Rich Newman

July 1, 2007

A Beginner’s Guide to the Black-Scholes Option Pricing Formula (Part 3)

Continued from part 2.


If you know a little about options already you will probably be aware that their values depend on something called volatility. Volatility is usually not needed to price derivatives that are not options.

Technically volatility is defined as the annualized standard deviation of the return on an asset (in our case Microsoft stock). They are expressed as percentages.

However, it’s easier to think of it intuitively as the amount that the price will swing around in a given period. Stocks with a high level of uncertainty surrounding them will have high volatilities. An example currently might be the stock of small Russian oil companies. Stocks that are relatively stable (e.g. Microsoft) will have lower volatilities.

Why does volatility affect the price of an option? Again this is because our payoff graph is not symmetrical. A stock that has a high volatility is more likely to swing around, and hence more likely to have a very high value or very low value at maturity. A stock with a low volatility is more likely to be close to its current value at maturity.

Now if the stock price at maturity is below our strike price we don’t care if it’s just slightly below or massively below. In both cases we don’t exercise the option and don’t make any money.

But if the share price at maturity is above our strike we really want it to be as far above the strike as is possible, since we make more money the higher the volume is.

So an option with a high volatility is more likely to make us lots of money if the price goes up, but won’t lose us lots of money even if the price goes down hugely.

As a result options with high volatility are more valuable than options with low volatility.

As we will see below both d1 and d2 in the values discussed above depend on volatility.

The Formula

Finally, note that if I have bought the call I am paying the cash amount in i) above and receiving the value of the stock ii). So we can say that the value, c, of a European call option on a non-dividend paying stock is:

Black-Scholes Call on European Stock

d1 and d2

As mentioned in the introduction the mathematics behind the calculation of the probabilities in the Black-Scholes formula is fairly complex. It turns out that if N() is the cumulative normal function (a statistical operator) then d1 and d2 can be expressed as below. I’ll just present the results without explanation here. Note that whilst these formulas are complicated, you can just plug in the underlying values and get a result: this is what is known as a ‘closed form’ solution.

Black-Scholes d1 d2

Here sigma (σ) is volatility, as discussed above.

Actual Derivation of the Formula

This article has attempted to provide an intuitive interpretation of the Black-Scholes formula, without going into the mathematics behind it. Such an interpretation inevitably glosses over some of the details. I have glossed over risk neutrality considerations above, for instance.

In particular the actual derivation of the Black-Scholes formula was not done directly using the intuitive ideas discussed here. I will discuss this in future articles.


John C. Hull. Options, Futures and Other Derivatives (Sixth Edition)



  1. Hi, I couldn’ find the better explaination than this, it was simply excellent way of defining the BS.


    Comment by Manoj Sharma — April 4, 2009 @ 10:05 am

  2. Really excellent way of explanation.Keep posting such blogs.

    Comment by P M W Ssawantth — May 2, 2009 @ 2:03 pm

  3. Thanks, that’s awesome. The cherry on top would be giving a walk-through example with given values.

    Comment by dee — June 12, 2009 @ 2:40 pm

  4. Thanks a lot!

    Comment by Sean — June 19, 2009 @ 10:39 am

  5. in entering the values, how do you deal with the 1n at the beginning of the equation? i know its a dumb question.

    Comment by pamelasuttonlegaud — November 3, 2009 @ 1:21 am

    • ln is the natural log.. just calculate it.

      Comment by YankeeFan#34521638 — December 5, 2009 @ 11:00 pm

  6. Thank you for saving my life. Having finals on the above model, and your explanation definitely helps! Best explanation ever.

    (P/S: Salute you for simplifying a complicated idea. I’m reading the same book (Hull) as you, and I had a hard time understanding it.)

    Comment by Erick — November 8, 2009 @ 7:21 am

  7. This is the first and only cogent non-math explanation I have ever found on the web of the BSM model. You have done a great job of separating the essential concept from the more complicated math which could confound and confuse someone who has no background in statistics.

    Comment by rds — February 26, 2010 @ 3:34 am

  8. This is a great start for anyone with a non-statistics background. A great way of introducing the model, even for those who intend to dig into the mathematics of it.

    Thanks, Great job!

    Comment by Njilefac — March 8, 2010 @ 12:16 pm

  9. thanks a lot……… one could have explained better than you sir………….

    Comment by ritu — April 20, 2010 @ 5:54 pm

  10. Why did you put KN(d2)e-rt together? This not how it’s writen in the formula.

    Comment by JOhn Cassidy — May 6, 2010 @ 6:49 pm

  11. hey, nicely done, but i have request – can you explain N(d1) and d1 intutively or in laymans terms.

    Comment by MK — June 15, 2010 @ 7:10 pm

  12. Super helpful, thank you. Wikipedia not understandable at all by non maths graduates!

    Comment by Louise — July 16, 2010 @ 10:55 am

  13. Great job indeed! It would be wonderful to close the chapter if d1 and d2 can now be explained in the same way (I know it is difficult to do so).

    Comment by AH — July 27, 2010 @ 2:57 pm

  14. thank you this helps a lot!

    Comment by Lenore — October 9, 2010 @ 1:13 pm

  15. Lovely man….. i hav no words to express my happy ness after reading and understandin The Bs….. Thanks a lot ……

    Comment by Swapnil S Anagal — March 28, 2011 @ 7:39 am

  16. thank you so much, it helps for my study 🙂

    Comment by Anonymous — April 4, 2011 @ 7:02 pm

  17. Great explanation. Thank you.

    Comment by Anonymous — July 11, 2011 @ 9:42 am

  18. awesome, man .. more is needed on d1, d2 and N()

    Comment by Anonymous — September 15, 2011 @ 9:10 pm

  19. another comment to tell you how great this was. as mentioned above more on the probabilites and what they mean but i guess thats deviating away from BS and more to stats. Thanks again

    Comment by Anonymous — October 9, 2011 @ 5:29 am

  20. Great stuff! Very good explanation of a very complex formula. Thanks a lot..

    Comment by Anonymous — October 13, 2011 @ 6:43 pm

  21. what exactly is d1 and d2?

    Comment by Anonymous — November 3, 2011 @ 7:12 pm

  22. “As a result options with high volatility are more valuable than options with low volatility.”

    No, because it will be more expensive. There’s no free lunch. Also, you forgot to mention that the formula is a complete dud due to the assumption of a normal distribution.

    Comment by Julian D. — December 9, 2011 @ 6:52 am

  23. Julian D – 1/ Options with high volatility are more valuable than options with low volatility in the same way that diamond necklaces are more valuable than glass necklaces. Of course a diamond necklace is more expensive, that doesn’t change the validity of the statement.

    2/ All mathematical models are approximations to what they are modelling. It’s true that the Black-Scholes model makes a number of assumptions that oversimplify how options markets really work. In practice options dealers will make some adjustments to the model to make it reflect the markets a little more closely.

    However, that doesn’t make the model a ‘dud’. When it was derived it provided a concrete basis for an options price where previously there was no easy way of working that out. Arguably its derivation started the whole of the derivatives trading industry that we see today.

    I may write a little more on point 2/ at some point since there seems to be some interest in the subject. When I wrote this article it got almost no hits relative to the rest of the blog, but now gets the most hits.

    Comment by richnewman — December 10, 2011 @ 5:10 pm

  24. Rich, very informative article, thanks. Did you go into more detail on the derivation of the formula in another article?

    Comment by JamesL — April 10, 2012 @ 3:02 pm

  25. However, it would be good to give definitions of the quantities! Like S and K for example ….

    Comment by Anonymous — May 9, 2012 @ 12:42 am

  26. Wow, very nice guide, thanks a lot. Saved me loads of time.
    Any chance for you to write a sequel for theses guides ? Or where to find such courses in layman’s term ?

    Comment by Salomon — May 21, 2012 @ 9:40 pm

  27. great article!!
    It would be better if you also explain the maths behind the formula of d1 and d2.
    but still because of that i memorized the formula easily

    Comment by Anonymous — September 25, 2012 @ 5:34 pm

  28. […] This is continued in part 3. […]

    Pingback by A Beginner’s Guide to the Black-Scholes Option Pricing Formula (Part 2) « xu1892 — September 29, 2012 @ 8:34 pm

  29. Excellent explanation in a lucid manner

    Comment by Anonymous — November 18, 2012 @ 9:40 pm

  30. Very easy to understand … !

    Comment by Anonymous — January 6, 2014 @ 6:12 pm

  31. Excellent, I learnt more from this than a master degree course that I am taking now (part of it introduced this theory but I did not have any clue afterward). that is why i am here.


    Comment by Levan — April 9, 2014 @ 1:44 am

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  33. excellent explanation of a complex topic.. bravo!!

    Comment by kashyap — January 28, 2015 @ 4:22 pm

  34. It is probably the best explanation, thank you.

    Comment by Gayane — September 16, 2015 @ 1:52 pm

  35. Hi, really nice article.

    I was wondering if you could explain why the probabilities involved are different, in an intuitive way?

    For instance, why are the variables d1 and d2 different?

    If we only exercise the option when the stock price is above the exercise price, and the probability of that occurring is N(d1), why is the probability of the stock being above the stock price in the stock term N(d2) and not N(d1)?

    Comment by Anon — September 19, 2015 @ 5:38 pm

  36. Thank you so much. This was of great help 🙂

    Comment by Amanjot K — November 15, 2015 @ 10:24 pm

  37. Excellent article! Quick question: based on this, why do ATM option prices move linearly with respect to volatility? Thank you!

    Comment by Jess — May 29, 2016 @ 6:02 am

  38. Actually i don’t even know how to start giving you my profound gratitude. never the less, thank you very much for your effort
    and concern

    Comment by Anonymous — June 15, 2016 @ 5:28 pm

  39. I don’t know what words to use to thank you for such a comprehensive and glorious explanation of Black Scholes option pricing model. I again thank you from the depth of my heart and say ” May you stay blessed”.

    Comment by Anonymous — September 28, 2016 @ 3:04 pm

  40. This clearly demonstrates that you can explain to others only what you yourself awesome manner of taking forward the explanation..Thanks!!!

    Comment by Akshay Jain — April 24, 2017 @ 1:34 pm

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