# Stock Market Models – Introduction to the Black-Scholes formula | Finance & Capital Markets | Khan Academy

## Stock Market Models – Most Popular?

Created by Sal Khan.

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### 50 Comments

1. ##### Elizabeta Eremieva
July 30, 2021 at 2:12 pm

Can you derive mathematically d1? Thank you! Or do you know where to find a derivation of d1? I mean why is it equal d2 Plus stand. Deviation and squared time until expiration

2. ##### HOBerry _
July 30, 2021 at 2:12 pm

Please do the whole series😭
My professor don't teach me anything
I am all dependent on you😭

3. ##### Krishna Banik Pushpita
July 30, 2021 at 2:12 pm

Thank you so much. you are a awesome teacher.

4. ##### Tsebo mahao
July 30, 2021 at 2:12 pm

Excellent explanation

5. ##### fahad waliany
July 30, 2021 at 2:12 pm

Anyone know how to get volatility for non-public start-ups or is there a different valuation method to calculate stock-option expense for start-up?

6. ##### Eliza Zhang
July 30, 2021 at 2:12 pm

Thank you soooo much!! you explain it very clear and very easy to understand. Even if I do not have a strong math background, I can still understand it. which is very helpful for my CMA study. I was trying a long time to find a good explanation online until I found you !!

7. ##### narleira
July 30, 2021 at 2:12 pm

A phenomenal lesson I’ve been watching this lesson overtime in my studies, and as I revisited, Always understand a little more. I have immense respect for Khan Academy and the work you guys do, really help me a lot. My humble contribution to this, 's in time 9:09, when you said that an increase in sigma (historical volatility), makes D1 go up and D2 down in value, which it's not quite accurate, I'm certain it's because you speaking in rough terms to be easier to understand.

As my understanding goes, an increase in volatility (sigma) makes BOTH D1 and D2 increase in value, but the ratio of this increase in D1 is greater. Which makes the difference, between them, and their fore between N(D1) and N(D2) greater. And that's why an increase in volatility alone makes the premium of a call option, as the example, have an increase in value. More specifically Extrinsic value. The outcome in the premium it's the same, but I humbly believe that's this's more the case in that situation.

Sorry for the bad English, I'm a foreigner and I love the videos. You guys should do a deep math course on Black and Scholes, I'm certain it's gonna be great! Best Wishes!

8. ##### Wharlhey Nunes
July 30, 2021 at 2:12 pm

Is there any Brazilians here?

9. ##### 69thAndYorkAve
July 30, 2021 at 2:12 pm

Sal, you ROCK!!! So thankful for this explanation and overview. Really appreciate bringing Black-Scholes back to practical use. Probably the best explanation on the web. No, make that in the entire world!

10. ##### Teodor Andersen
July 30, 2021 at 2:12 pm

This is way to easy. Please fix

11. ##### Susan wolf
July 30, 2021 at 2:12 pm

I'm such an idiot. I thought it said the Back to School formula XD 🤦‍♀️ (low key clicked because of that) 🙃

12. ##### Renee Fong
July 30, 2021 at 2:12 pm

Hands off my uni lecturer is the worst, this simple 10 minutes saved my life

13. ##### George Hamilton
July 30, 2021 at 2:12 pm

Who is this guy? How does he know all this stuff?

14. ##### Frederick Farias
July 30, 2021 at 2:12 pm

I did this in college, 43 years ago with select stocks in S&P 500. The problem with it is that the distribution of risk is not representative to reality, it’s only an attempt or approximation. For example a standard dev. gives equal weights to up and downside.

15. ##### Sayali Patil
July 30, 2021 at 2:12 pm

Thank you so much for this. I have immense respect for Khan Academy and the work you guys do. It's quality and life-changing.

16. ##### Dedicated Motion
July 30, 2021 at 2:12 pm

Thank you!

I want to Algebraic Isolate the Stock Price from the Black Scholes formula.

I have not been successful yet. Is there a way?

17. ##### Frank Mathews
July 30, 2021 at 2:12 pm

great post

18. ##### Johnny Joseph
July 30, 2021 at 2:12 pm

That was the best nights sleep I've had in a long time! Thank you!!!

19. ##### Utkarsh Singh
July 30, 2021 at 2:12 pm

i thought beta was volatility

20. ##### Dan Chatka
July 30, 2021 at 2:12 pm

Sal Khan is a national treasure.

21. ##### Boubacar Bâ
July 30, 2021 at 2:12 pm

I guess it should also be mentioned that Merton and Scholes went on to create a hedge fund (LTCM) which almost melted down the economy and was bailed out by tax payer money to avoid a contagion to the overall economy that would have been in the order of trillions of USDs. The math might have been elegant, but risk pricing (the actual reason why they have received the nobel prize) was not really the thing these gentlemen were good at ironically.

22. ##### Preeti Saraf
July 30, 2021 at 2:12 pm

Very well explained. I finally understood the logic with the help of your video. Thank you for making it so simple.

23. ##### The Commercial Drone Solution Jamie Waller
July 30, 2021 at 2:12 pm

Can the N(d1) and the N(d2) be replaced with just the formulas for d1 and d2?

24. ##### DerpySeaTurtle
July 30, 2021 at 2:12 pm

I'm just curious, but is anyone a high school student like me just learning about options during there free time? lmao

25. ##### Qi KONG
July 30, 2021 at 2:12 pm

It's clearly explained! Thanks :).

26. ##### Kinga Bazan
July 30, 2021 at 2:12 pm

Great, thank you ! ! ! 🙂 🙂 🙂

27. ##### Brendan
July 30, 2021 at 2:12 pm

So simple and clear. Thanks Sal

28. ##### April Zhao
July 30, 2021 at 2:12 pm

Thank you so much! I'm preparing for my CFA level 2 and was stuck on the black-sholes model. You saved me so much time of head-scratching!

29. ##### Teeteu P
July 30, 2021 at 2:12 pm

Wow thank you I intuitively understand the formula now

30. ##### Abdulrahman Oubari
July 30, 2021 at 2:12 pm

How to calculate N(d1)? if we have for example d1=-0,22

31. ##### Gerardo Moscatelli
July 30, 2021 at 2:12 pm

d2 is wrong there

32. ##### verdadero amor
July 30, 2021 at 2:12 pm

This is by far the easiest way to explain the B&S formula ! Thank you so much for this instructive video !

33. ##### j shrinu
July 30, 2021 at 2:12 pm

Great explanation thank you so much

34. ##### Kanji Story
July 30, 2021 at 2:12 pm

S1 displays serial correlation

35. ##### Hugo Casal
July 30, 2021 at 2:12 pm

Thank you so much

36. ##### Tom_MKW
July 30, 2021 at 2:12 pm

is there also a formula like this for american options?

37. ##### D BCCRC
July 30, 2021 at 2:12 pm

There's no 'Nobel Prize' in economics. They did not win a Nobel price. They won a 'Nobel Memorial Prize'.

38. ##### XalX
July 30, 2021 at 2:12 pm

from where would i derive the risk free interest rate or the SD?

39. ##### Long Lucas
July 30, 2021 at 2:12 pm

WHy it has no sound?

40. ##### Wolfgang Icarus
July 30, 2021 at 2:12 pm

Intuitively why is it hat the higher the stock price the higher the call option is? I thought it'd be the opposite intuitively. Because at a lower stock price the room for increasing stock price is higher thus when you buy a call option, the lower the strike price is the more the potential earnings will be. Can someone explain to me please!!!

41. ##### Mirza Adil
July 30, 2021 at 2:12 pm

Lovely

42. ##### Robbie Kalshaw
July 30, 2021 at 2:12 pm

My god, thank you for existing. I love you. Thank you thank you for making finance fun!

43. ##### Mahesh Naik
July 30, 2021 at 2:12 pm

Great explanation!!!

44. ##### Kelly KitKat
July 30, 2021 at 2:12 pm

You do not know the value of a call option – its true value, could be zero – until that date arrives, and the price of that share is revealed, whether it be, "in the money" or not. This formula is "voodoo economics".

45. ##### Kelly KitKat
July 30, 2021 at 2:12 pm

Value is a SUBJECTIVE notion. This is an attempt to get many to agree on the value of an option, and to treat it as its true value, price, or cost. That is dangerous. It is merely an educated guess, or guesstimate.

For accounting purposes , rather than use this formula, if a company issues call options to its employees it should purchase countervailing put options, and record that purchase as the cost of those call options. Simple, right? You wish to give X to somebody, purchase X first, for say C dollars, and record \$ C as the true cost of X.

That said, the more shares a company issues the less each share outstanding is worth – unless the company is issuing and selling those shares on the market to raise cash – in which case, value per share MIGHT be growing. But creating shares to give to employees, who pay not for them, tends to decrease value per share for all shareholders. The more persons have an equal share in something, the less each share is worth.

46. ##### Eunice K
July 30, 2021 at 2:12 pm

Thank you so much

47. ##### Bao Dang
July 30, 2021 at 2:12 pm

This guy teaches by day and trades FD’s by night.

48. ##### Enosh Subba
July 30, 2021 at 2:12 pm

👍

49. ##### Sukhwinder Pal
July 30, 2021 at 2:12 pm

Excellent!! Nothing will be able to substitute your way of explaining…. Keep it up, Thanks!!

50. ##### Implantica Media Library
July 30, 2021 at 2:12 pm

Hello, thank you for the video. I am struggling to understand the difference and definitions of d1 and d2. Any chance to explain please? Thanks!