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Confidence intervals for the Kelly criterion

Published in Automated Trader Magazine Issue 39 Q2 2016

Trade-sizing under constant probabilities and fixed payoffs is straight-forward. But having varying probabilities and payoffs introduces uncertainty into how much to bet and therefore requires additional inputs.

John Kelly answered the question of how a gambler with an edge should act in order to maximize his bankroll growth. The resulting trade sizing scheme (the Kelly criterion) shows the optimal fraction of the bankroll to be allocated to each opportunity.

The Kelly criterion has a number of desirable properties:

  • As we only ever invest a fraction of our wealth, we can never go bankrupt.
  • The strategy is guaranteed to asymptotically outperform any essentially different strategy.
  • The time for the bankroll to reach any fixed amount is asymptotically smallest with this strategy.

However, the scheme also possesses some undesirable properties. The biggest concern is overestimating the sizing fraction which can lead to disaster. There is a critical level of the invested fraction (approximately equal to twice the optimal proportion) where the growth rate of the portfolio becomes negative. In this case, the bankroll asymptotically tends toward zero.

AUTHOR'S BIO

Euan Sinclair is the co-founder and CEO of ­factorwave.com, a quantitative stock and option advisory service. He holds a PhD in theoretical physics from the University of Bristol and has written two books, "Volatility Trading" and "Option Trading" , as well as numerous papers and articles.

In some situations, the Kelly ratio can be calculated exactly. However, in most cases this is not possible. When applying the Kelly criterion to sports gambling or trading financial instruments, the optimal ratio needs to be estimated by analyzing historical data.

As we will only ever have a finite amount of data, this is a case where we are attempting to estimate a population parameter from a sample. Inevitably, this means our estimate will have some degree of uncertainty.

Traders know this. They often arbitrarily reduce the size recommended by the Kelly criterion. This is known as fractional Kelly betting. This idea reduces risk, but doesn't eliminate it, because we are still just scaling the estimated fraction. There is even the possibility that the true bet size should be negative, and we should not bet at all. Also, arbitrarily reducing the fraction like this does not take into account the actual uncertainty of our estimate of the betting fraction.

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