In the last module, we introduced the concept of the volatility surface, and we saw that volatility surfaces in practice tend to have specific shapes. In particular, if you fix a time to maturity and you look at the slice of the volatility surface, then you will typically see that implied volatilities increase as the strike decreases. This is known as the volatility skew or the smile. In this module, we're going to discuss some reasons why we see a skew or smile in practice. So recall, this is our example of an implied volatility surface. This is just the volatility surface for a particular moment in time for a particular underlying security. In this case it was the EURO STOXX index in November 2007. We mentioned before that the way this volatility surface is constructed is we have a set of options which strikes on maturities K_1, T_1 up to say K_n, T_n. What we do is we figure out the implied volatility for each of these strike maturity pairs, and we do that, as we said, by equating the market price for the option C_mkt. So the market price of the option K_i, T_i with the Black Scholes price of the option. So at the current price S, R, C, K_i, T_i and we get Sigma K_iT_i. What we do is we see this in the marketplace, we know all of these parameters S, R, C can be estimated, K_i and T_i, and so we know the Black-Scholes formula. So the only thing we need to calculate is this, and we explained why we can get a unique solution to this when there's no arbitrage. So what we do is we get the implied volatility at all of these strike maturity pairs that are traded in the marketplace. Maybe there are these quantities here that I am plotting. Then I actually fit a surface to all of these points. So that's how I get my implied volatility surface. We mentioned as well that one striking feature of implied volatility surfaces in general is the so-called skew. That is if I fix a particular time to maturity, maybe 2.5, I will see that the implied volatilities tend to increase as the strike decreases. So this is my slice of the volatility surface at T equals 2.5, and I can see that these volatilities are increasing as the strike decreases. So that's called a skew or a smile. After the Wall Street crash of 1987, this skew or smile behavior started to appear in the marketplaces for various derivatives markets, and people started wanting to understand why these skews were there, and they also wanted to be able to build models that produce these skews. So the skew or smile that you see in options markets is a very important feature of those markets. So we are going to discuss a couple of reasons for why a skew actually exists in practice. There are at least two principal explanations for the skew. The first explanation is risk aversion, and this explanation can appear in many guises. For example, security prices often jump, and jumps to the downside tend to be larger and more frequent than jumps to the upside. Another guise is that as markets go down, fear or panic sets in and volatility goes up. The third reason is simply supply and demand. Investors like to protect their portfolio by purchasing out-of-the-money put options. So there is more demand for options with lower strikes. So if there's more demand for options with lower strikes, then the prices of these options with lower strikes will actually increase and therefore they will have higher implied volatilities. Note that in making this argument I'm using the fact that a European option price increases as the Sigma parameter increases. So all of these three commons here or three points here reflect risk aversion in some sense. The fact that when markets go down, people get more worried, markets become more volatile, therefore options become more expensive. Supply and demand, people want to protect their portfolios against the downside or against negative returns in the marketplace. One way to protect your portfolio in that situation is to buy out-of-the-money puts. So there's a natural demand for out of the money puts in the marketplace. Again, that pushes those option prices up which is reflected in higher volatilities for these out of the money options. So these points all reflect risk aversion in some form or another. A second explanation is the so-called leverage effect. The leverage effect is based on the fact that the total value of the company assets are, i.e. debt plus equity, is a more natural candidate to follow geometric Brownian Motion or at least to have IID returns.
