Leverage Using The Kelly Criterion

I will start this post by reminding you that trading with leverage carries significant risk.

This post is about the algorithmic use of leverage in day trading. The goal is to optimize the growth of the trading capital. In Part I it was shown that day-trading SP500 can be described in terms of a binominal game. Parts II & III showed how to make this game in favor of the trader. The known solution to favorable binomial games is The Kelly criterion.   Kelly's solution involves a mathematical idealization that the capital can be divided infinitely, i.e. a bet less than 1 cent is possible, which is not the case in real trading. In this post, an example of how the Kelly criterion can be used to control leverage in real trading is provided.

Part  II  describes a generic algorithm for trading SP500 which ensures a positive expectation of return. Averaged over the years 2007 to 2017 this algorithm has the win ratio w = 0.38 and payout p= 2.7.  Kelly is given by k = w-(1-w)/p = 0.15. Thus, for the optimal growth of the trading capital, the fraction of the capital to wager (f) has to be 0.15. Recall that the bot uses 1/4 of the daily Standard Deviation of SP500 as the stop loss. Consequently, to start trading one Emini contract at the beginning of 2007, it was necessary to wager $280= 1/4*1.25%*SP500*50 and keep $280/k=$1800  as the initial capital. In the simulation, the initial capital was set to $5400 which is 3x of the precalculated starting capital. The goal was to decrease the volatility of the trading capital.  The initial bet remained $280, i.e. f=0.05 in the first trading event.  Unlike the Kelly solution, the bet will not be changed at every step, therefore f varies with the trading capital. The algorithm of adapting f is as follows:

if  (f < k/6)  number of contracts =  number of contracts* 2;
if (f >  k/2)  number of contracts =  number of contracts / 2;

 The result of the trading from the long side ($5,000 to $5,000,000 in about two years) is shown below.

DISCLAIMER
The result presented here is a theoretical study that was conducted out of my personal curiosity. The algorithms described here may not be used as trading advice.