Calculating ITM/OTM Probs, POP, POT

Could you suggest any understandable read on this topic?

My advanced mathematical grasp is very limited, hoping for something comprehensible. Any author who took the pain to explain it step-by-step from basics to advanced.

Thank you!

@Lennon if you have the patience to view some lengthy videos (with a lot of almost useless chatter) this guy covered a lot of stuff:

The Skinny On Options Math

The nice part, if you sign up (free), is the possibility to download the "slides" that recap what was discussed in the video (but frequently some familiarity with math formulas is required).

This book has a short chapter (#9) on "Probabilities" where the author discusses both POT (Probability of Touching) and POP (Probability of Profit).

Essential Option Strategies: Understanding the Market and Avoiding Common Pitfalls Hardcover – September 19, 2016
by J. J. Kinahan (Author)

but on the other hand, this content is probably too lightweight.

I hope someone else has better suggestions for you!


Dear Mr. Bonariva,

Thank you very much for your reply.

  1. Those videos are sort of promotional in nature for Thinkorswim. Actually watched them earlier. TOS provides these features in-built in their platform. And these videos are their effort to make others aware of the concepts. But does not discuss the math.

  2. Was very disappointed by J.J. Kinahan's book. He has just written some passages without explaining the underlying maths for understanding these beautiful numbers.

Recently I succeeded to accurately calculate implied volatility in AmiBroker for European Calls/Puts using excel examples shown in So was wondering whether I could go a step further and learn the real applications of implied volatility.

Has been curiously searching libraries and book stalls for few weeks now. Still no luck on something comprehensive on this subject.

So far have learned that Delta can be used as a proxy but its just an approximation. My quest is to understand the math behind the calculation of these probabilities. The best that I have found is this InvestExcel spreadsheet but has tons of doubts for what has been shown.

Thank again!

@lennon maybe you should evaluate:

"The complete guide to Option Pricing Formulas" by Espen Gaarder Haug (Mc Graw Hill) - 2nd Edition

This (IMHO very good) book comes with a CD with almost all the formulas in VBA (embedded in Excel files) that are not too difficult to port to AFL.

I see that there is a section (2.8) for "Probabilities Greeks" that opens with these words:

In this section we look at risk-neutral probabilities in relation to the BSM formula. Keep in mind that such risk-adjusted probabilities could be very different from real-world probabilities

The chapter provides some formulas for ITM probabilities (2.8.1) but at a glance, I do not see OTM ones.

1 Like

Thank you @beppe,

You are very kind. I am amazed by your knowledge. I do not have that book.

Probability% of a strike price expiring ITM is the mother of other probabilities. 100-ITM Prob% will tell us the probability of a strike expiring OTM.

POT is approximately twice of ITM Prob%. Not aware of the actual formula.

I don't know how to calculate POP. And neither I am aware of the mathematical variations used for calculating ITM Prob%. Until I know what variations are used to discern ITM Prob% how will I figure out which one to use?

Will try to finish your recommended book by this weekend.

Many thanks again!

What is disheartening is that mostly authors speak about the probabilities in terms of BSM.

But on practical outset binomial probability distribution appears to be more promising to me.

And @beppe I cannot finish your recommended book by this weekend - it's like The Everest, extremely academic in nature. Not able to find much matter or a book for the study of options probability using binomial probability distribution.

I would again need to go back to my shell. Aahhh trading.... So easy to get in, so hard to get out......

Statistics is such a broad topic! 1) See note at bottom

To keep it "options" related here are some additional books titles that may have some relevant content for you:

How to Price and Trade Options: Identify, Analyze, and Execute the Best Trade Probabilities - by Al Sherbin
Chapter 3/4: Building the Foundation/Trade Probabilities: What to Look For

The distribution curve that feeds our pricing model and probability assumptions when trading options is the curve defined by the implied volatility of the options in an option chain. This volatility is driven by the distribution curve, which in turn drives our probability calculation. ... The Distribution Curve As we discussed, all option models have the underlying’s distribution curve at its heart

Option Volatility and Pricing: Advanced Trading Strategies and Techniques 2nd Edition - by Sheldon Natenberg
Chapter 19: "Binomial Option Pricing"

If we want binomial values to approximate Black-Scholes values, u and d must be chosen in such a way that the terminal prices approximate a lognormal distribution. But why should the probabilities be the same? Perhaps the probability of movement in one direction is greater than the probability of movement in the other direction.

Systematic Options Trading: Evaluating, Analyzing, and Profiting from Mispriced Option Opportunities - by Sergey Izraylevich , Vadim Tsudikman
Chapter 5. Selection of Option Strategies

Normal distribution has nonzero probability of occurrence of very high and very low variable values. ... Therefore, we cannot fully rely on the results of statistical tests assuming that probability distribution of the analyzed variable is normal. ... The nonparametric test based on binomial distribution analyzes the difference between the two frequencies observed in the experiment and compares it to the frequency assumed under the null hypothesis.

Option Pricing and Estimation of Financial Models with R - by Stefano M. Iacus
Chapter 2: Probability, random variables and statistics

Similarly, it is possible to prove that the Binomial distribution is not infinitely divisible, while the Gamma, the exponential, the Meixner and many others are. The log-Normal distribution is sometimes called the Galton's distribution. More precisely where Bin(n, p) stands for Binomial law3 with parameters n and p, which is the following discrete distribution.

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach - by Guojun Gan , Chaoqun Ma , Hong Xie
Chapter 41: Discrete-Time Models

As the number of time steps increases, the binomial price of an European option converges to the Black-Scholes price (Follmer and Schied, 2004). The dynamics of the riskless asset is given by The random variables μ1, μ2,…, μN are assumed to be independent and identically distributed on the probability space (Ω, , P): where 0 < d < u and p (0,1). Definition 41.17 (Binomial Model).

(Subset of results from a search on SafariBooksOnLine for "binomial probability distribution options". This is a service I recommend to every programmer/trader; you can access all the above books, and many others from over 200 publishers, also in the free 10 days trial period. No affiliation, I'm just a normal subscriber).

1) The "Statistics" search, limited to books in the "Math & Science" and "Financial and Economics" section return over 2000 titles...


@beppe thanks for recommending the books. Would like to acknowledge that Sheldon Natenberg's Option Volatility and Pricing is a must for such endeavor. Its truly an eye opener.