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In the last couple of modules we introduced the volatility surface.

We saw how to construct the volatility surface, and we also discussed the skew

and why we might see a skew in practice. In this module we're going to discuss

what the volatility surface tells us. We will see that the volatitlity surface

gives us the marginal risk neutral distributions of the stock price.

It does not tell us anything about the joint risk neutral distributions of the

stock price at various times. So that is the key part of the volatility

surface. It is very important to appreciate it.

It only tells us the marginal risk neutral distributions of the stock price

at a given fixed time. It tells us nothing about the joint risk

mutual distributions. And we will emphasize that in this module

and indeed in later modules. So recall again this is an example of

implied volatility surface and just remind yourselves again to make sure we

don't forget. It is constructed as follows, we see a

set of strike expiration pairs in the market place.

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So we have k1 t1 up to k n t n. We see the option prices in the

marketplace for all of these. So, we actually see the caller put price,

let's say call price c subscript mkt for market of kiti, and that's true for i

equals 1 to n. So, we see these prices in the

marketplace, and what we do is, we set these prices equal to the Black-Scholes

price with S, r, K, T, C, and sigma of kt.

So we set this market price equal to the Black Shoal's price.

We know the left hand inside, we know the Black Shoal's formula, we know s r kt.

We can estimate C, and so there's just one unknown in this equation, and we can

actually back out of this unknown for sigma k i t i.

And that will give us the implied volatility at the strike k i and

expiration t i. So that will give us a number of points

on our surface here. And then we fill in the rest of the

surface using some sort of interpolation or extrapolation procedure.

I didn't really discuss how we would do this interpretation or extrapolation, but

one has to be careful when doing it. So we're going to continue to assume that

the volatility surface has been constructed from European option prices.

We're going to discuss now what the volatility surface tells us and what we

can use it for. Certainly we can use it for risk

management purposes. I've mentioned that already.

We can do scenario analysis. We can actually stress the volatility

surface by moving it up or down, or moving parts of the volatility surface up

or down. Recomputing the value of a portfolio,

computing the pnl, and so on. So, the volatility surface is certainly

used for risk management purposes. What we're going to discuss in the next

couple of slides, is what can it tell us in terms of being able to price

derivative securities beyond call and put options.

So, to answer this question, let's first of all consider, a butterfly strategy.

Now a butterfly strategy centered at k, does the following.

It buys a call option with strike k minus Delta K.

It buys a call option with strike k plus Delta K.

And then it sells two call options with strike k.

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The value of the butterfly, B0, we'll say, at time t equals 0, is therefore

given to us by this expression here. Where C is the call option price at the

strike K minus Delta K, and maturity T, and so on.

And in fact, in practice what we'll be doing is, doing this, using the market

prices, so if you like you can assume that these are market prices.

MKT, being shorthand for market. let's also see what the payoff of this

butterfly strategy is at maturity. So maturity is capital T.

Let's get an idea of what this looks like at maturity.

So, let's draw a plot. So we will call this the payoff, and we

will call it B capital T, for the payoff of the butterfly strategy at maturity.

And along the x axis we will have the underlying security price, which is st,

at maturity capital T. And, let's, mark off k.

k minus Delta K. And k, plus Delta K.

Well, it's pretty straight forward, to see, that this strategy earns nothing, if

the stock price at time capital T is less than k minus Delta K.

It also earns nothing if the stock price at time capital t is greater than k plus

k Delta K. Moreover, is easy to see that the maximum

payoff of this strategy is equal to k and it occurs if the stock price itself at

maturity is equal to k. And it grows linearly for values of stp

low k and then it decreases down to zero, at k plus Delta K.

So, in fact this is the payoff of the butterfly strategy at maturity as a

function of st. Now, a couple of things to keep in mind,

the maximum payoff is k. And, if you like, if you're inside this

interval where you do get a payoff at time capital t, the average payoff will

be k over 2. So the average payoff, if you're paid

off, will be k over 2. Alright another thing to keep in mind, we

know from risk mutual pricing that the fair value of this payoff.

This is a payoff at maturity. So the fair value of this payoff at

maturity is the current value of the butterfly today which is B0.

We know B0 is equal to the right inside of 6.

But from risk mutual pricing this is also equal to the expected value of time 0.

Using risk mutual probabilities e to the minus r times t times the payoff.

And the payoff we will call B capital T. Now I know I've used b in the past to

refer to the cash account. Here it's referring to the butterfly

payoff here, and this is our butterfly payoff.

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So keeping this in mind we're going to get an alternative expression for b0.

We have one expression for B0 here in equation 6.

On the next line we're going to get an alternative expression for B0 using this

representation here. So what we can say is that B0 is equal to

e to minus rt or rather is. It is approximately equal to e to the

minus rt. Times the risk neutral probability of st

being between k minus Delta K, and k plus Delta K times Delta K over 2.

Now, where does that came from? Well, if you think about it, it comes

from this idea here. So the payoff occurs if st is in k minus

Delta K. Up to k plus Delta K.

So the risk neutral probability of that is q of k minus Delta K being less than

or equal to st, being less than or equal k plus Delta K.

So, that's the probability that s t is inside this interval here.

Now, were imagining Delta K being small, by the way.

In fact, soon we're going to let Delta K go to zero.

So we can imagine Delta K is very small. So, this is the probability, the risk

neutral probability that st is inside this interval here.

We already explained that if s t is in this interval then the pay off you expect

to get is k over 2. And indeed that is why we multiply by the

k over 2 here. So we have our e to minus r t term, the

probability that st is inside this interval, times the average payoff in

this interval. And so that's how we get this first line

here. It's an approximation, but it is a very

good approximation for small Delta K. Now, if you recall something about

density functions, then you will understand why we're letting q.

The risk neutral probability that st is in this interval is equal to the risk

neutral density times the width of the interval.

So we're saying the risk neutral probability that st is between k minus

Delta K and k plus Delta k. That is approximately equal to the

density, risk neutral density evaluated at k times the width of the interval to

Delta K. And that just follows from a property of

PDFs. We actually explain this in one of the

additional modules on, on probability that we also recorded, they're also

available on the, on the plat, course platform.

So, remember, if you've got a PDF, in general.

So if this is our PDF, f of x. And suppose we want to compute the

interval that the random variable x is inside x0 to x 0 plus Delta X.

Well, the density satisfies that the probability, that the random variable x

is in, x0 plus Delta x where Delta x small.

That's approximately equal to f of x 0 times the width of the interval which is

Delta X. This is a standard property of

probability density function and that's all we're using here.

So we're saying the portability that st is inside this interval here is equal to

the density which is ft, times the width of the interval which is 2 Delta K.

So therefore we're going to get B0 is approximately equal e to the minus rt.

Times ft of k, times Delta K squared. We have a two here, but that counts as

with a two there, and we get a Delta K times Delta K, which is delta k squared.

So what we've done now is we've come up with 2 expressions for the value of the

butterfly strategy. We have this expression here in equation

6 which is exact. I'm here with this expression here which

is an approximation. But as Delta K goes to zero, this

approximation also becomes exact. So, what we're going to do is, we're

going to equate equation 7 with equation 6 and then solve for ft of k.

Or in other words, bring ft of k over to the left hand side.

So we will see ft of k, is approximately equal to e to the rt, times this

expression on the right hand side of 6 here, divided by Delta K squared.

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If we now let Delta K go to 0 in 8. Well, if you recall your, your calculus,

you'll see that all you're doing when you do this is actually computing the second

partial derivative of the call option price with respect to the strike.

And so what we're seeing is that by constructing a butterfly strategy where

Delta K goes to 0. We're actually able to come up with a

risk neutral probability density function for st, f t is equal to e to the r t,

Delta 2 c, Delta k squared. And so.

The volatility surface gives us the marginal risk-neutral distribution of the

stock price, st, for any fixed time, t. So this is a really interesting

observation. We see option prices for finite number of

strikes and maturities. We compute those implied volatilities.

We then actually fit the volatility surface to those finite number of points.

I mentioned earlier that we need to fit the volatility surface very carefully.

And the reason is if we want to be able to compute something like f t of k, then

we're computing partial derivatives and second partial derivatives.

So we need to make sure we do things, we fit things very smoothly.

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This means that, given the implied volatility surface, sigma kt.

We can compute the price p0 of any derivative security whose payoff f only

depends on the underlying stock price of the single and fixed time capital T.

Now, maybe I've chosen f. Unfortunately here because I used f

subscript t. F subscript t to denote the risk neutral

PDF of st. Here, f, this f here has got nothing to

do with this. This is the risk neutral PDF of st.

F, here, is just some arbitrary function representing the payoff of some

derivative security. So just to be clear, the f that I'm using

here has nothing to do with the f subscript t on the previous slide.

Which was the risk neutral probability density function of st.

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However, knowing the volatility surface tells us nothing about the joint

distribution of the stock price at multiple times t1 up to tn.

And this is not surprising since the volatility surface is constructed from

European option prices. And European option prices only depend on

the marginal distributions of st. Just to be clear, the joint distribution

of the stock price at multiple times t1, tn, what I'm referring to there is the

following. It would be this distribution, t1, up to

tn, of s, t1, up to stn. So this is the risk neutral joint

distribution of the stock price at times t1 up to tn.

And what we're saying here is that we don't know anything about this joint

distribution. The only thing that we can learn from the

volatility surface is the marginal distribution for each individual time t1

up to tn. Here's an example.

Suppose we wish to compute the price of a knockout put option with time t payoff

given to us here. So, it's this piece here is like a

regular put option. So this is a like a regular European put

option, the maximum of k minus s, t, and 0.

However we only get that payoff if the minimum stock price over the interval of

0 to capital T is greater than or equal to B.

So B here represents a barrier. If the stock price ever falls below B,

then this indicator function here becomes 0, and so we get nothing.

Remember the indicator function, this indicator function, can take on two

possible values. It takes on the value 1, or 0.

It takes on the value 1, if the minimum of st is greater than or equal to B.

And that's the minimum over 0 less than or equal to little t, less than or equal

to capital T and it takes on the value 0 otherwise.

So this is an example of a knockout put option.

And the point I'm trying to make here is that, we cannot compute the process of

this option just using the implied volatility surface.

The implied volatility surface will only give us the marginal distribution,

marginal risk neutral distribution of the stock price.

It doesn't give us the joint distribution.

And in order to evaluate this, I would need to compute the following.

So if the value of the security is p0. It will be equal to the expected value

using risk-neutral probabilities e to the minus or t times the maximum of k minus

st and 0 times the indicator function of st being greater than or equal to B.

And the point I'm trying to make is, in order to compute this expectation.

I would need to know the joint risk neutral distribution for the stock price

at all times between 0 and capital T. But I don't know that.

I can not compute that risk neutral distribution from the, from the implied

volatility surface. And so in practice and we'll come back to

this soon. We would need to use some sort of model,

some arbitrage free model to estimate the price of this quantity.