Silly Wall Street Sayings:

Delta is Probability

I've always wanted to write this article though it is unlikely everyone has been waiting wiith bated (or even baited) breath to read it.

Delta is defined as the dollar amount the option price will change if the stock moves one dollar.

[Superfluous complicated equations]
δ  = N (d 1)   
w here:d 1=ln(S  K    )+(r+σ  2  2   )t σ t          
 K           Option strike price
 N           Standard normal cumulative distribution function
 r          Risk free interest rate
 σ          Volatility of the underlying
 S           Price of the underlying
 t          Time to option's expiry

As I'm sure is obvious from observing this equation, Delta will occur as an 'S' shaped curve, increasingly linear as the time to expiry increases. Here is an actual graph of the same strike option with differing times to expiration.

Option Delta changes with time

It is important to note that Delta is a mathematical calculation about the price movement of an option relative to the underlying stock. At no time does probability enter into the math. Misusing math is the province of the con man; or, as my Dad was fond of saying, "Figures don't lie but liars can figure".

The red curve depicts the Delta of a call with a shorter time to expiry than the blue curve. With a little imagination, you could see the blue curve continue to straighten as time to expiry increase.

Similarly, a little imagination could see the red curve approach a discontinuity at the 'At the money (ATM)' level such that you are left with a coin flip probability whether your option will be in or out of the money on the last day, last hour, last minute of expiration. That is not entirely real, of course, as other factors are at play.

The clear lesson here is that Delta changes with the time remaining to expiration and the value of underlying.

Our experience in the real world has prepared us for static probabilities: flipping a coin or playing cards. None of our experience has prepared us to handle variable probability.

On a coin flip, you do not have to check the currency market for the value of a penny; you do not have to consider how long it might be until you spend it. None of that will affect the expected value of flipping the coin.

Why does it matter? Our internal biases are based on our experience. When our mental model does not match real world events it is inevitable that we will make poor decisions. And that is the entire problem with a short cut to probability using Delta.

Let's take a simple example of selling a covered call on AAPL. Selling a covered call is a way to bring in some cash on long position we hold so long as the call we sell remains higher than the value of the strike price at expiration or whenever we close it. As you will learn in our TAP training, writing short term covered calls in a bull market may not be a wise choice.

As nothing is simple, we have a procedure for writing covered calls:
  • Select an expiration: we will look out to next Friday, 26 Oct.
  • Look at the Average True Range: Which is about $5.7 for AAPL. With the current price at $219.31, we can expect AAPL to rise to $225 or fall to $213.6 on a normally active day over the short term
  • Select a strike price: we want something that is out of the money that yields some return. Stay tuned for a future article on "Getting Paid Enough for the Risk". In this case, we like the 23 Delta strike at 225 but it is too close to being within the ATR. We'll work the $227.5 strike, 14 Delta call that ranges from $0.50 - $0.54 today. For our purposes, let's suppose we were able to sell it at $0.52.
  • Calculate the maximum loss: Sadly, the maximum risk on selling a covered call is an infinite loss as AAPL goes to infinity. More realistically, should we not be paying attention while AAPL jumps $10, we'll suddenly find ourselves with about a $10 loss on our $0.52 gain from selling the call.
  • Calculate the probability of winning: If the Delta-as-Probability were true, we would think we have an 86% chance of a winner on this position.
  • Calculate the expected value: Using Delta, this would maybe be (0.86 * $0.52) + (0.14 * (-10.00)) or -$0.95 or about an 80% loss on the position.
If we take a different approach to what it might take to make money on this trade, it is quite a bit worse. We have to get the following things right, where we have estimated our probability of being right on each of them:
  • Market needs to be flat-ish over the next week: 50% probability
  • Segment needs to be flat-ish over the next week: 50% probability
  • AAPL needs to be flat-ish over the next week: 50% probability
  • Calculate the expected value: (0.5 * 0.5 * 0.5) * $0.52 or $0.065. Well, at least were we to take this trade a few thousand times, we could come out ahead by enough money to pay the tax on a bottle of sparkling water.
You can play around with the probabilities based on your own perspective and market view.

Which one of these is more correct? Your guess is as good as mine and that is the entire point. The talking heads on "Options Action" would have you believe there is an 86% chance of a winner on this trade using Delta as a proxy for probability. Which is entirely incorrect.

There is no doubt that we would take more 86% probability of success positions than we would 12.5% probability of success trades. Using Delta as a surrogate probability will lead to low probability of success investments, something we need to avoid in aspiring to 10alpha.

Another view of this, however, is to look at very long term options. Suddenly, Delta isn't so bad as a proxy. That is all due to the straightening of the curve as time to expiry increases. Effectively, buying an 80 Delta call on AAPL that expires in January of 2020 may well have something near an 80% probability of success for some near term yet indeterminate period of time.

Now that is something we can work with and our TAP training takes advantage of this scenario in a relative sense when comparing investment alternative.

Sometimes we want to believe something so badly that we allow ourselves to be taken advantage of Aaron B. Powell, Doomsday Diaries III: Luke the Protector