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 and the shapes they 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 Brownian 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 and 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 while, but it was maybe after this crash that for many people it became clear that the assumptions of Geometric Brownian 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 and 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 those 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 are going to use call options but I can tell you that if 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 prices 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 all of these parameters S, we see that it's the current stock price, time to maturity is known, risk-free interest rate is known, that even dividend yield can be estimated, the strike is known and all we're left with is the implied volatility sigma KT 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-Scholes 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 will generally depend on K and T. That is why we have written it as sigma of K and T. Here is an example of the implied volatility surface as of the 28th of November 2007 for the Euro Stoxx 50 index. This is an index of stocks traded in the Euro zone, there are 50 securities in the index and that is the analog if you like of the S& P 500 in the US. So there are several points 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 marketplace 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 determine the implied volatility. 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 like in Oral, 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. 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 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 both in the strike dimension and in the time to maturity dimension. So that is how I construct my implied volatility surface. As I mentioned on 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 the 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 the 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 0 value, than maybe the option starts off here or I'll start off with zero if it's out of the money but I'll start off let's say at this point here as a function of sigma and then it's going to grow someway like this. How do I know it's going to increase? Well I know is 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 is my sigma K T and that is how I know there will be a unique solution to that equation 2. 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 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 of 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 was 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 volatility surface on day 1, if I look at it on the next day, I should still see the same constant segments. So that's what I mean when I say constant through time.
