The Kelly Criterion – A Game Changer

mathsSuccessful options trading is based on understanding the statistical probabilities and using those probabilities to create an “edge”.  The Kelly Criterion is one method that traders use to determine the appropriate position sizing for a given trade. The underlying principle is that you should not put all of your money into a single trade, but rather put in an amount that is appropriate given the probable outcome of the trade and the impact that it may have on the overall account. However, position sizing is not the only use for the Kelly Criterion in trading options.

The Kelly Criterion (or Kelly formula) emerged from statistical work done by John Kelly at Bell Laboratories in the 1950’s to help figure out better ways to deal with signal-noise issues in long-distance telephone communications. Soon after it was published, the formula became very popular with gamblers who found that it could be used as a consistent money management system.

Calculating the Kelly Criterion relies on two basic components: a trading strategy’s percentage of winning trades (or probability of profit) and its win/loss ratio. The win/loss ratio is equal to the average profit from winning trades divided by the average loss from losing trades.  With these two components, it is easy to calculate what percentage of a trading account should be risked on any given trade.

The formula for the Kelly Criterion is:

Kelly % = W – [(1 – W) /R]

where

W = Win percentage (probability of profit)

R = Win/ loss ratio

For example, let’s assume that a hypothetical trade has a 60% probability of profit and will generate a maximum of a $600 profit or a $400 loss. The calculation for the Kelly percentage is as follows:

W = 0.60 (or 60% probability of profit)

R = 1.5 (or Max Profit of $600 divided by Max Loss of $400)

Kelly % = 0.60 – [(1 – 0.60) /1.5]

Kelly % = 0.33 or 33%

Based on the formula, up to 33% of the account equity can be risked on this trade. Unfortunately, in trading it is rare to find trades with this good of a risk/reward ratio.

Let’s look at another example which will utilize the formula in a real world options trade. With IWM trading at 117 and 38 days to expiration, we will sell a vertical credit spread for 1/3 the width of the strikes.

Sell 1 IWM Dec 121/122 Call Vertical @ $0.33

The probability of profit on this trade is 67%. Let’s plug this information into the Kelly formula.

W = 0.67 (or 67% probability of profit)

R = 0.493 (or Max Profit of $0.33 divided by Max Loss of $0.67)

Kelly % = 0.67 – [(1 – 0.67) /0.493]

Kelly % = 0

The Kelly formula is telling us that 0% of our capital should be allocated to this trade! Why? Because statistically, over time, this trade as no “edge” and therefore zero expectancy for success. Many options traders do not hold trades through expiration, but choose to take profits earlier (i.e. 50% of max profit). Does this improve the situation? Quite the contrary. If losing trades are allowed to run to maximum loss, then winning trades must also be allowed to run to maximum profit in order to just break even. If winners are taken off earlier (while still allowing losers to run to max loss), it results in a negative expectancy (or negative Kelly %). This is a key takeaway that runs counter to what many experts recommend with regards to defined-risk trades. If we are going to manage winners, we must also manage losers

This is where I feel the Kelly Criterion is a game changer. Rather than solving for the Kelly %, we can use the formula to solve for the point at which to close a losing trade. Having an exit strategy is a critical part of trading.  I have had no problems exiting winning trades. I typically exit credit spreads and strangles when I have received 50% of the maximum profit. However, at least for me, determining the exit point on a losing trade has been a bit more ambiguous. By using the Kelly Criterion we can calculate the point to exit a losing trade that will provide the trader with an “edge.” First, we will re-arrange the formula to determine where to take losses in order to break even (Kelly % = 0).

R = (1 /W) -1

In the prior example we sold the IWM 121/122 call credit spread. Let’s now assume that we plan to take profits at 50% of maximum profit. The spread that we sold for $0.33 will be closed for $0.17 ($0.33 * .50 = $0.165).  W equals the probability of profit (in this case, 67%).

R = (1 / .67) -1

R = 0.493

R is the profit divided by the loss and, in this case, is equal to 0.493. By dividing the 50% max profit by the value of R (0.493) we can determine that the max loss should equal 1.01 times (or 101%) the total credit received. We would exit the trade when the loss was $0.33 per contract (1.01 * 0.33).

Let’s look at another real-world example from a trade I placed this week.

On Monday, I sold the /ES Mar 16 2250/2300/1725/1675 iron condor for $6.75 ($337.50 credit received). At the time the trade was placed, it had a probability of profit (POP) of 84.37%. I plan to close the trade at 50% of max profit (Pmax).

R = (1 / POP) -1

R = (1 / .8437) -1

R = 0.185

Average Loss = Pmax / R

Average Loss = 0.50 / 0.185

Average Loss = 2.70 (or 270%)

This establishes the average loss at $18.23 (or 270% of the credit received). By adding the initial premium ($6.75) to the max loss ($18.23) we are able to determine the price at which to close the trade ($24.97) to break even over time.

However, we don’t want to just break even. By closing losing trades prior to the breakeven point provides the trader with the “edge” necessary for long-term profitability. In this particular trade, we would want to take our loss before it reached 270% of max profit.

How much edge you wish to give yourself is a personal preference. In “How to Price and Trade Options” by Al Sherbin, it is suggested that an edge of 10-15% is appropriate in most cases. For this next example, we will use a 15% edge. We subtract the edge from the probability of profit when solving for R. For example,

R = [1 / (POP – Edge)] -1

R = [1 / (.8437-.15)] -1

R = 0.442

Average Loss = 0.50 / 0.442

Average Loss = 1.13 (or 113%)

For the same /ES iron condor with a 15% edge on probabilities, we will now exit the trade if the loss exceeds 113% of the credit received ($6.75 * 1.13 = $7.63). Adding the original trade price (credit received) of $6.75 to the max loss of $7.63 sets a price of $14.38 at which we will exit the trade.

The math may seem a bit daunting at first, but a simple spreadsheet makes calculating profit at and loss levels a breeze. I have included the formulas below that I utilize in my spreadsheet.

Pmax = % of max profit to close profitable trade

Lmax = % of credit received to close a losing trade (average loss)

POP = % probability of profit

E = % edge on probilities

C = credit received per contract

Lmax = Pmax / ((1 / (POP – E)) -1)

Take profits at = C * (1 – Pmax)

Take losses at = C * (1 + Lmax)

Being able to place a trade with known exit points that provide a statistical edge to the trader is going to be a real game changer for my own trading. When placing trades, I have always entered a GTC order to close the trade at the target profit level. In addition, I am now also setting an alert in the trading platform which notifies me via text message when a loss approaches the point determined by the Kelly formula.

If you wish to read further on the subject, I would highly recommend Al Sherbin’s book, How to Price and Trade Options.

I have included an interactive calculator in the Members section which can be utilized to experiment with the formula. Special thanks to Henrik Santander of The Lazy Trader for writing this script for me.

  • The Lazy Trader

    Ufffffffff, solid solid solid!!!!
    LT

    • Aram Basmadjian

      Thanks, LT! Also, thanks for taking the time to convert the spreadsheet to javascript so that it could be incorporated into this post!

  • Thomas Serafini

    Hi,

    thank you for the interesting article, however it seems that your analysis is not taking into account an important fact. It seems that you are computing the POP using the options delta, which by definition, gives the probability if you keep options until expiration. However if you close the trade before expiration, probabilities change. The smaller the max loss is, the smaller the POP becomes.

    Look at your first IWM example: if you close the trade when the loss becomes $0.33 you have more probability to close at a loss than to take the full $0.67 loss. (as a paradox, if you set the take loss at $0.01 you turn the Kelly formula hugely at your favour, but it is obvious that with a $0.01 stop loss the probability of profit is almost 0).

    The same is for closing the trade early when you reach the 50% of max profit. This will increase the probability of profit to you advantage.

    For this reason I believe you analysis is not completely correct: in the final formula, you set a POP and compute an exit level. But with that exit level the POP is different so the so the level you have just computed does not hold. From a mathematical point of view you should setup this problem as an optimization problem.

    By the way, the POP changes over time: as time passes, the trade becomes more and more in your favour, so you may want to adjust the exit level accordingly over time.

    • Aram Basmadjian

      Thomas, you bring up some excellent arguments. The formula only provides a snapshot based on the probabilities at the time the trade is actually placed. Additionally, the formula does not take into consideration situations where volatility may expand resulting in a loss that approaches the max loss determined by the formula. In a situation like this, the formula would have you closing a trade for a loss that still has an extremely high probability of success. That is one of the reasons that I won’t use a stop/loss order to exit the trade based on the Kelly formula. I simply have an alert set and then make a determination at that point whether the loss should be taken or not. However, the formula DOES provide a line in the sand where the trade can be re-evaluated and a decision made.

      As for the mathematics of this, I am utilizing the formula exactly as it has been presented in various research with regards to trading. Keep in mind that the formula is based on the law of large numbers (the classic coin toss example).

      I am certainly not promoting that this is the silver bullet of trading. There is no such thing as alchemy and no system will work all the time. My point is to simply present a method that I am now using in my own trading to determine a point at which consideration should be given to exiting a losing trade. I have found it helpful from the standpoint that it takes the emotion (and hope) out of staying in a losing trade. Some traders will exit when the option price is twice the credit received. This is a valid method as well.

      I am hoping to have a live calculator on the site to assist those who prefer to experiment a bit. At some point, I also plan to have a follow-up article to address some of your issues as well as a discussion on how to deal with trades that suddenly gap down and cannot be mechanically exited, the implications of using more or less “edge” in the calculation, etc.

      • Thomas Serafini

        Thank you Aram for clarifying the inner meaning of your article!
        Keep up the good work!

      • drmark27

        I agree with Thomas and I don’t think you responded to his point. In the first example, how do you figure a 67% probability of profit when you plan to close for half the credit sold?

        • Aram Basmadjian

          Probability of profit is determined from the trading platform (ThinkorSwim). You are correct in that the POP is based on holding through expiration. For more in-depth analysis with the research to back it up, I would suggest reading Chapter 7 of How to Price and Trade Options by Al Sherbin.

          • drmark27

            I agree that POP is based on holding through expiration. You therefore should not be using 67%. What number should you use? I would agree with Thomas that the number is constantly changing. Without a reasonable mathematical model though, I don’t understand why you are using a static equation like this. Can we even begin to estimate how far off your solution will be?

          • Aram Basmadjian

            As I have stated before, if you would prefer to read more comprehensive and authoritative research on this, please take a look at Al Sherbin’s book. I am the first to admit that I do not have the knowledge or trading experience that he does. Mr. Sherbin’s resume is not too shabby (over 25 years of options trading experience, MBA in finance and minor in mathematics, former CBOE/CME floor trader, etc.). He uses these formulas in his own trading to determine exit points and has had a solid track record.

  • Very interesting indeed. I agreed that most traders are not able to take losses gracefully. It is human nature to want to be right. By taking a loss, we have to admit we were wrong with out original bet. When you make trading more mechanical, like what you are doing, it minimizes emotional trading. Your success as a trader will depend on whether you ask yourself how you plan on mitigating losses to your trading capital, rather than how you will get rich quickly.

  • Tony

    Many thanks for the article Aram! Gives me a lot to think about!

    • Aram Basmadjian

      Glad you found it thought provoking, Tony!

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