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In the last module we introduced the concept of the volatility surface.

And we saw that volatility surfaces in practice tend to have specific shapes.

In particular if you fix a timed maturity and you look at the slice of the

volatility surface. Then you will typically see that implied

volatilities increase as the strike decreases.

This is known as the volatility skew or the smile.

In this module we're going to discuss some reasons why we see a skew or smile

in practice. So recall, this is our example of an

implied volatility surface. This is just the volatility surface for a

particular moment in time for a particular underlying security, in this

case is the Euro stocks index in November, 2007.

Martin mentioned before that the way this volatility surface is constructed is we

have a set of options with strikes, and maturities K1, T1, up to say, KN, TN.

And what we do is, we figure out the implied volatility.

For each of these strike maturity pairs. And we do that, as we said, by equating

the market price for the option, CMKT. So the market price of the option.

KITI, with the Black-Scholes price of the option.

So at the current price srckiti and we get sigma.

K I T I. And what we do is, we see this in the

marketplace, we know all of these parimeters, S R C, can be estimated, K I

and T I. And so we know the Black-Scholes forumla,

so the only thing we need to calculate is this.

And we explained why we can get a unique solution to this when there's no

arbitross. So what we do is we get the implied

volatility at all of these strike maturity pairs that are traded in the

marketplace. Maybe they are these quantities here that

I am plotting. And then I actually fit a surface to all

of these points. So that's how I get my implied volatility

surface. We mentioned, as well, that one striking

feature of implied volatility surfaces, in general, is the so called skew.

That is, if I fix a particular time to maturity.

Maybe 2.5. I will see, that, the implied

volatilities tend to increase. That's it here.

They tend to increase as the strike decreases.

So this is my slice of the volatilities surface of T equals 2.5.

And I can see that these volatilities are increasing, as the strike decreases.

So that's called a skew or smile. And after the, the Wall Street crash of

1987, this skew or smile behavior started to appear in the marketplaces for various

derivatives markets. And people started wanting to understand

why these skews were there. And they also wanted to able to build

models that produced these skews. So the skew or smile that you see in

options markets is a very important feature of those markets.

So we're going to discuss a couple of reasons for why a skew actually exists in

practice. there are at least two principal excuses

for the skew. First explanation is risk aversion.

And this explanation can appear in many guises.

For example, security prices often jump, jumps from a downside tend to be larger

and more frequent than jumps from the upside.

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Another guise is that as markets go down fear or panic sets in and volatility goes

up. A third reason is simply supply and

demand. Investors like to protect their portfolio

by purchasing out-of-the-money put options, and so there is more demand for

options with lower strikes. So if there's more demand for options

with lower strikes, then the prices of these options with lower strikes will

actually increase. And therefore they will have higher

implied volatilities. Note that in making this argument I am

using the fact that our European option price increases as the sigma parameter

increases. So, all of these three comments here or

three points here reflect risk aversion in some sense.

The fact that when markets go down, people get more worried, markets become

more volatile, therefore options become more expensive.

Supply and demand. People want to protect their portfolios

against the downside or against negative returns in the marketplace.

One way to protect your portfolio, in that situation is to buy out-of-the-money

puts. And so there's a natural demand for out

of the money puts in the market base. Again, that pushes those option prices

up, which is reflected in higher volatilities for these out of the money

options. So these points all reflect risk aversion

in some form or another. A second explanation is the so-called

leverage effect. The leverage effect is based on the fact

that the total value of the company assets, i.e debt plus equity, is a more

natural candidate. Is a more natural candidate to follow

geometric grounding motion, or at least to have IID returns.

So let's spend a little bit of time talking about the leverage effect.

Let V, E and D Denote the total value of a company, the companies equity and the

companies staff, respectively. Then, the so called fundamental

accounting equation states that V is equal to D plus E.

So on the left hand side we have V, the value of the firm.

This is the value of all of the assets that accompany a firm house.

Well, if you think about it for a moment you will see that all of those assets,

all the cash flows produced by those assets.

Must go to the debt-holders and the equity holders.

So therefore, we get V equals D plus E. One way to see this visually as well is

to break up the total value of the firm into an equity piece, which we will have

down here. And up here we've got a debt piece.

So this is the total value of the company.

We've got it split up into equity and debt.

And indeed, equation three is the basis for many classical structural models.

So we won't be discussing structural models in this course, but I can tell you

that these models are sometimes used to price risky.

Or default able debt, and indeed credit default swaps as well.

Merton in the 70s was the first to recognize that equity could be viewed as

a call option on V withs trike equal to D.

And this is valid because debt holders get paid before equity holders.

So what Merton was saying is that we can view equity, the equity piece of a firm,

or certainly at maturity if you like, imagine that there's some maturity here.

Then the equity value at maturity is equal to the maximum of 0 and V minus D.

And so what this is doing is a, it's reflecting the fact that the debt-holders

get paid off first. So equity's always the riskiest part of

the capital structure of a company. So equity holders actually incur losses

before debt-holders. So, if a company is being liquidated at

time capital 'T' say, then the debt holders must get their money first.

And only after debt holders get their money do the equity holders get paid.

What they get paid then is the residual, they get V minus D.

They only get that if D is less than V. Otherwise the limited liability of shares

And equity holders means that they would get zero.

So Martin was the first to actually make this point, he then actually was abel to

say, we'll lets maybe model the dynamics of V.

Instead of saying let E, the equity piece or the stock price follow a geometric

grounding motion. Maybe we could let V follow a geometric

grounding motion. And then use risk-neutral pricing to

actually get the value of the equity. And in turn, use that to get the value of

the debt as well. So this gave rise to what I called

structural models for pricing the components of the capital structure in a

company. The capital structure being the equity,

the debt, and so on. And by the way, this way of looking at

things is very important. It's playing out right now in the global

fianncial crisis as people are talking about banks failing.

And whether equity holders or deposit holders incur the losses.

So all of these ideas we're talking about here are actually very relevant to what's

going on in the world right now. To see how the leverage effect can

actually give rise to the skew, let's do the following.

Let delta v, delta e, and delta d be the change in values in v, e, and d

respectively. So this might be over some time horizon.

T to T plus delta T then the fundamental accounting equation again state that this

condition. This equation must be satisfied and we'll

assume that delta t is fairly small, relatively small so that delta V is also

relatively small. So now if we divide across this equation

by V we get the following here. And then all we're doing is rearranging

them. We're going to take an E outrside and

bring it down here. And take a D outside this term and divide

by D over here. So equation four is a way of writing the

return on the value of the company. So this here is the return on the value

of the company. So if I say, or V for the return in V, so

or v, this is return on the equity piece and this is the return on the debt piece.

So we see that rv is equal to E over V times rE plus D over V times rD.

So, in other words we can actually say that the return on the company.

The return on the assets of the company 'rv' Is a weighted combination of the

return on the equity part of the company and the return on the debt part of the

company. Now by the way just as an aside for those

of you that might have studied corporate finance before and capital structure

before. We're not going to go into taxes and

benefits from taxes on debt and so on. That's another matter entirely.

What we're doing here is just trying to understand how the leverage effect can

give rise to the skew that we see in implied volatility surfaces in practice.

Alright, so let's, let's come back to, to this.

So, what we will do is we'll assume the following.

Suppose that the equity piece is substantial, so that it absorbs almost

all the losses. So remember, this is how we're thinking

of, of, of our capital structure. We've got our equity piece down here.

We've got our debt piece down here. V is equal to D plus E.

Now, if E is substantial enough, so that any of these changes in v losses or gains

can be absorbed by e, then that means that delta d will be very small.

If delta D is very small then we can do the following.

Let's take variances across equation 4. If we do that we'll get the following.

We'll get sigma squared v. So this is the variance of the return on

the value of the firm. Is equal to E over V to 2b squared times

sigma squared E. This is the variance of rE.

Plus d over v 2b squared. Times sigma squared D.

Your sigma squared d is the variance of r d plus twice e over v times d over v, the

covariance of rE and rD. However, if the equity component is very

substantial, so that it absorbs almost all of the loses, and so the debt is not

very risky, then delta D will be very small.

And in particular, sigma square D on the covariants of rE with rD will be very

small in comparison with sigma square E. So in particular, in this situation, this

will be approximately equal to 0. And so therefore I can get sigma V as

approximate equal to E over V times sigma E.

I can rearrange to get sigma E equal V over E times sigma V, remember V equals E

plus D. So if I substitute E plus d in for V.

I will get sigma e equals 1 plus D over E times sigma V.

And so if sigma V, is a constant. Imagine the value of the assets of reform

following geometric value in motion. So, in that case Sigma V is a constant,

we'll see that naturally Sigma E will actually increase as V decreases.

In other words, even as Sigma V is a geometric grounding motion, then as V

goes up or goes down, Sigma E will actually change.

So, sigma V can be constant, but sigma E will therefore be stochastic, and we will

see that sigma E will increase as the equity piece decreases.

And, so, this also explains why you would see a skew in the marketplace.

Why you would see volatilities, implied volatilities, being higher for lower

strikes than for higher strikes. This is called the leverage effect.