## The Kelly criterion

Today , I’m going to talk about a formula that came from information theory. Don’t be intimidated.
Based on the experience I have had writing spreads out of the money, and one standard deviation away from the current stock price, I decided to investigate the best method for money management to obtain an exponential growth over time.
If you remember, in the early posts I spoke about the Martingale method, and also about the gambler’s fallacy. The fact is that with any strategy, there is always a sequence that will cause you to lose everything. Naturally, most traders use some percentage of their capital dedicated to trading, say 10%, which means that initially things move slowly but eventually the earnings accelerate. The other advantage of this is that you get to stay in the game longer, because it can take a while to get you drawn down to where it is no longer possible to trade.
The Kelly Criterion is a well known formula discovered by J.L. Kelly in 1956. See here for details.
Kelly used the information theory of Claude Shannon to estimate the ratio of capital a hypothetical gambler should wage given a stream of binary bets, with probability p, and odds paying b on every win, and total wager loss with probability q=1-p.

The formula says the ratio of capital to wager is: f = (bp-q)/b.

Case study.
My favorite strategy is to sell a put or a call, close to the one standard deviation line (Bollinger band set at 2, 20) and to buy another call or put a little bit lower or higher than the first. The strategy is called selling a spread, because the purchase of the second option still leaves you with a credit from the sale of the first option. If you do both calls and puts at the same time it’s called an iron condor. You sell the spread outside the one standard deviation because this way, if the stock inches slowly toward your spread before expiration, there is a good chance it won’t reach it. It will make you nervous but you should just set a stop order to buy back the short leg and accept the loss. You can decide to keep the long leg of the position depending on the situation. When selling put spreads, because stocks can drop faster than they go up, I can sometimes risk the long option premium in addition to the loss I took buying back the short leg, if I think the stock will drop far below the long put and yield a nice profit.
I also like to sell puts using weekly option for three reasons. First the time decay pays much faster than only monthly options, leaving you exposed for a shorter time. The second reason is that it corresponds to a much shorter span of days than the 20 days I use on my Bollinger bands, so the bet is more conservative. Ideally you should adjust your Bollinger settings to the number of days you plan to sell. I leave mine at 20 days and sell anything less than 20 days from expiration. Finally, when strikes available are 0.5 apart, two \$50 spreads often pay better than a single \$100 spread.

Given this strategy, how much should we risk ?
We estimated the probability before and I will repost the picture I used at the time.

Based on the normal distribution, and a +1 sigma position, the probability of the stock ending within the blue band is 68.26%, and if you count the probability that it will end up in the green band favorable to you that’s another 13% or so. I will assume the split is 81% in your favor and 19% against you. It’s actually 0.8413 for the exact Normal distribution, but we can’t bank on the stock following the Normal law anyway, so I’m purposely reducing the probability to stay conservative.
The payoff is the premium received for the spread. I’ll assume a \$100 risk. Luckily you won’t lose all that, so the ratio will be better.
Let’s say the credit/payoff is \$44. The odds are 44 to 100.
So, we have p=0.81, q=0.19 and b=0.44. The Kelly criterion says f=(bp-q)/b = 0.378 or 37% of the capital should be risked. But for a payoff of less than \$23/\$100 no wager should be made.
So in summary, if you write one sigma out of the money spreads, make sure they pay \$44 to each \$100 spread and risk at most 37% of your capital.
Considering that when volatility is low few spreads bring in more than \$25, it highlights the need to set tight stops in case the position goes against you.

You can’t change the probability, but you can change the odds a little bit by setting the stop where your loss will be less than the full spread risk. By setting a stop at \$50 instead of \$100 for the spread, you get different odds, say \$25 : \$50 and you improve your chance at exponential growth.

Here is a quick chart of what the Kelly criterion says you should risk based on b, the odds and p, the probability:

Red cells obviously say you should stay out or take the opposite bet.