Trading of Stocks using Kelly Criterion
Team Latte
Sep 15, 2005

Let us say you have $100,000 to invest in a certain stock and you are faced with this question: you want to trade the stocks consistently over a period of one year but you don't know how much to put into the market at any one period, say, every month given the market's direction. As a day trader, you want to enter the market in the morning and exit at the close taking profits or losses with you. It is like a bet: you bet an amount in the market and at the end you either make money or lose money.

Kelly criterion can, with certain very simplistic assumptions, help you answer your question to a very great extent. Let us say that your investment horizon is three months and over the last three months the stock has had a mean monthly return of 0.2% and a monthly volatility of 7%, and that these numbers are calculated using monthly closing prices of the stock. You don't know which way the market will move, and you cannot short sell the stock, therefore, you would ideally want to bet a fraction of your total the total amount (that you hold) every month. You start with $100,000 on day zero and you want to bet a fraction q of that amount on day one of the first month. Now since markets follow a random walk the stock's closing price on the last trading day of the month would be a random outcome; let's denote this by a series of random numbers (one for each day) (z₁,z₂,z₃,..............z_N). Thus after the first month's bet (investment in the stock) you will have in your hand (at the end of the first month) an amount equal to:

On the first day of the second month, regardless of the outcome of the first day, you will bet the same fraction q of whatever is left from your investment from the previous month. Then, after the second month you will have in your hand an amount equal to:

After 12 months you will have the following amount:

Your problem is how do you choose the quantity q so that your strategy has the optimum payoff? In fact you are going to determine the quantity q such that it maximizes your expected long term growth rate. That growth rate is:

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