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## Finance and capital markets

### Unit 9: Lesson 7

Black-Scholes formula# Implied volatility

Created by Sal Khan.

## Want to join the conversation?

- What is meant by a "log return" ?(12 votes)
- A log return is another way of describing when interest is continuously compounded. It is a measure of exponential growth.

For example, if something grows at 5% continuously compounded, it will grow at an exponential rate because each additional 5% will be bigger than the previous 5%.(38 votes)

- I'm looking for information on the Newton / Raphson method. I entered those words into your search field and was led to this video. And although the video is interesting it seems to have little relation to my search string.(12 votes)
- If the market is anticipating a possible crash, then Puts should be priced higher than Calls. The implied volatility from looking at Puts would therefore be higher than the implied volatility from looking at Calls. How are these 2 different volatility values reconciled, given that the volatility variable is a single scalar value, and is directionless?(5 votes)
- The implied volatility is the level of ”sigma” replaced into the BS formula that will give you the lowest difference between the market price (that you already know) of the option and the price calculated in the BS model. The thing is, that the implied volatility shoud be calculated with the newton-raphson algoritm, in a more difficult way. If you go backwards in the BSF, you will only get the real volatility from the market (of that moment you calculated it). So, the point is, the implied volatility is a theoretical level of the volatility that, replaced into the BSF will give you the most reality-like price. Please, tell me if I am wrong. Sorry for my bad english.(7 votes)

- so is sal saying here that the B.S.F isn't the only way options are priced? I went into this thinking it was a formula for pricing options, based on using historical volatility. But now he's saying you can figure out volatility based on options prices! So which came first...the price of the option (using this formula) or the volatility?

Are options originally priced using this formula, and then subjected to market forces?(5 votes)- People forecast voltatilty hence you can see what the market is implying by reverse engineering BS equation. Analyst will all have there own idea of stock forecast and its volatility - these assumptions are in the call price. That's my understanding. So in a way you can see what the market is implying and you can also come up with your own forecasts.(4 votes)

- Sal,

First of all, thank you... you provide great instruction. My question comes into play around3:25. If we know S(o), X, r, T, AND C(o), then we can obviously determine sigma, as you indicated. My question is this: if the value of the option C(o) is constantly changing based on market transactions, then why is sigma, which is an estimate, even included in the formula? Why does it matter? Sure, we can determine an implied volatility and apply it across various markets, but why does that matter when we already know the market value of all securities?(1 vote)- The option formula is trying to use a statistical approach to figure out the likelihood that the market price will go above the exercise price. Imagine a call option that is out of the money, Let's say the stock is at 15 and the strike price is 20. If the stock normally moves up and down from day to day by 1, then you are not going to be very excited about the chance that the option will finish in the money (ie with the stock above 20). But let's suppose that the stock price is very volatile. It still averages 15, but some days it's 25 and some days it's 5. Isn't the option worth more in that second situation? A lot more, right?(7 votes)

- Is there a formula that can be used to determine the implied volatility, or can it only be estimated using an iterative approach?(1 vote)
- What Sal is saying is that, if we have the actual market price of the option, we can then use Black Scholes to calculate the value of implied volatility. So the value of implied volatility for a security is constantly being determined by market forces. If we do not have the market price of the option then we have to estimate volatility using the standard deviation.(1 vote)

- options use days to expiry rather than years, so how would time be enter in these conditions.(1 vote)
- Time is often just a percentage of a year. For example, 61 days would be 61/365.

Sometimes a 360 day year is used in finance to split everything up equally (30 day months, 90 day quarters etc).(1 vote)

- secondly can you use 10 yr treasury as the risk free rate..(1 vote)
- Yes. Often times people will use a treasury security that corresponds to the time frame of their investment and/or the time frame of the data they are using in their calculations.(1 vote)

- Will there be more videos on the Black-Scholes Formula coming soon?(1 vote)
- Something's not quite making sense to me.

I've been trying to look for a good explanation of implied vol but still can't find it.

The Call REQUIRES five parameters: S(0), X, T, R and Sigma. (ASSUMPTION)

Of these five, Sigma is hard to find, so we get the value of C and "reverse engineer" to fnd Sigma.

Hold on!

In the line I wrote above, labelled ASSUMPTION, to find C, we REQUIRE all 5 parameters, INCLUDING Sigma.

So if we use C to "reverse engineer" to find Sigma, that must mean we already had Sigma in the first place (or else, how did we manage to find C)?

Furthermore, we also said we need all 5 parameters to find C but if we somehow have found C, why do we care about "reverse engineering" to find Sigma?(1 vote)- To find implied volatility, you need C.

Options are not priced using implied volatility. That's why it's called "implied".

But sometimes people find it interesting to see what volatility you would need to have in order to explain the current price with the B-S model.(1 vote)

## Video transcript

Voiceover: In the last video,
we already got an overview that if you give me a stock price, and an exercise price, and a risk-free interest rate, and a time to expiration and the volatility or
the standard deviation of the log returns, if you give me these six things, I can put these into the
Black-Scholes Formula, so Black-Scholes Formula, and I will output for
you the appropriate price for this European call option. So it sounds all very straightforward, and some of this is straightforward. The stock price is easy to look up. The exercise price, well, that's part of the contract. You know that. The risk-free interest rate, there are good proxies for it, money market funds, there's government debt, things like that, so that's pretty easy to figure out, or at least approximate. The time to expiration, well you know that, you know what today's date is. You know when this
thing's going to expire, so that's pretty straightforward. Now let's think a little
bit about volatility, so how do you actually
measure the standard deviation of log returns. Now, one of the assumption
about Black-Scholes Formula is that this is a constant thing. This is just some intrinsic
propety of this security. Well, the only way that
you can at least attempt to estimate it is by
looking at the history of the standard deviation of log returns. The way that people
would normally do it is they'll say, "Okay, what
has historically been "the standard deviation of log
returns over some time period "where that security has not
changed in some dramatic way?" And then use that as the input, and then they would
come up with some price. Well, that's all interesting,
but it's very important to know that this is an estimate. This right over here is an estimate. There's no way of us actually
knowing the actual intrinsic and it's even up for
grabs whether there is, whether you can even as assume that there's some constant
intrinsic property as this volatility that's
going to be constant over the life of this option. So this is just an estimate.
It's important to know that. But what is interesting is that
these things are being traded. These call options are being
traded all of the time, and so you could actually look
up the price of this call option. You could look up a call
option with this stock price, this exercise price. You know what these two things are, and you could say, "Hey, look.
This traded for $3 just now." So you actually can
figure out what this is, which raises a very, very
interesting question. If you know exactly what this is because you know what the
market is pricing this at, so let me write this. You know what the market
believes this price should be, so the market belief, and it's based on their
actual transcations, so it's based on transactions. This is what the market is
saying the correct price is. You can figure that out,
you can just look that up. You can figure out all
of this other stuff. Can you then take this
output plus all of these to work backwards through
Black-Scholes to figure out what the market is guessing about this, or what the market is
estimating about that. The answer is yes. This is where this whole idea about when people talk
about what is the volatility in the market, or even where
are carton volatility rates, or even what does the market
expect volatility to be? How do we know what the market
expects volatility to be? Well, we can look at what
markets are trading options at. We could look at all of
this other information that would be inputted into
Black-Scholes equation, and we can say, "Hey,
look. Based on the fact "that the market is
paying $5 for this option, "and all of these other variables, "they must assume that the
standard deviation of log returns "for this security is now this." Now, let's say that
things get really scary. The market becomes a
lot dicier and choppier. Well then, people are gonna
pay more for this option. It's gonna drive the
implied volatility up. So when you hear people talk
about implied volatility, or implied vol, and there are even people who will actually trade
on implied volatility, This is what they're talking about. They're saying, "Look. Options
are trading all the time." Can we use that price, the market belief of what those prices should be, and then work backwards
through Black-Scholes to figure out, because we
know these are all facts. We can look these things up, but based on what the market
is trading these options at, can we figure out what
the implied volatility, what the implied market
belief about volatility for that security is, and then we can actually aggregate it across many, many securities, and come up with an implied volatility for given markets at a time. So it's a very, very,
very interesting idea, but in some levels, it's
kind of a basic one. You're just working backwards
through Black-Scholes.