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Introduction to the Black-Scholes formula | Finance & Capital Markets | Khan Academy

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Created by Sal Khan. Watch the next lesson: https://www.khanacademy.org/economics-finance-domain/core-finance/derivative-securities/black-scholes/v/implied-volatility?utm_source=YT&utm_medium=Desc&utm_campaign=financeandcapitalmarkets Missed the previous lesson? Watch here: https://www.khanacademy.org/economics-finance-domain/core-finance/derivative-securities/interest-rate-swaps-tut/v/interest-rate-swap-2?utm_source=YT&utm_medium=Desc&utm_campaign=financeandcapitalmarkets Finance and capital markets on Khan Academy: Interest is the basis of modern capital markets. Depending on whether you are lending or borrowing, it can be viewed as a return on an asset (lending) or the cost of capital (borrowing). This tutorial gives an introduction to this fundamental concept, including what it means to compound. It also gives a rule of thumb that might make it easy to do some rough interest calculations in your head. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. For free. For everyone. Forever. #YouCanLearnAnything Subscribe to Khan Academy’s Finance and Capital Markets channel: https://www.youtube.com/channel/UCQ1Rt02HirUvBK2D2-ZO_2g?sub_confirmation=1 Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy
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Text Comments (96)
Waikoh (8 days ago)
My god, thank you for existing. I love you. Thank you thank you for making finance fun!
Mahesh Naik (20 days ago)
Great explanation!!!
Kelly KitKat (28 days ago)
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".
Kelly KitKat (28 days ago)
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.
Eunice K (1 month ago)
Thank you so much
Bao Dang (2 months ago)
This guy teaches by day and trades FD’s by night.
Nidhi Shah (2 months ago)
Thank you much love & respect!
Enosh Subba (2 months ago)
Sukhwinder Pal (3 months ago)
Excellent!! Nothing will be able to substitute your way of explaining.... Keep it up, Thanks!!
Bricktop1006 (3 months ago)
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!
Max Bg (4 months ago)
eloteh (5 months ago)
"most famous equation in finance", lol. a certain CAPM may be a good rival.
mhaddadi (7 months ago)
Where is the Volatility?
Christian Prior (3 months ago)
mhaddadi At 8:22, the sigma.
Niall English (9 months ago)
Any chance you could put this in lay man's terms without the scribbles so it might make some sense?
Nick Voss (9 months ago)
The fact that they had to add the caveat that their formula only works if the option is a European option, there is a constant risk-free rate of return and constant volatility, and that trading must be continuous and costless proves that the math is useless in reality. The fact that the formula is based on a metaphor between stock prices going up and down in a market and physical particles' random motion in a vacuum proves both that the math is useless and the theory behind it is pseudoscience. .
Vilela One (9 months ago)
Where I can find the "future videos"?
UmTheMuse (3 months ago)
He's got his own website in the description. He and his crew teach all sorts of topics (history, math, computer science, etc).
21cabbage RS (9 months ago)
this saved me in my financial theory and practice exam
Jacob G (10 months ago)
so can i use this formula for American options or no?
SEO experimentations (1 year ago)
This is an ok formula except that it assumes a normal distribution which is scary for option writers in the event of a black swan
SEO experimentations (1 year ago)
RIP fischer black
Ethelya Vlogs (1 year ago)
This is just superb, I been tryin to find out about "trade mispricing transfer pricing" for a while now, and I think this has helped. Have you heard people talk about - Genubrey Mispriced Infiltration - (Have a quick look on google cant remember the place now ) ? Ive heard some incredible things about it and my friend got cool results with it.
Kirsten Sais (1 year ago)
Thank you.
TheBoredEngineer (1 year ago)
This video makes me take my subject Financial Engineering seriously. I mean, I just find it interesting on how this mumbo-jumbo of variables makes so much sense in going to more advanced financial modelling techniques.
Antoine Compagnie (1 year ago)
If d_1 and d_2 get larger, doesn't SN(d_1) and Xe^{-rT}(d_2) annulate themselves ? And what is the point therefore ?
Meme Team (1 year ago)
Not sure if you care after six months, but it's because of the square As the standard deviation gets larger, the difference in the two parts of the equation also gets larger.
Mia Chiu (1 year ago)
Thank you for saving me heaps time in studying!!!!
MolecularArchitect (1 year ago)
Honestly, I was about to click away, but then I stuck it out. Sal's just a genius at teaching.
Bastian Ehler (1 year ago)
Hello people, It's nice to have those videos here, but could you please PROVIDE A NORMAL, NEUTRAL ENGLISH and not this American DIRTY SLANG ??? The rest of the world is not keen on this kind of English. I don't understand a word.
Stephanie Susilo (1 year ago)
Thank you ! This helps me a lot. Keep up the good work :)
Sebastian Roest (1 year ago)
for d1 ...Would we not need to discount the X (exercise price) by e^-rT ?
Ashish Purohit (1 year ago)
Sebastian Roest it's market stock price, the strike price is in future
Oscar Kr (2 years ago)
I have been watching these kind of videos for 3 years now for my uni and i have to say, i found no one close to your broad range of videos and high quality standards. You made me pass a lot of courses. I sincerely hope you earn a lot of youtube money.
Kyyp3r (2 years ago)
I thought the third guy was going to be called Formula
Football Freaks (2 months ago)
Dude that's an epic comment! You deserve a Nobel prize for this!!
vermasean (6 months ago)
LOL'ing a year later! HA HA!
Ronnie Sokha (6 months ago)
V H (8 months ago)
haha stop it!
Alejandro Tovar (8 months ago)
Or model
Rabie HABTAH (2 years ago)
kindly be advised date pf expiration T is expressed in days or weeks or what? if I'm calculating C0, thank u
David Nightingale (2 years ago)
i would imaging you convert r to daily rate, and then t would be the number of days.
Vicki Bee (2 years ago)
Great. Another one? So there are Futures Traders and now Options Traders? I have a semi-friend who was a Futures Trader but he also calls himself a broker. No idea if that's the same as a Futures Trader. He worked at a place with Futures in its name. He has a fit if I tell online in a social media the exact place of where he worked. No idea why that is either but I don't think I'll ever understand what he does. I've been trying to for over 5 years now. He's done work in all parts of finance though. He was also a hedge funds manager for awhile. He started as a floor trader in the 80's.
Vicki Bee (2 years ago)
I call it a semi-friend bc someone like him never would have been friends with me (never would have met me) if I didn't know someone who died in the World Trade Center. He had a fund for victims' families. That's how I met him but I never used the fund. Eric wasn't a stockbroker. He was a recruiter for professional businesses. I have no idea what that means. Unfortunately I didn't ask and he didn't offer to explain what he does. He was too busy being excited about the Sphere that was in the World Trade Center (donated by someone in 1970) than he was about his actual work.
Vicki Bee (2 years ago)
According to doctor I used to live with that means they spent all their time, "every waking hour and to the exclusion of everything else," trying to win the Nobel prize. I told him he should do it. All his parents ever did was boast about how wonderful he was, how genius he was, how he was a genius when he was 6 years old. Going on and on about it every time we ever saw them, I didn't see why he shouldn't also get a Nobel prize to go with it. That's when he said you have to spend more time trying to win the prize than you do with anything else in your life.
Felicitas Schulze (2 years ago)
There is no Noble Price in Economics. This was never intended by Mr. Nobel, as economics is a science with hughly vested interests (c.f. for instance Eugene Fama & Efficient Market Hypothesis). It was "invented" from the Schwedish Central Bank as a meassure of losing its independence or better getting under political investigation - done by buying economics a near-natural science reputation. All to get there own vested interests being done and keep critics outside.
Vicki Bee (2 years ago)
Really? I thought Paul Krugman won a Nobel prize. My friend who's an economist used to like him until he started liking Piketty better. He spelled it for me. He almost died laughing when I pronounced it Skoles. He finds me frequently amusing.
Md.Mijanur Rahman (2 years ago)
Sigma squared simply means the variance. Excellent explanation
Nitesh Prabhu (1 month ago)
Roughly speaking
N. Werum (3 years ago)
Fantastic!!! with some real world introduction of the people how introduced this calculation, then some basic intuition and then a perfect break down to how it works (although the explanation is not very in depth, which is, at this point, absolutely ok!) Thank you very much!
Derek Yue (3 years ago)
Really appreciate with your effort in making this video. I could really get the meaning and applied to tests or exams. THX!
Víctor Cortés (3 years ago)
I like the "rough speaking", it made things more clear to me.
Tim Fox (1 year ago)
Talk rough to me
MarvelGirl100 (3 years ago)
Thank you for Posting. You have done what many teachers do not do today. That of breaking the down the equation. Describing the outcome of the variable in a clear way. Your speaking voice is excellent and engages the listener. Do you know how hard it is to find teachers that want to teach because they are passionate teaching. You have unraveled our FATHER OF PRICING Option"s Formula and have done him a great service.
wronski11 (3 years ago)
Shall it be possible for you to recommend a book on the topic.
Tyko Brian (2 years ago)
+Alexandru Dumitras writer is Hull
Alexandru Dumitras (3 years ago)
+wronski11 Options, futures, and other derivatives :)
Gregorezzi (3 years ago)
Edward Thorp was originally the pioneer of this formula but hadn't publicise it straight away
GD (1 year ago)
That was his mistake hiding it from others. He made tons of money using the formula but missed out on Nobel prize. His choice.
Terrence Alexander (3 years ago)
Honorable mention to Louis Bachelier who did the preliminary research that led to the Black Scholes Formula.
Van Trinh (3 years ago)
Thank you so much for the video. I have a question, can you tell where can we get the FX option prices history. I doing my thesis and Black Scholes is in my theoretical background, except I will actually use Garman Kohlhagen model, which is basically Black Scholes but is used for FX options. 
Zoonova.com (4 years ago)
This is a great video, but if you want hands on pricing Options using the Black Scholes model, and other option models, check out Zoonova.com. Freemium financial web app.
Rich Yoe (4 years ago)
Fantastic video!
marryson123 (4 years ago)
What about american options?
Brian Dolan (3 months ago)
It applies to american options too, you just need to adjust the time variable.
Faraaz Farooqi (3 months ago)
One of BSM's many assumptions is that it does not apply to American options
Martin Pinto (4 years ago)
Hi Khan, where can I find your video for the explanation of the log returns?
mootaz shukair (4 years ago)
Great Video!!!!!
Dung Luu (4 years ago)
Thank you. It's interesting and much more easier to understand/remember than textbook :))
Kevin Parsley (4 years ago)
Didn't Edward Thorp really develop this?  
Humberto Arroyo (4 years ago)
I can use this in forex
HuffyTheMan09 (3 years ago)
+Humberto Arroyo no you can't lmao
Abdullah Ibrahim (5 years ago)
Great video as usual sal! Can you by any chance make videos in preparation for the CFA?
TurtleSlayer94 (5 years ago)
Great video! I'm studying for an exam and your videos really are a lot clearer and easier to remember than the rambling of my professor or the 800 pages of examples in our books. Life saver!
Ran Duan (5 years ago)
Thanks! Very helpful! Looking forward to the derivation!
Jon Cederqvist (5 years ago)
There is no Nobel Prize in economics. There is a Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, but that's not issued by the Nobel foundation.
Joey Numbers (5 years ago)
I really enjoyed this video. Thanks.
Vicki Bee (5 years ago)
According to the doctor I used to live with, getting a Nobel prize in something means you have to spend every waking hour of your day for years on end studying your profession, which leaves little to no time for your family or pursuit of other social interests. I had asked him why he doesn't get a Nobel prize so the world would recognize his gr8 genius his parents never shut up telling to people. The sarcasm was lost on him, b/c he answered the question seriously.
ZnorfRat (5 years ago)
I remember Sal saying that he wants too but he has to get an intuition on the subject before he can teach it.
axe863 (5 years ago)
Black Scholes is extremely inferior 1)The process is actually multifractional (MPRE-multi-fractional with random exponent) multi-tempered stable motion 2).Even if the stochastic process is just a geometric Brownian motion, the nonzero quadratic variation of the aforementioned process coupled w/ non-zero proportional trading costs implies perfect dynamic hedging is infinitely costly. 3)There is endogenous price impact non-neutrality (endogenous reflexivity) 4). There is endogenous incompleteness.
Remembering 1992 (11 months ago)
@axe Feel better now?
lola fick (1 year ago)
U been reading George Soros' stuff m8?
Ming Huei Leong (5 years ago)
Thanks for the video! As an Actuarial Student, this video is a pretty accurate and informative explaination video about the B-S Formula. 1 thing I would like to ask though, why didn't you include the divident factor into the equation?
OmegaCraftable (5 years ago)
I've never done any finance or economics, nor seen any previous videos on the subject, I just watched it because it sounded interesting :p
Steven_Seagan _fan (5 years ago)
great video for economics and finance students like me!!! thanks, Sal!!!
David (5 years ago)
Options are my bread and butter, calls/puts/credit spreads, always wondered about this formula but never dared to try to understand it! Very interesting video, not for everyone (your younger audience) I suspect, thank you !
TY GY (5 months ago)
haha 26 yr old that was an insurance agent. :) #europeantrades are my bread and butter when i plummit my futures lol i learned the hard way.
omali leg (5 years ago)
I also think we need a number theory and proofing playlist.
wmackr (5 years ago)
Great video Sal! thank you
Carl Campbell (5 years ago)
I think we need a quantum mechanics playlist

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