more final wealth) whose mean cannot exceed what the leverage ratio was at the start of the model (say, $2:1$). Although the ratio of the gambler's wealth to that of an equivalent buy-and-hold investor is a submartingale (always expected to increase), his realized compound growth rate converges in mean square to $\nu$. This happens because the equilibrium leverage ratio converges to $1:1$ in lockstep with the gradual rise of margin loan interest rates.