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