Kelly Criterion

This is a concept well-known to gamblers. The analysis below is based on a reading of “Option Trading Pricing and Volatility Strategies” by Euan Sinclair.

Consider a binary game where the return on capital at risk is $W$ in the event of a win and $-L$ in the event of a loss. Then the return $R$ is a binary( $-L,W$ ) random variable. If a fraction $f$ of the capital $C$ is placed at risk, then the capital after one play of the game is $C' = (C - Cf) + Cf(1+X) = C(1 + fX)$ . Assuming in $n$ plays of the game there are $m$ wins, then the capital gain after $n$ steps is $g_n = \frac{C_n}{C_0} = (1 + fW)^m (1 - fL)^{n-m}$ . The incremental gain per step is $g_n^{1/n} = (1 + fW)^{m/n} (1 - fL)^{1-m/n}$ . As $n \rightarrow \infty$ , $m/n \rightarrow p$ almost surely, where $p = Pr[X = W]$ is the probability of a win. Hence $g_n^{1/n} \rightarrow g := (1+fW)^p (1-fL)^{1-p}$ almost surely, and so the capital gain $C_n/C_0 \rightarrow g^n$ almost surely.

For large $n$ , we may ostensibly safely assume that $C_n = C_0 g^n$ , and therefore the issue of selecting the fraction $f$ of capital to risk that maximizes the gain after $n$ steps is roughly equivalent to the maximizing the limiting gain per play $g$ . Let $q = 1 - p$ . Then

\frac{d g}{d f} = g[pW(1+fW)^{-1} -qL(1-fL)^{-1}]

and so the extreme value satisfies

f = \frac{p}{L} – \frac{q}{W}

Kelly Criterion Example

Given $c > 0$ , consider games parameterized by win return $r > c$ where the return $R$ is a binary( $-1,r$ ) random variable such that $P[R = r] = \frac{1+c}{1+r}$ . Then the expected return $\mathbb{E} R = c$ is independent of the win return. So for these games, a moon-short lottery has the same expected return as a game with a high win probability. The Kelly Criterion optimal fraction $f$ is

f(r)=p-qr^{-1} = \frac{c}{r}

Which indicates that the capital at risk is the expected return in units of win return. If investor preference is based on expectation alone, all games are equivalent, but with regards to capital accumulation the betting fraction will decrease with win return.

Criticism of the Kelly Criterion

The stationarity of the game parameters is a requirement. If this model is adopted to characterize returns in a dynamically changing world the underlying assumption is incorrect and the parameters cannot be meaningfully estimated.

Defense of the Kelly Criterion

The analysis does indicate that over some sufficient small time horizon and for sufficiently large number of plays, the criterion could apply. Therefore it may be useful as a guide for active traders if they roughly know the win and loss return and probabilities.

The takeaway for active traders is that position sizing should decrease if risky high-return trades are considered. Since these are typically rarer than low-return trades with better win probabilities, greater capital gain appreciation may be more feasibly be achieved with the latter and identification of such opportunities is more valuable to a trader than lotteries.

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