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 timed 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 is the Euro stocks index in November, 2007. Martin mentioned before that the way this volatility surface is constructed is we have a set of options with strikes, and maturities K1, T1, up to say, KN, TN. And 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, CMKT. So the market price of the option. KITI, with the Black-Scholes price of the option. So at the current price srckiti and we get sigma. K I T I. And what we do is, we see this in the marketplace, we know all of these parimeters, S R C, can be estimated, K I and T I. And so we know the Black-Scholes forumla, 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 arbitross. So what we do is we get the implied volatility at all of these strike maturity pairs that are traded in the marketplace. Maybe they are these quantities here that I am plotting. And 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. That's it here. They tend to increase as the strike decreases. So this is my slice of the volatilities surface of T equals 2.5. And I can see that these volatilities are increasing, as the strike decreases. So that's called a skew or smile. And after the, 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 able to build models that produced these skews. So the skew or smile that you see in options markets is a very important feature of those markets. So we're going to discuss a couple of reasons for why a skew actually exists in practice. there are at least two principal excuses for the skew. First explanation is risk aversion. And this explanation can appear in many guises. For example, security prices often jump, jumps from a downside tend to be larger and more frequent than jumps from the upside. Another guise is that as markets go down fear or panic sets in and volatility goes up. A third reason is simply supply and demand. Investors like to protect their portfolio by purchasing out-of-the-money put options, and 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 am using the fact that our European option price increases as the sigma parameter increases. So, all of these three comments 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. And so there's a natural demand for out of the money puts in the market base. 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, i.e debt plus equity, is a more natural candidate. Is a more natural candidate to follow geometric grounding motion, or at least to have IID returns. So let's spend a little bit of time talking about the leverage effect. Let V, E and D Denote the total value of a company, the companies equity and the companies staff, respectively. Then, the so called fundamental accounting equation states that V is equal to D plus E. So on the left hand side we have V, the value of the firm. This is the value of all of the assets that accompany a firm house. Well, if you think about it for a moment you will see that all of those assets, all the cash flows produced by those assets. Must go to the debt-holders and the equity holders. So therefore, we get V equals D plus E. One way to see this visually as well is to break up the total value of the firm into an equity piece, which we will have down here. And up here we've got a debt piece. So this is the total value of the company. We've got it split up into equity and debt. And indeed, equation three is the basis for many classical structural models. So we won't be discussing structural models in this course, but I can tell you that these models are sometimes used to price risky. Or default able debt, and indeed credit default swaps as well. Merton in the 70s was the first to recognize that equity could be viewed as a call option on V withs trike equal to D. And this is valid because debt holders get paid before equity holders. So what Merton was saying is that we can view equity, the equity piece of a firm, or certainly at maturity if you like, imagine that there's some maturity here. Then the equity value at maturity is equal to the maximum of 0 and V minus D. And so what this is doing is a, it's reflecting the fact that the debt-holders get paid off first. So equity's always the riskiest part of the capital structure of a company. So equity holders actually incur losses before debt-holders. So, if a company is being liquidated at time capital 'T' say, then the debt holders must get their money first. And only after debt holders get their money do the equity holders get paid. What they get paid then is the residual, they get V minus D. They only get that if D is less than V. Otherwise the limited liability of shares And equity holders means that they would get zero. So Martin was the first to actually make this point, he then actually was abel to say, we'll lets maybe model the dynamics of V. Instead of saying let E, the equity piece or the stock price follow a geometric grounding motion. Maybe we could let V follow a geometric grounding motion. And then use risk-neutral pricing to actually get the value of the equity. And in turn, use that to get the value of the debt as well. So this gave rise to what I called structural models for pricing the components of the capital structure in a company. The capital structure being the equity, the debt, and so on. And by the way, this way of looking at things is very important. It's playing out right now in the global fianncial crisis as people are talking about banks failing. And whether equity holders or deposit holders incur the losses. So all of these ideas we're talking about here are actually very relevant to what's going on in the world right now. To see how the leverage effect can actually give rise to the skew, let's do the following. Let delta v, delta e, and delta d be the change in values in v, e, and d respectively. So this might be over some time horizon. T to T plus delta T then the fundamental accounting equation again state that this condition. This equation must be satisfied and we'll assume that delta t is fairly small, relatively small so that delta V is also relatively small. So now if we divide across this equation by V we get the following here. And then all we're doing is rearranging them. We're going to take an E outrside and bring it down here. And take a D outside this term and divide by D over here. So equation four is a way of writing the return on the value of the company. So this here is the return on the value of the company. So if I say, or V for the return in V, so or v, this is return on the equity piece and this is the return on the debt piece. So we see that rv is equal to E over V times rE plus D over V times rD. So, in other words we can actually say that the return on the company. The return on the assets of the company 'rv' Is a weighted combination of the return on the equity part of the company and the return on the debt part of the company. Now by the way just as an aside for those of you that might have studied corporate finance before and capital structure before. We're not going to go into taxes and benefits from taxes on debt and so on. That's another matter entirely. What we're doing here is just trying to understand how the leverage effect can give rise to the skew that we see in implied volatility surfaces in practice. Alright, so let's, let's come back to, to this. So, what we will do is we'll assume the following. Suppose that the equity piece is substantial, so that it absorbs almost all the losses. So remember, this is how we're thinking of, of, of our capital structure. We've got our equity piece down here. We've got our debt piece down here. V is equal to D plus E. Now, if E is substantial enough, so that any of these changes in v losses or gains can be absorbed by e, then that means that delta d will be very small. If delta D is very small then we can do the following. Let's take variances across equation 4. If we do that we'll get the following. We'll get sigma squared v. So this is the variance of the return on the value of the firm. Is equal to E over V to 2b squared times sigma squared E. This is the variance of rE. Plus d over v 2b squared. Times sigma squared D. Your sigma squared d is the variance of r d plus twice e over v times d over v, the covariance of rE and rD. However, if the equity component is very substantial, so that it absorbs almost all of the loses, and so the debt is not very risky, then delta D will be very small. And in particular, sigma square D on the covariants of rE with rD will be very small in comparison with sigma square E. So in particular, in this situation, this will be approximately equal to 0. And so therefore I can get sigma V as approximate equal to E over V times sigma E. I can rearrange to get sigma E equal V over E times sigma V, remember V equals E plus D. So if I substitute E plus d in for V. I will get sigma e equals 1 plus D over E times sigma V. And so if sigma V, is a constant. Imagine the value of the assets of reform following geometric value in motion. So, in that case Sigma V is a constant, we'll see that naturally Sigma E will actually increase as V decreases. In other words, even as Sigma V is a geometric grounding motion, then as V goes up or goes down, Sigma E will actually change. So, sigma V can be constant, but sigma E will therefore be stochastic, and we will see that sigma E will increase as the equity piece decreases. And, so, this also explains why you would see a skew in the marketplace. Why you would see volatilities, implied volatilities, being higher for lower strikes than for higher strikes. This is called the leverage effect.
