The Kelly criterion maximizes the long-run growth rate of capital by betting a specific fraction of wealth on each opportunity. Under log-normal returns, the fraction is μ/σ² — expected return divided by variance. The proof assumes continuous trading and fixed volatility. Both assumptions fail in real markets.
The paper examines what happens when returns follow a time-changed process — a random clock that stretches or compresses calendar time. In quiet periods, the clock runs slowly (few effective bets per unit time); in volatile periods, it runs fast (many effective bets). Stochastic volatility is a time change: the volatility process determines how many “effective coin flips” occur per day.
The Kelly criterion breaks under time change. It maximizes growth rate only for normally distributed log-returns — the one case where the time change is deterministic (Gaussian ≡ Brownian motion at a constant rate). For any heavier-tailed distribution (which time-changed processes always produce), the Kelly fraction is too large. You overbET.
The magnitude of the failure scales with the variance of the stochastic clock. A more variable clock means heavier tails, which means a larger gap between the Kelly position and the optimal position. The correction is always in the direction of betting less — the time change adds risk that the Kelly formula doesn't account for.
Thorp's ruin threshold — the maximum bet that avoids certain eventual bankruptcy — provides a tighter bound but still doesn't optimize growth. The optimal position sits between the two: less aggressive than Kelly, more aggressive than the ruin limit.
The clock is a risk factor. Kelly optimizes over the cards; it forgets about the timer.