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, 2013I think we need a quantum mechanics playlist

## wmackr

July 29, 2013Great video Sal! thank you

## omali leg

July 29, 2013I also think we need a number theory and proofing playlist.

## David

July 29, 2013Options 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, 2013great video for economics and finance students like me!!! thanks, Sal!!!

## OmegaCraftable

July 29, 2013I'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, 2013Thanks 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, 2013Black 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, 2013I 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, 2013According 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, 2013I really enjoyed this video. Thanks.

## Jon Cederqvist

August 1, 2013There 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, 2013Thanks! Very helpful! Looking forward to the derivation!

## TurtleSlayer94

December 12, 2013Great 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, 2013Great video as usual sal! Can you by any chance make videos in preparation for the CFA?

## Humberto Arroyo

February 15, 2014I can use this in forex

## Kevin Parsley

February 24, 2014Didn't Edward Thorp really develop this?

## Dung Luu

March 6, 2014Thank you. It's interesting and much more easier to understand/remember than textbook :))

## mootaz shukair

May 19, 2014Great Video!!!!!

## Martin Pinto

July 28, 2014Hi Khan, where can I find your video for the explanation of the log returns?

## marryson123

August 5, 2014What about american options?

## Rich Yoe

October 25, 2014Fantastic video!

## Zoonova.com

October 29, 2014This 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.

## Van Trinh

March 18, 2015Thank 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, 2015Honorable mention to Louis Bachelier who did the preliminary research that led to the Black Scholes Formula.

## Craneformers

June 3, 2015Edward Thorp was originally the pioneer of this formula but hadn't publicise it straight away

## Jim

August 17, 2015Shall it be possible for you to recommend a book on the topic.

## MarvelGirl100

August 31, 2015Thank 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, 2015I like the "rough speaking", it made things more clear to me.

## Derek Yue

December 2, 2015Really 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, 2016Fantastic!!! 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, 2016Sigma squared simply means the variance. Excellent explanation

## Felicitas Schulze

July 1, 2016There 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, 2016According 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, 2016Great. 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, 2016kindly be advised date pf expiration T is expressed in days or weeks or what? if I'm calculating C0, thank u

## Kyyp3r

December 6, 2016I thought the third guy was going to be called Formula

## Oscar Kr

January 12, 2017I 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, 2017for d1 …Would we not need to discount the X (exercise price) by e^-rT ?

## Stephanie Susilo

March 30, 2017Thank you ! This helps me a lot. Keep up the good work 🙂

## Bastian Ehler

May 5, 2017Hello 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, 2017Honestly, I was about to click away, but then I stuck it out. Sal's just a genius at teaching.

## Pin-Chi Chiu

June 18, 2017Thank you for saving me heaps time in studying!!!!

## Antoine Compagnie

July 13, 2017If 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, 2017This 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, 2017Thank you.

## Ethel Suarez

November 28, 2017This 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, 2017RIP fischer black

## SEO experimentations

December 10, 2017This 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, 2018so can i use this formula for American options or no?

## 21cabbage RS

April 1, 2018this saved me in my financial theory and practice exam

## Vilela One

April 4, 2018Where I can find the "future videos"?

## Nick Voss

April 10, 2018The 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, 2018Any chance you could put this in lay man's terms without the scribbles so it might make some sense?

## mhaddadi

June 4, 2018Where 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, 2018Thanks.

## IM Film Library

October 5, 2018Hello, 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, 2018Excellent!! Nothing will be able to substitute your way of explaining…. Keep it up, Thanks!!

## Enosh Subba

October 17, 2018👍

## Bao Dang

October 24, 2018This guy teaches by day and trades FD’s by night.

## Eunice K

November 30, 2018Thank you so much

## Kelly KitKat

December 19, 2018Value 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, 2018You 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, 2018Great explanation!!!

## Risk Assessment

January 8, 2019My god, thank you for existing. I love you. Thank you thank you for making finance fun!

## caesar augustus

January 26, 2019Lovely

## Wolfgang Icarus

February 3, 2019Intuitively 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, 2019WHy it has no sound?

## XalX

April 4, 2019from where would i derive the risk free interest rate or the SD?

## D BCCRC

April 5, 2019There's no 'Nobel Prize' in economics. They did not win a Nobel price. They won a 'Nobel Memorial Prize'.

## tomschrauwen

April 12, 2019is there also a formula like this for american options?

## Hugo Casal

April 22, 2019Thank you so much

## Quantum Curiosity

April 29, 2019S1 displays serial correlation

## j shrinu

September 25, 2019Great explanation thank you so much

## USWMO

October 20, 2019great video!