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We're now going to discuss the volatility surface.

If the Black-Scholes model was correct, the volatility surface would be flat.

In practice, it is anything but flat. And we're going to see, in this module,

what the volatility surface is, how it is constructed.

And we're also going to see some of the arbitrage constraints that restrict the

volatility surface on the shapes it can take on in practice.

The Black-Scholes model is a very elegant model.

But for several reasons, it does not perform very well in practice.

The first reason is that security prices often jump.

However, this is not possible with geometric grounding motion.

A second reason is that security price returns tend to have fatter tails than

those implied by the log-normal distribution.

By fatter tails, I mean the fact that extreme returns are more likely in

practice than you would expect if the security prices actually had logged

normal distributions as implied by geometric Brownian motion.

Returns also are clearly not IID in practice.

So, if I'm to break any time period, if I was to break any time period up into

finite intervals of time, then if the security price follows a geometric

Brownian motion, then log returns would be IID.

And this is clearly not the case in practice.

By the way, if you want to learn more about geometric Brownian motion, there is

a module that we have recorded on geometric Brownian motion that can be

found on the course platform. Anyway, for all of these reasons, we know

that security prices in practice do not follow geometric Brownian motion.

And market participants are well aware of the fact that the Black-Scholes model is

a very poor approximation to reality. They've certainly known this since the

Wall Street crash of 1987. I will return to discussing the crash of

87 in a, in a, in a while. But it was, maybe, after this crash that

for many people it became clear that the assumptions of geometric barrier motion

did not hold and that, in practice, people would have to adjust the

Black-Scholes model in an appropriate way in order to trade options.

All of that, having been said, we have to point out that the Black-Scholes model

and the language of Black-Scholes is still pervasive in finance.

Most derivatives markets use aspects of Black-Scholes to both quote option prices

as well as to perform risk management. So, even though the Black-Scholes model

is clearly not a good approximation to market dynamics, it is still very

necessary to understand the Black-Scholes model if you want to understand

derivatives pricing and how derivatives are used in practice.

The incorrectness of Black-Scholes is most obviously manifested through the

volatility surface. This is a concept that is found also

throughout derivatives markets. The volatility surface is constructed

using market prices of European call and put options.

Now they can also be constructed using American option prices, but it's a little

trickier. So, we're going to stick with the case of

European call and put option prices. So, these include, for example, options

on foreign exchange and options on the most commonly traded market indices, such

as, the S&P 500, the Eurostoxx, the Dax , the Nikkei, and so on.

So, all of these indices have options traded on them that are European options,

and so everything we say here will apply to these indices and, indeed, foreign

exchange options as well. The volatility surface, sigma K, T, is a

function of the strike K and the expiration, T.

It is defined implicitly through this equation here.

Where c subscript mkt stands for the market price of the call option.

And c subscript bs stands for the Black-Scholes price of a call option.

Now, in this definition we're going to use call options but I can tell you that

in the case of European options we could just as easily use put options, we're

going to get the exact same volatility surface.

So we will stick with call options here but again we could just as easily use put

options as well. So what have we got here?

On the left hand side we have the market price of a call option when the current

stock price is s, the strike is k, and the time to maturity is capital T.

We can see this in the marketplace. We can go into the marketplace and see

how much this call option is worth. That is the left hand side, over here.

On the right hand side, we want to use the Black-Scholes formula for the price

of a call option. Now, we're going to know that all of

these parameters s, we see that's it's the current stock price timed maturities

known, risk free interest rate is known. The dividend, dividend yield can be

estimated. The strike is known.

And all we're left with, is the implied volatility, sigma K, T, or simply Sigma,

as we've been calling it, up until now. So, what we do, is as follows, we equate

the market price of the option with the Black-Schole's price of the option.

And we solve for the one unknown parameter sigma.

So, when we solve this equation for sigma, we are getting what is called the

implied volatility for the option. Note also that this implied volatility

would generally depend on K and T. And that is why we've written it as sigma

of K and T. Here's an example of the implied

volatility surface, as of the 20th of November, 2007, for the Euro Stocks 50

index. This, is an index of stocks traded in the

Euro zone. It is, there are 50 securities in the

index. And it is, the analog, if you like, of

the S&P 500 in the US. So, there are several points to, to keep

in mind. First of all, we don't see this surface

in practice. What we actually see, is the following.

We see a finite number of options in the market place, which strikes and

maturities K1, T1, up as far as, let's say Kn, Tn.

So these are the strike maturity pairs for which options are traded in the

marketplace. Maybe these values here represent these

values of K and T. So, I might see a finite number of strike

maturity pairs and I'm plotting them here in the figure.

Now, what is done at this point is for each option price, I actually determined,

the implied volatility. [SOUND].

And I do that by working with this equation here.

I have my Black-Scholes formula coded up. I can have it coded up in any language I

like, in Ora, Python, or Excel. And I see the market price as well.

And what I do is, I run a simple calibration or root finding algorithm to

determine what value of sigma will make this equation correct.

And that's how I calculate these values here.

So, I can get all of these values here and then I can plot these points here as

I have shown. At that point, I now have a finite number

of, of these points. What I do is I fit a surface to this

point. And that gives me my implied volatility

surface. I need to fit my surface carefully, I'll

use some sort of regression or interpolation, extrapolation procedure to

complete the surface. And that gives me the implied volatility

surface. Now, in practice, to get a surface like

this, I would also need to have additional points, as well.

And, I might make some assumptions, in order to extrapolate out to the extreme

edges, edges both in strike, in the strike dimension and in the time to the

maturity dimension. So, that is how I construct my implied

volatility surface. And as I mentioned in the previous slide,

for European options, it doesn't matter whether I use call or put options, I'm

going to get the same surface. Now, here's a question.

Why will there always be a unique solution, sigma K, T to this equation

here? This is equation 2.

How do I know that I will always find a unique solution to this equation here?

Well, here's why. The first thing to remember is vega, if

you recall vega. Vega of a call option was equal to Delta

c, Delta sigma. And we mentioned that this is always

strictly positive. So, now you can imagine drawing the

following graph. On the x-axis we will plot sigma.

And on the y-axis, we will plot the option price C of sigma.

Now, if this is the zero value then maybe the option starts off here or it starts

off with zero. If it's out of the money but it will

start off, let's say, at this point here, as a function of sigma.

And then it's going to grow someway like this.

Okay? How do I know it's going to increase?

Well, I know it's going to increase because vega equals delta c, delta sigma,

strictly positive. So, c is an increasing function of sigma.

Now, if I go into the marketplace, I will see the option price in the market.

And maybe this will be the option price in the marketplace.

So, therefore, all I need to do to find my unique value of sigma is to come

across here, find out what this value is. And this my sigma K, T.

And that is how I know there will be a unique solution to that equation too.

I am assuming, of course, that there is no arbitrage with the market price of the

option. Now, if the Black Scholes model were

correct, then we should have a flat volatility surface with sigma K, T equals

sigma for all K, T. After all, remember, the Black-Scholes

formula is, is based on the Black-Scholes model.

And the Black-Scholes model assumes that the stock price follows a geometric

Brownian motion, so that the price at time T in the stock is given to us by

this quantity that I'm writing here, where wT is a Brownian motion.

And here, it is assumed that sigma is a constant.

So, if the Black-Scholes model is correct, and indeed the price dynamics of

the underlying security follow geometric Brownian motion, then sigma would be a

constant. And I would get sigma K, T equals sigma

for all K and T. And indeed, it would be constant through

time. As I compute the vol surface, the

volatility surface on day one, if look at it on the next day, I should still see

the same constant sigma. So, that's what I mean when I say

constant through time. In practice, however, volatility surfaces

are not flat. And they move about randomly.

Indeed, options with lower strikes tend to have higher implied volatilities.

And we can see this here. Note that the lower strikes are down in

this direction. So, we see the lower strikes tend to have

higher implied volatilities than higher strikes.

For a given maturity T, this feature is typically referred to as the volatility

skew or the smile. Notice for any fixed timed maturity T,

suppose I take T equals two years, and I look at the slice corresponding to T

equals two years, I'll still see this behavior, where the implied volatilities

rise as the strike decreases. So, the fact that the volatility surface

is not constant is another way to recognize the fact that the Black-Scholes

model is incorrect. It is not close to being right.

And the market knows it is not correct. For a given strike K, the implied

volatility can be either increasing or decreasing with time to maturity.

In general, for a fixed K, sigma K, T converges to a constant as T goes to

infinity. Of course, I should mention, in practice

we will only see options with maturities after 2 or 3 years.

So, in general, you actually don't observe sigma K, T for T being very

large. It is also worth mentioning that when T

is small, you often observe an inverted volatility surface, with short term

options having much higher volatilities than longer term options.

12:50

And indeed, we see that here to some extent as well.

We see that for very small times maturity and for strikes that are fairly low or

least moderate to low, we see the implied volatilities are higher than for longer

times maturity. This actually is often true in times of

market stress. In times of market stress there's a lot

of worry and concern in the market. People are risk averse.

There's a lot of volatility. And, as we know, option prices increase

with volatility. And so when there is market stress, we

can't see short term options having higher volatility than longer maturity

options. Single stock options are generally

American. And in this case, call or put options

typically give rise to different surfaces.

But I mentioned a moment ago, we're not really go into this.

The general ideas behind the volatility surface can be found by just discussing

the case of European options. And indeed that is what we will stick to.

Now, the fact of the volatility surface is not constant, emphasizes just how

wrong Black-Scholes is. And in particular, how wrong the

geometric Brownian motion model, for security dynamics is.

That said, pretty much every equity and foreign exchange derivatives trading desk

computes the Black-Scholes implied volatility surface for all of the markets

they're trading in. So, these could be foreign exchange

rates, like dollar versus euro or dollar v.

yen or euro v. pound and so on.

And also, for all the main equity in the states, like the S&P 500, the Eurostoxx,

the Nikkei, the Dow Jones, the FTSE, and so on.

Not only are the volatility surfaces calculated for all, in all of these

markets, they also calculate the Greeks. So, remember, the Greeks are the

sensitivities of the option prices with respect to parameters.

So we have the delta, the gamma, the vega, the theta, and so on.

We can still calculate all those Greeks using the Black-Scholes formula.

But we just have to make sure now that when we use Black-Scholes formula, we're

using the correct volatility, which is a function of the strike and time to

maturity. So, it is, it is, it is interesting to

note how the Black-Scholes formula is wrong, the Black-Scholes model is wrong.

Everybody knows it's wrong. It is wrong for a number of reasons.

That said, the Black-Scholes model is still used everywhere.

And indeed, use of the Black-Scholes formula is often likened to using the

wrong number in the wrong formula to obtain the right price.

Where does that come from? Well, if we go back to equation 2, this

is what I'm getting at. So, the wrong number is this, sigma K, T.

After all, the Black-Scholes model would assume sigma is a constant.

So, the wrong number is going into the wrong formula.

The wrong formula is the Black-Scholes option price.

And it's the wrong formula because, you know, the Black-Scholes model doesn't

hold. So, the wrong number goes into the wrong

formula to give the right price. The right price is the market price.

And of course that is the right price because that's the price of the option in

the marketplace. The shape of the implied volatility

surface is constrained by the absence of arbitrage.

And it is worth making this point here. For example, we know that implied

volatilities must be greater than or equal to zero for all strikes K and

expirations T. So, therefore, we must have this

condition here. It is also true that at any given

maturity the skew can not be too steep. Otherwise, arbitrage opportunities, such

as a put spread arbitrage, would exist. Now, what do I mean by put spread

arbitrage? Well, I'll answer that here.

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So, lets fix T and lets look at a slice on the volatility surface.

So, here I'm going to show you a slice. So, T is fixed.

Here is K, the strike. And up here, I therefore have sigma K.

I'm going to exclude T from the argument of sigma because we have it fixed in this

picture here. Now, I said that the skew can not be too

steep. Well, what would be too steep?

Well, maybe this would be too steep. Why can we not get a skew that is too

steep? Well, the reason is as follows.

Imagine we've got two strikes. So, these two strikes, we'll call them K1

and K2. Now, imagine we buy a put.

Maybe I'll write it over here. Imagine, we buy a put with strike K2 and

we sell a put with strike K1. Well, such a strategy is actually known

as a put spread. So, if I buy a put with strike K2 and

sell a put with strike K1, then this is going to have a positive cost.

The value of this will be greater than or equal to zero, and that's because K2 is

greater than K1. And so the payoff of the put strike K2

will always be greater than or equal to the payoff of the put with strike K1.

So, it's value today must be greater than or equal to zero as well.

So, therefore, my put spread must have a positive price in the marketplace, if

there's no arbitrage. However, in my Black-Scholes volatility

world, if I have a volatility surface like this, then this is going to be sigma

K2 and this will be sigma K1. So, clearly sigma K1 is greater than

sigma K2. Remember however, the price of a put

option, so the Black-Scholes price of a put option would be increasing in sigma.

And if this skew is to steep, then sigma K1 will be much larger than sigma K2.

And the put option with strike K1 will be more expensive than the put option with

strike K2. And that would introduce an arbitrage

because as I said, in the marketplace, the put with strike K2 must be more

valuable than the strike with K1. But if this gets too steep, then in fact,

that would be violated and there would be an arbitrage in the volatility surface.

So, that's what I mean by put-spread arbitrage.

Likewise, the term structure of implied volatility cannot be too inverted.

What do I mean by that? Well, again, we can draw another picture,

but this time we will keep K fixed. So, T is actually our variable on the

x-axis here. This is sigma of T and I'm keeping K

fixed here. Well, this would be an inverted, what

would be called an inverted term structure of implied volatilities.

This is just a slice of volatility surface I showed you a while ago.

So, for example, if I come back for a picture, I would fix K.

So, maybe I would fix K at 4000. And then I would get this slice here.

And this would be the term structure of implied volatilities when K equals 4000

so it would be this guy here. If I'm looking at K equals 3800, say,

here, I get this kind of term structure implied volatilities and we see that they

have, it, at and we see that it is inverted at K equals 3800.

So, returning to this here, the best way to explain what I mean by call spread

arbitrager is just to make the following point.

Suppose [SOUND] r equals c, the dividend yield equals zero.

Well, then, in that case, it can be shown mathematically.

If there's no arbitrage, then, an option price, let's say, a call option price

would strike c2 must be greater than the value of a call option price, sorry, a

call option price with expiration T2 must be greater than a call option price with

expiration T1, where T2 is greater than T1.

So, maybe this point here is T1 and this point here is T2.

But if it gets too inverted, the implied volatility for T1 is here.

It is sigma T1, and for T2, it is here. And it's the same sort of argument as we

used up here for the put spread. If it gets too inverted, then sigma T1 is

too large, relative to sigma T1, relative to sigma T2.

And the call option price with maturity T1 would be greater than the call option

price with maturity T2. And that would be an arbitrage.

That only holds mathematically when r equals c equals zero, but the same

intuition holds more generally. you might want to think, by the way, if

you're interested, why in this situation this must hold.

Maybe we'll address that in the forums. So, to summarize, in practice, the

implied volatility surface will not violate any of these restrictions.

One, two, and three. Otherwise, there would be an arbitrage in

the market. These restrictions can be difficult to

enforce, however, when we are stressing the volatility surface.

What we mean by stressing the volatility surface?

Well, stressing a volatility surface is something that is often performed in risk

management applications. What we do is we have a portfolio of

derivatives, maybe a portfolio of options.

We know the current value. And we want to see what will be the new

value of this portfolio if the volatility surface changes.

So, what we might do is shift the volatility surface from its current

surface to a new surface. That will give us a new value of the

portfolio, from which we can calculate the profit and loss.

So, this is something people often do in practice.

So, it's also an example of a scenario analysis which we saw in an earlier

module, where we shift the underlying security by various percentages.

We also shift the implied volatilites by various percentages and recalculate the

value of our portfolio in these scenarios.

And then, compute the profit and loss in these scenarios.

So, as I said, this is a very important task for risk management.

In derivatives portfolios, we need to be able to stress volatilities.

We want to be able to stress volatilities in a manner that is consistent with no

arbitrage. And so, when we are moving an entire

surface, we need to do so in such a way that it doesn't violate these no

arbitrage conditions.