  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.
