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 volatility 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 a key part of the volatility surface. It is very important to appreciate. 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 neutral distributions, and we will emphasize that in this module and indeed in later modules. So recall again this is an example of an implied volatility surface and just to remind ourselves again to make sure we don't forget. It is constructed as follows. We see a set of strike exploration pairs in the marketplace. So we have K1, T1 up to Kn, Tn. We see the option prices in the marketplace for all of these. So we actually see the call or put price let's say call price C subscript MKT for market of Ki,Ti and that's true for i equals one 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 or K, T,C and sigma of K, T. So we set the market price equal to the Black-Scholes price. We know the left-hand side. We know the Black-Scholes formula. We know s,r, k,t. We can estimate C and so there's just one unknown in this equation and we can actually back out this unknown for sigma Ki, Ti and that will give us give us the implied volatility at the strike Ki and expiration Ti. 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 interpolation 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, you 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 and moving parts of the volatility surface up or down, recomputing the value of a portfolio, computing the P and L 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 to call option with strike K. 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 M, K, T being shorthand for market. Let's also see what the payoff of the butterfly strategy 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'll 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 straightforward 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 Delta K. Moreover, it 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 ST below K and then it decreases down to 0 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 your inside this interval where you do get a payoff at time capital T, the average payoff will be K over two. So the average payoff if you're paid off will be K over two. All right. Another thing to keep in mind, we know from risk-neutral 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-hand side of six, but from Risk-neutral pricing which is also equal to the expected value at time 0 using risk-neutral 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 be in the past to refer to the cash account. Here it's referring to the butterfly payoff here and this is our butterfly payoff. So keeping this in mind, we are going to get an alternative expression for B0. We have one expression for B0 here in equation six. On the next slide, 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 the 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 two. Now where does that come 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 upto 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 to K plus Delta K. So that's the probability that ST is inside this interval here. Now, we're imagining Delta K being small by the way. In fact, soon we're going to let delta K go to 0. 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 ST is in this interval, then the payoff you expect to get is K over two and indeed that is why we multiplied by the K over two here. So we have our e to the minus rt term. The probability that ST is inside this interval times the average payoff in the central 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.
