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 time to 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 KT converges to a constant as T goes to infinity. Of course, I should mention, in practice we will only see options with maturities out to two or three years. So in general, you actually don't observe Sigma KT for T being very large. It is also worth mentioning that when T is small, you often observe an inverted volatility surface, which short term options having much higher volatilities than longer-term options. Indeed, we see that here to some extent as well. We see that for very small times to maturity and for strikes that are fairly low or at least moderate to low, we see the implied volatilities are higher than for longer times to 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. So when there is market stress, we tend to see short term options having higher volatility than longer maturity options. Single stock options are generally American, and in this case, call and put options typically give rise to different surfaces. But I mentioned a moment ago, we're not really going to 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 that 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's it. 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. So these could be foreign exchange rates like dollar versus euro, or dollar v yen, or euro v pound, and so on. Also, for all the main equity indices, the S&P 500, the Euro Stoxx, the Nikkei, the Dow Jones, the FTSE, and so on. Not only are the volatility surfaces calculated 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 the Black-Scholes formula, we're using the correct volatility which is a function of the strike and time to maturity. So it is interesting to note how the Black-Scholes formula is wrong, the Black-Scholes model is wrong. Everybody knows it's wrong, and it's wrong for a number of reasons. That said, the Black-Scholes model is still used everywhere. 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 two, this is what I am 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 formulas the Black-Scholes option price, and it's the wrong formula because we 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. 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 expiration 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. So let's fix T and let's look at a slice of 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 at fixed in this picture here. Now, I said that this 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. So the payoff of the put with strike K2 will always be greater than or equal to the payoff of the put with strike K1. So its 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 will be increasing in Sigma. If this skew is too 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 put with strike K1. But if this gets too steep, then in fact that would be violated and there will 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 to 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 or 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 the volatility surface I showed you a while ago. So for example, if I come back to our picture, I would fix K, so maybe I would fix K at 4,000. Then I would get this slice here, and this would be the term structure of implied volatilities when K equals 4,000. So it will be to this guy here. If I'm looking at K equals 3,800 say here, I get this term structure of implied volatilities and we see that it is inverted at K equals 3,800. So returning to this here, the best way to explain what I mean by counter spread arbitrage is to make the following point. Suppose, 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 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 use up here for the put spread. If it gets too inverted, then Sigma T1 is too large relative to Sigma T2, and the call option price with maturity T1, will 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 no arbitrage in the market. These restrictions can be difficult to enforce however when we are stressing the volatility surface. What do we mean by stressing the volatility surface? Well, stressing the 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 volatilities by various percentages, and we 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. 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.
