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.