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 company's equity, and the company's debt, 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 a company or a firm has. Well, if you think about it for a moment, you'll 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. Indeed, equation 3 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 defaultable debt, and then the credit default swaps as well. Merton in the '70s, actually was the first to recognize that equity could be viewed as a call option on V with strike equal to D. 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. So what this is doing is, A, it reflecting the fact that the debt holders get paid off first. So equity is 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. 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, but they only get that if D is actually less than V. Otherwise, the limited liability of shares and equity holders means that they will get zero. So Merton was the first to actually make this point. He then actually was able to say, "Well, let's maybe model the dynamics of V." Instead of saying let E, the equity piece or the stock price follow a geometric Brownian motion, maybe we could let V follow geometric Brownian motion, and then use risk-neutral pricing to actually get the value of the equity. In turn, use that to get the value of the debt as well. So this gave rise to what are 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. By the way, this way of looking at things is very important. It's playing out right now in the global financial 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 for 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 of 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 states that this condition, this equation must be satisfied. 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. Then, all we're doing is rearranging things. We're going to take an E outside here and bring it in down here, and take a D outside this term, and then divide by D over here. So equation 4 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 r_V for the return on V. So or V, this is return on the equity piece, and this is the return on the debt piece. So we see that r_V is equal to E over V times r_E plus D over V times r_D. So in other words, we can actually say that the return on the company, the return on the assets of the company, r_V, 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. All right. So let's come back to this. So what we will do is we will 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 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 will 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 be squared times Sigma squared E. This is the variance of r_E plus D over V to be squared times sigma squared D, where sigma squared D is the variance of r_D plus twice E over V times D over V, the covariance of r_E and r_D. However, if the equity component is very substantial so that it absorbs almost all of the losses, and so the debt is not very risky, then Delta D will be very small. In particular, Sigma squared D and the covariance of r_E with r_D will be very small in comparison with Sigma squared E. So in particular, in this situation, this will be approximately equal to zero. So therefore, I can get Sigma V is approximately equal to E over V times sigma E. I can rearrange to get Sigma E equals 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. So if Sigma V is a constant, imagines the value of the assets of a firm following geometric Brownian motion. Since in that case Sigma V is a constant, we will see that naturally, Sigma E will actually increase as E decreases. In other words, even if Sigma V is a geometric Brownian 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. We will see that Sigma E will increase as the equity piece decreases. 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.
