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

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

Voiceover: We’re now gonna talk about probably the most famous
formula in all of finance, and that’s the Black-Scholes Formula, sometimes called the
Black-Scholes-Merton Formula, and it’s named after these gentlemen. This right over here is Fischer Black. This is Myron Scholes. They really laid the
foundation for what led to the Black-Scholes Model and
the Black-Scholes Formula and that’s why it has their name. This is Bob Merton, who really
took what Black-Scholes did and took it to another level to really get to our
modern interpretations of the Black-Scholes Model
and the Black-Scholes Formula. All three of these
gentlemen would have won the Nobel Prize in Economics, except for the unfortunate fact that Fischer Black passed away
before the award was given, but Myron Scholes and Bob Merton did get the Nobel Prize for their work. The reason why this is such a big deal, why it is Nobel Prize worthy, and, actually, there’s many reasons. I could do a whole
series of videos on that, is that people have been
trading stock options, or they’ve been trading options
for a very, very, very long time. They had been trading them, they had been buying them, they had been selling them. It was a major part of
financial markets already, but there was no really good way of putting our mathematical minds around how to value an option. People had a sense of the
things that they cared about, and I would assume
especially options traders had a sense of the things
that they cared about when they were trading options, but we really didn’t have an
analytical framework for it, and that’s what the
Black-Scholes Formula gave us. Let’s just, before we dive into
this seemingly hairy formula, but the more we talk about it, hopefully it’ll start
to seem a lot friendlier than it looks right now. Let’s start to get an intuition for the things that we would care about if we were thinking about
the price of a stock option. You would care about the stock price. You would care about the exercise price. You would especially care
about how much higher or lower the stock price is relative
to the exercise price. You would care about the
risk-free interest rate. The risk-free interest
rate keeps showing up when we think about taking a
present value of something, If we want to discount the value
of something back to today. You would, of course, think
about how long do I have to actually exercise this option? Finally, this might look a
little bit bizarre at first, but we’ll talk about it in a second. You would care about how
volatile that stock is, and we measure volatility
as a standard deviation of log returns for that security. That seems very fancy, and we’ll talk about that in
more depth in future videos, but at just an intuitive level, just think about 2 stocks. So let’s say that this is
stock 1 right over here, and it jumps around, and I’ll make them go flat, just so we make no judgment about whether it’s a good investment. You have one stock that kind of does that, and then you have another stock. Actually, I’ll draw them on the same, so let’s say that is stock 1, and then you have a
stock 2 that does this, it jumps around all over the place. So this green one right
over here is stock 2. You could imagine stock 2 just in the way we use the word
‘volatile’ is more volatile. It’s a wilder ride. Also, if you were looking at
how dispersed the returns are away from their mean, you see it has, the returns have more dispersion. It’ll have a higher standard deviation. So, stock 2 will have a higher volatility, or a higher standard deviation
of logarithmic returns, and in a future video, we’ll talk about why we care about log returns, Stock 1 would have a lower volatility, so you can imagine,
options are more valuable when you’re dealing with, or if you’re dealing with a
stock that has higher volatility, that has higher sigma like this, this feels like it would drive
the value of an option up. You would rather have an option when you have something like this, because, look, if you’re owning the stock, man, you have to go after,
this is a wild ride, but if you have the option,
you could ignore the wildness, and then it might actually make, and then you could exercise the option if it seems like the right time to do it. So it feels like, if you
were just trading it, that the more volatile something is, the more valuable an
option would be on that. Now that we’ve talked about this, let’s actually look at
the Black-Scholes Formula. The variety that I have right over here, this is for a European call option. We could do something very
similar for a European put option, so this is right over here
is a European call option, and remember, European call option, it’s mathematically simpler
than an American call option in that there’s only one time
at which you can exercise it on the exercise date. On an American call option, you can exercise it an any point. With that said, let’s try to
at least intuitively dissect the Black-Scholes Formula a little bit. So the first thing you have here, you have this term that involved
the current stock price, and then you’re multiplying
it times this function that’s taking this as an input, and this as how we define that input, and then you have minus the exercise price discounted back, this discounts
back the exercise price, times that function again, and now that input is slightly different into that function. Just so that we have a
little bit of background about what this function N is, N is the cumulative distribution function for a standard, normal distribution. I know that seems, might
seem a little bit daunting, but you can look at the
statistics playlist, and it shouldn’t be that bad. This is essentially saying for
a standard, normal distribution, the probability that your
random variable is less than or equal to x, and another way of thinking about that, if that sounds a little, and it’s all explained in
our statistics play list if that was confusing, but if you want to think about
it a little bit mathematically, you also know that this is going to be, it’s a probability. It’s always going to be greater than zero, and it is going to be less than one. With that out of the way, let’s think about what
these pieces are telling us. This, right over here, is dealing with, it’s
the current stock price, and it’s being weighted by
some type of a probability, and so this is, essentially,
one way of thinking about it, in very rough terms, is this
is what you’re gonna get. You’re gonna get the stock, and it’s kind of being
weighted by the probability that you’re actually
going to do this thing, and I’m speaking in very rough terms, and then this term right
over here is what you pay. This is what you pay. This is your exercise
price discounted back, somewhat being weighted, and I’m speaking, once again, I’m hand-weaving a lot of the mathematics, by like are we actually
going to do this thing? Are we actually going
to exercise our option? That makes sense right over there, and it makes sense if the
stock price is worth a lot more than the exercise price, and if we’re definitely going to do this, let’s say that D1 and D2 are
very, very large numbers, we’re definitely going to do
this at some point in time, that it makes sense that
the value of the call option would be the value of the
stock minus the exercise price discounted back to today. This right over here,
this is the discounting, kind of giving us the present
value of the exercise price. We have videos on discounting
and present value, if you find that a little bit daunting. It also makes sense that the more, the higher the stock price is, so we see that right over here, relative to the exercise price, the more that the option would be worth, it also makes sense that
the higher the stock price relative to the exercise price, the more likely that we will
actually exercise the option. You see that in both of
these terms right over here. You have the ratio of the stock
price to the exercise price. A ratio of the stock price
to the exercise price. We’re taking a natural log of it, but the higher this ratio
is, the larger D1 or D2 is, so that means the larger the input into the cumulative
distribution function is, which means the higher
probabilities we’re gonna get, and so it’s a higher chance
we’re gonna exercise this price, and it makes sense that then this is actually going to have some value. So that makes sense, the relationship between the stock price and the exercise price. The other thing I will focus on, because this tends to be a deep focus of people who operate with options, is the volatility. We already had an intuition, that the higher the volatility, the higher the option price, so let’s see where this factors
into this equation, here. We don’t see it at this first level, but it definitely factors into D1 and D2. In D1, the higher your standard
deviation of your log returns, so the higher sigma, we have a sigma in the
numerator and the denominator, but in the numerator, we’re squaring it. So a higher sigma will make D1 go up, so sigma goes up, D1 will go up. Let’s think about what’s happening here. Well, here we have a sigma. It’s still squared. It’s in the numerator, but we’re subtracting it. This is going to grow faster than this, but we’re subtracting it now, so for D2, a higher sigma
is going to make D2 go down because we are subtracting it. This will actually make, can we actually say this is going to make, a higher sigma’s going to make the value of our call option higher. Well, let’s look at it. If the value of our sigma goes up, then D1 will go up, then this input, this input goes up. If that input goes up, our cumulative distribution
function of that input is going to go up, and so this term, this whole term is gonna
drive this whole term up. Now, what’s going to happen here. Well, if D2 goes down, then our cumulative distribution
function evaluated there is going to go down, and so this whole thing
is going to be lower and so we’re going to have to pay less. If we get more and pay less, and I’m speaking in very hand-wavy terms, but this is just to understand that this is as intuitively
daunting as you might think, but it looks definitively, that if the standard deviation, if the standard deviation
of our log returns or if our volatility goes up, the value of our call option, the value of our European
call option goes up. Likewise, using the same logic, if our volatility were to be lower, then the value of our
call option would go down. I’ll leave you there. In future videos, we’ll think about this in a little bit more depth.


  • Carl Campbell

    July 29, 2013

    I think we need a quantum mechanics playlist

  • wmackr

    July 29, 2013

    Great video Sal! thank you

  • omali leg

    July 29, 2013

    I also think we need a number theory and proofing playlist.

  • David

    July 29, 2013

    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 !

  • Steven_Seagan _fan

    July 29, 2013

    great video for economics and finance students like me!!! thanks, Sal!!!

  • OmegaCraftable

    July 29, 2013

    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

  • Ming Huei Leong

    July 29, 2013

    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?

  • axe863

    July 29, 2013

    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.

  • ZnorfRat

    July 30, 2013

    I remember Sal saying that he wants too but he has to get an intuition on the subject before he can teach it.

  • Vicki Bee

    July 30, 2013

    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.

  • Joey Numbers

    July 30, 2013

    I really enjoyed this video. Thanks.

  • Jon Cederqvist

    August 1, 2013

    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.

  • Ran Duan

    October 2, 2013

    Thanks! Very helpful! Looking forward to the derivation!

  • TurtleSlayer94

    December 12, 2013

    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!

  • Abdullah Ibrahim

    December 18, 2013

    Great video as usual sal! Can you by any chance make videos in preparation for the CFA?

  • Humberto Arroyo

    February 15, 2014

    I can use this in forex

  • Kevin Parsley

    February 24, 2014

    Didn't Edward Thorp really develop this?  

  • Dung Luu

    March 6, 2014

    Thank you. It's interesting and much more easier to understand/remember than textbook :))

  • mootaz shukair

    May 19, 2014

    Great Video!!!!!

  • Martin Pinto

    July 28, 2014

    Hi Khan, where can I find your video for the explanation of the log returns?

  • marryson123

    August 5, 2014

    What about american options?

  • Rich Yoe

    October 25, 2014

    Fantastic video!


    October 29, 2014

    This is a great video, but if you want hands on pricing Options using the Black Scholes model, and other option models, check out Freemium financial web app.

  • Van Trinh

    March 18, 2015

    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. 

  • Terrence Alexander

    May 22, 2015

    Honorable mention to Louis Bachelier who did the preliminary research that led to the Black Scholes Formula.

  • Craneformers

    June 3, 2015

    Edward Thorp was originally the pioneer of this formula but hadn't publicise it straight away

  • Jim

    August 17, 2015

    Shall it be possible for you to recommend a book on the topic.

  • MarvelGirl100

    August 31, 2015

    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.

  • Víctor Cortés

    November 14, 2015

    I like the "rough speaking", it made things more clear to me.

  • Derek Yue

    December 2, 2015

    Really appreciate with your effort in making this video. I could really get the meaning and applied to tests or exams. THX!

  • N. Werum

    January 13, 2016

    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!

  • Md.Mijanur Rahman

    June 2, 2016

    Sigma squared simply means the variance. Excellent explanation

  • Felicitas Schulze

    July 1, 2016

    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

    September 16, 2016

    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.

  • Vicki Bee

    September 16, 2016

    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.

  • Rabie HABTAH

    November 23, 2016

    kindly be advised date pf expiration T is expressed in days or weeks or what? if I'm calculating C0, thank u

  • Kyyp3r

    December 6, 2016

    I thought the third guy was going to be called Formula

  • Oscar Kr

    January 12, 2017

    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.

  • Sebastian Roest

    February 5, 2017

    for d1 …Would we not need to discount the X (exercise price) by e^-rT ?

  • Stephanie Susilo

    March 30, 2017

    Thank you ! This helps me a lot. Keep up the good work 🙂

  • Bastian Ehler

    May 5, 2017

    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.

  • MolecularArchitect

    May 30, 2017

    Honestly, I was about to click away, but then I stuck it out. Sal's just a genius at teaching.

  • Pin-Chi Chiu

    June 18, 2017

    Thank you for saving me heaps time in studying!!!!

  • Antoine Compagnie

    July 13, 2017

    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 ?

  • TheBoredEngineer

    September 14, 2017

    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.

  • Kirsten Sais

    November 27, 2017

    Thank you.

  • Ethel Suarez

    November 28, 2017

    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.

  • SEO experimentations

    December 10, 2017

    RIP fischer black

  • SEO experimentations

    December 10, 2017

    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

  • Jacob G

    March 12, 2018

    so can i use this formula for American options or no?

  • 21cabbage RS

    April 1, 2018

    this saved me in my financial theory and practice exam

  • Vilela One

    April 4, 2018

    Where I can find the "future videos"?

  • Nick Voss

    April 10, 2018

    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. .

  • Niall English

    April 14, 2018

    Any chance you could put this in lay man's terms without the scribbles so it might make some sense?

  • mhaddadi

    June 4, 2018

    Where is the Volatility?

  • eloteh

    August 1, 2018

    "most famous equation in finance", lol. a certain CAPM may be a good rival.

  • Max Bg

    August 29, 2018


  • IM Film Library

    October 5, 2018

    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!

  • Sukhwinder Pal

    October 14, 2018

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

  • Enosh Subba

    October 17, 2018


  • Bao Dang

    October 24, 2018

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

  • Eunice K

    November 30, 2018

    Thank you so much

  • Kelly KitKat

    December 19, 2018

    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.

  • Kelly KitKat

    December 19, 2018

    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".

  • Mahesh Naik

    December 26, 2018

    Great explanation!!!

  • Risk Assessment

    January 8, 2019

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

  • caesar augustus

    January 26, 2019


  • Wolfgang Icarus

    February 3, 2019

    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!!!

  • Long Sagad

    March 4, 2019

    WHy it has no sound?

  • XalX

    April 4, 2019

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


    April 5, 2019

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

  • tomschrauwen

    April 12, 2019

    is there also a formula like this for american options?

  • Hugo Casal

    April 22, 2019

    Thank you so much

  • Quantum Curiosity

    April 29, 2019

    S1 displays serial correlation

  • j shrinu

    September 25, 2019

    Great explanation thank you so much


    October 20, 2019

    great video!


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