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## Black-Scholes formula

Current time:0:00Total duration:5:00

# Implied volatility

## 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.