# Delta = probability of excercise?

Does the delta of an option equals the probability of exercise of this option?

Not exactly, but almost. For a quick look/check it can be considered that way.

There are textbooks that make this connection but the caveat is that the relationship becomes more broken the lower the delta (anything below 50)… ALSO it does not consider the probability that an option will go into the money (but expire worthless)… for trading purposes this is likely more important.

for example… a 10 delta option would have a 10% chance of expiring in the money (based on this concept) but the actual probability is likely much lower… you could buy 10 delta puts on 100s of different stocks for weeks on end in a rising market and they may all expire worthless.

As I recall this is exactly what cfai say about fx options. In fact they even say you can call your FX broker and express the option as a delta rather than a strke price.

meanwhile, back in the real world, of course they are never the same. implied volatility is rarely the same as actual volatility. and ‘exercise’ is something the back office do (as opposed to ITM).

That doesn’t have anything to do with whether the delta is a probability of exercise.

Personally, I’ve never heard of the option delta being a probability of exercise. For put options, of course, it’s absurd.

I’m not going to open the book, but there is definitely something written about a PM wanting to protect the 10% risk event of a change in FX rates and calling his broker for a 0.1 delta option.

(and for OTC FX options, you don’t need to consider puts).

cfai absurdity not mine!

FYI, for whatever its worth… Just wanted to add that delta is N(d1) in black-scholes’ formula while the probability of exercise is N(d2) - without including the price of the option. That is, probability of exercise would be N(d1)/ { E[ST | ST>K] }.

This is actually one of the few times I disagree with S2000. Yes, the absolute value of delta is used by many as a probability of the option finishing in-the-money.

One of the best books on options ever written, “The Bible of Option Strategies,” mentions this in its preface. Additionally, I’ve taken option courses on both the undergrad and graduate levels and both instructors have taught this. I was always taught this is a rule-of-thumb probability rather than a proven theory.

However, for the purposes of the exam, the CFAI doesn’t go into detail too much about this outside to say a .10 Delta strike has a lower chance of finishing in-the-money than a .25 Delta versus a .50 Delta and a .50 Delta call is currently at the money.

From the book: “Delta is another way of expressing the probability of an option expiring in-the-money. This makes sense because an ATM call option has a Delta of 0.5; i.e., 50%, meaning a 50% chance of expiring ITM. A deep ITM call will have a Delta of near 1, or 100%, meaning a near 100% chance of expiration ITM. A very out-of-the-money call option will have a Delta of close to zero, meaning a near zero chance of expiring ITM.”

Page 34 (PDF counter), graf 2.

Nobody mentioned the absolute value of delta.

I made a statement about delta, which stands: because it’s negative for puts, using it as a probability is absurd.

Using the absolute value of delta as a probability isn’t absurd, of course. I’ve simply never heard of anyone using it that way.

I’m afraid S2K is right.

Without going in details that I also assume is outside of his statistical knowledge. It is just not that simple.

The closest thing you could come up with is saying that N(d2) is the probability of expiring ITM for a binary option with an expected return on the underlying equal to the risk-free rate. Too many assumptions, while also assuming BS model holds true (and it doesn’t).

The simplest way to counter the book’s delta argument is that options are not normaly distributed, and probably not lognormal either, but it’s closer. Take 1,000 European options with a delta of 0.1 and 1 year to expiry, this means that with enough iterations, you will always get ~100 options expiring ITM, but this is not how options are priced.

If it were as simple as estimating the probability of ITM as taking delta for the actual (expected) price drift, we’d all be sitting on mountains of gold.

It accidentaly happens sometimes.

Don’t let it scare you.

This seems like we’re arguing a technicality, but you did not say absolute value. That said, it is known in the industry that Delta is an approximate figure of in-the-money probability.

Dude, I’m pretty sure the Options Industry Council, the industry-sponsored organization, knows more about options than you do. I know you have a following here, but don’t delude yourself into thinking you know more than you do. First, S2000 did not dispute my post was incorrect, but rather I mischaracterized his argument by adding “absolute.” I’m actually providing links rather than sycophantly posting somebody else is right. As I said, it’s an *approximation* that is widely-known in the industry.

Here’s the quote from the OIC:

“Looking at the Delta of a far-out-of-the-money option might give an investor an idea of its likelihood of having value at expiration. An option with less than a .10 Delta (or less than 10% probability of being in-the-money) is not viewed as very likely to be in-the-money at any point and will need a strong move from the underlying to have value at expiration.”

I, for one, wasn’t arguing a technicality.

As I said, I’d never heard of the delta of an option being its (approximate) probability of exercise, so clearly I wasn’t aware of the subtleties of language used in this context. In my ignorance, I said that the delta of a put (being negative) cannot be a probability.

That’s an interesting quote. I certainly don’t dispute that the smaller the delta, the less likely the option will expire in the money and, therefore, the less likely that it will be exercised. I’m less sanguine about the delta being an approximation of that probability, however.

First, we’d need to know how close the delta has to be to the actual probability to considered “approximate”. If the delta is 0.3 and the probability is actually 0.2, is that close enough? How about 0.3 and 0.1?

Second, I’d love to see a study that’s been done that compares deltas with the ultimate frequency of expiring in the money. I’m not aware of any such study (obviously: I wasn’t aware of any of this two days ago). Is there such a study?

I wouldn’t dispute that the industry uses such an approximation. I’d like to know whether it’s justified or not.

I think I’ll run some Monte Carlo simulations and see where they get me. It may be useful data for an article.

“I’d like to know whether it’s justified or not.”

I’m quick to admit what I don’t know. And this, sir, is your area of expertise (hypothetical vs. theoretical). I don’t know of any study that compares the percentage of finishing in-the-money and Delta. But considering Delta changes with every move of the underlying I can see, conceptually, the positive relationship. Whether the probabilities end up being totally correct at the end or are simply the best probability/Delta estimate at that time under the Black Sholes pricing model is beyond me. As I said, I’m quick to admit what I don’t know.

If you have some huge breakthrough please keep me in the loop.

Here are the parameters for the B-S-M model I used to run the simulations:

• Strike price = \$20/share
• Time to expiration = 0.5 years (I used a 360-day year, so this is 180 days)
• Annual (effective) risk-free rate = 2%
• (Annual) volatility of (continuously compounded) returns of the underlying: 10%

I used @Risk in Excel to run the Monte Carlo simulations: 100,000 iterations for each delta. I used Solver to determine the (spot) price of the underlying for each delta. I compounded daily returns, with serial correlations of zero.

These are the results:

• Delta = 0.00, P(exercise) = 0.0%
• Delta = 0.10, P(exercise) = 0.0%
• Delta = 0.30, P(exercise) = 0.0%
• Delta = 0.30, P(exercise) = 0.0%
• Delta = 0.40, P(exercise) = 0.0%
• Delta = 0.41, P(exercise) = 0.0%
• Delta = 0.42, P(exercise) = 0.0%
• Delta = 0.43, P(exercise) = 0.0%
• Delta = 0.44, P(exercise) = 0.0%
• Delta = 0.45, P(exercise) = 0.1%
• Delta = 0.46, P(exercise) = 0.5%
• Delta = 0.47, P(exercise) = 1.8%
• Delta = 0.48, P(exercise) = 5.0%
• Delta = 0.49, P(exercise) = 12.4%
• Delta = 0.50, P(exercise) = 25.0%
• Delta = 0.51, P(exercise) = 42.3%
• Delta = 0.52, P(exercise) = 61.0%
• Delta = 0.53, P(exercise) = 77.6%
• Delta = 0.54, P(exercise) = 89.2%
• Delta = 0.55, P(exercise) = 95.8%
• Delta = 0.56, P(exercise) = 98.6%
• Delta = 0.57, P(exercise) = 99.6%
• Delta = 0.58, P(exercise) = 99.9%
• Delta = 0.59, P(exercise) = 100.0%
• Delta = 0.60, P(exercise) = 100.0%
• Delta = 0.70, P(exercise) = 100.0%
• Delta = 0.80, P(exercise) = 100.0%
• Delta = 0.90, P(exercise) = 100.0%
• Delta = 1.00, P(exercise) = 100.0%

If the delta were a good approximation to the probability that the option would be exercised, a graph of P(exercise) vs. delta would look like this:

/

In fact, the graph looks like this:

_/¯

I can run some more simulations, particularly with different volatilities on the underlying returns, but the results here are pretty clear: the option delta isn’t remotely a good approximation to the probability that the option will be exercised.

By the way, another interesting result (about which I already knew) is that the delta of an at-the-money call option isn’t necessarily 0.50. Here, the delta for an at-the-money call option is 0.57. You get a 50-delta call when the spot price is \$19.75 and the strike price is \$20.00.

I ran another set of simulations with σ = 30%:

• Delta = 0.00, P(exercise) = 0.0%
• Delta = 0.10, P(exercise) = 0.0%
• Delta = 0.30, P(exercise) = 0.0%
• Delta = 0.30, P(exercise) = 0.0%
• Delta = 0.40, P(exercise) = 0.0%
• Delta = 0.41, P(exercise) = 0.0%
• Delta = 0.42, P(exercise) = 0.0%
• Delta = 0.43, P(exercise) = 0.0%
• Delta = 0.44, P(exercise) = 0.0%
• Delta = 0.45, P(exercise) = 0.0%
• Delta = 0.46, P(exercise) = 0.0%
• Delta = 0.47, P(exercise) = 0.0%
• Delta = 0.48, P(exercise) = 0.1%
• Delta = 0.49, P(exercise) = 0.7%
• Delta = 0.50, P(exercise) = 2.2%
• Delta = 0.51, P(exercise) = 6.3%
• Delta = 0.52, P(exercise) = 14.4%
• Delta = 0.53, P(exercise) = 28.1%
• Delta = 0.54, P(exercise) = 45.7%
• Delta = 0.55, P(exercise) = 64.6%
• Delta = 0.56, P(exercise) = 80.4%
• Delta = 0.57, P(exercise) = 91.0%
• Delta = 0.58, P(exercise) = 96.6%
• Delta = 0.59, P(exercise) = 98.9%
• Delta = 0.60, P(exercise) = 99.8%
• Delta = 0.70, P(exercise) = 100.0%
• Delta = 0.80, P(exercise) = 100.0%
• Delta = 0.90, P(exercise) = 100.0%
• Delta = 1.00, P(exercise) = 100.0%
Again, the results here are pretty clear: the option delta isn’t remotely a good approximation to the probability that the option will be exercised.

^^He wins

Sure, you should publish your work and let the greater options community debate this. All I know is in the option community delta is considered an approximate of in-the-money probability, obviously you say this should not be used. Or, you should trade in these markets to take advantage of this.