My Life as a Quant (25 page)

Various academics had attempted to alter some of the assumptions about the future evolution of stock prices in the Black-Scholes model so as to make it better mimic the bond price distribution of
Figure 10.2
. In a similar spirit, but with greater practicality since he was part of a business, Ravi had invented the then-current Goldman bond options model by cleverly modeling the future behavior of bond yields rather than bond prices. A bond's yield is the average annual percentage return you will earn if you buy it at its current price and then hold it to maturity while collecting all the coupons and the final repayment of principal. Ravi simply assumed that a bond's yield rather than its price obeyed the ever-widening Black-Scholes distribution of
Figure 10.1
. In that case, as time passes and the bond gets very close to its maturity, the future value of its yield, no matter how high or low it ranges, becomes irrelevant to the bond price; too little time is left until maturity for the yield to have an effect. Therefore, although the distribution of bond yields in Ravi's option model grew limitlessly high or low as time passed and resembled
Figure 10.1
, the bond prices computed from these yields looked much like those of
Figure 10.2
.

Ravi's model did a reasonable job of capturing the future behavior of a bond's price. It also matched the intuition of the traders, who were already used to thinking about bonds in terms of their yields, and so took naturally to thinking about the range or volatility of yields. The same good idea often occurs to several people simultaneously, and versions of the model soon popped up independently at other Wall Street firms. When I came to work at Salomon Brothers a few years later, I found they had a similar scheme.

But there were deeper, subtler problems with the model that arose from its origin in the Black-Scholes framework. Just as the Black-Scholes model treats each stock as an independent variable, so Ravi's model treated each bond as independent. Doing so led to a host of conceptual inconsistencies. For, while there is nothing obviously wrong in imagining that the future prices of a share of IBM and a share of AT&T can vary independently, it is inconsistent to model the future prices of, say, a five-year bond and a three-year bond as varying independently. If you do, things fall apart.

Bonds are connected to each other. The future behaviors of a five-year bond and a three-year bond are
not
independent, but overlapping: Two years from now, the five-year bond will be a three-year bond, so you cannot model one bond's future without implicitly modeling another. In fact, it is impossible to model one bond without modeling all of them.

A five-year bond and a three-year bond have other commonalities, too. You can think of a five-year Treasury bond that pays interest every six months as a collection of ten zero-coupon bonds with maturities spread six months apart over the next five years. Similarly, a three-year Treasury bond is a collection of six zero-coupon bonds respectively maturing every successive six months over the next three years. Decomposed in this way, the bonds' ingredients are shared: Both contain the first six zero-coupon bonds. Therefore, in modeling the three-year bond, you are also implicitly modeling parts of the five-year bond.

In essence, Ravi's model allowed impermissible violations of the law of one price that lies beneath all rational financial modeling. This law demands that any two securities with the same final payouts must have the same current value. Now, there is a combination of short-term options on long-term bonds that has exactly the same payout as a short-term bond, and therefore, the combination of options should have the same theoretical value as the short-term bond, despite their formally different names. But Ravi modeled long-term bonds independently of short-term ones, and so the model had no way of enforcing this equivalence.

Every hard look at Ravi's model led to the same insight: It is impossible to model bonds piecemeal, one bond at a time. You must build a model of the future evolution of all bonds, that is, of the yield curve itself. This was our aim.

I had left Fischer's office a little chastened by his sharp remarks about the names of the fields on my calculator, but in a few days he let me know that I could join him and Bill Toy in their effort to create a new bond options model. It was a singular opportunity that had a large and beneficial effect on my life.

That spring of 1986 I attended my first options conference, an annual event organized by Howard Baker, Menachem Brenner, and Dan Galai at the Amex. I was one of about a hundred participating quants, traders and academics, all of us actively involved in the field at a time when options meetings were still a rarity, before the conference-for-profit organizations like
Risk
magazine began to dominate the market and eventually put the Amex options conference out of business. I recall several presentations on new models for valuing bond options, one in particular by Rick Bookstaber, then at Morgan Stanley. You could sense a rising urgency, almost a race, to solve the problem. Fischer told Bill and me that Bob Merton was working on the same problem as a consultant for another investment bank.

The Goldman contingent at the conference had more than an academic interest—our traders were making daily markets in long-expiration options on long-dated bonds, precisely the domain where the contradictions in Ravi's model were most severe. The traders were aware of their need for a better model, and as such were at the forefront of the impetus to replace it.

We knew that we had to model the future behavior of all Treasury bonds, that is, the evolution of the entire yield curve. How to set about it was neither obvious nor easy. A stock price is a single number, and when you model its evolution, you project only one number into an uncertain future. In contrast, the yield curve is a continuum, a string or rubber band whose every point, at any instant, represents the yield of a bond with corresponding maturity. As time passes and bond prices change, the yield curve moves, as illustrated in
Figure 10.3
. To evolve the entire yield curve forward in time is a much more difficult task: Just as you cannot move the different points on a string completely independently of each other, because the string must stay connected, so bonds close to each other must stay connected, too.

Figure 10.3
Yield curves can vary during the day.

How, then, to project bond prices into the future? Fischer, Bill, and I were pragmatists. We were building a model for traders, and we wanted it to be simple, consistent, and reasonably realistic. Simple meant that only one random factor drove all changes. Consistent meant that it had to value all bonds in agreement with their current market prices; if it produced the wrong bond prices, it was pointless to use it to value options on those bonds. Finally, realistic meant that the model's future yield curves should move through ranges similar to those experienced by actual yield curves.

When physicists build models, they often first resort to a toy representation of the world in which space and time are discrete and exist only at points on a lattice—it makes picturing the mathematics much easier. We built our model in the same vein. We imagined a world in which the shortest investment you could make lasted exactly one year, and was represented by the one-year Treasury bill interest rate. Longer-term rates would then be a reflection of the market's perception of the probable range of future short-term (that is, one-year) rates.

In this spirit, we built a simple model of future one-year rates that resembled a discrete version of the stock price distributions of
Figure 10.1
. The initial one-year rate, as shown in
Figure 10.4
, was known from the current yield curve. As you looked further out into the future, rates could range over progressively wider values.

Figure 10.4
The Black-Derman-Toy model focuses on the distribution of future short-term rates. Here, each dot corresponds to a particular value of the future one-year rate. The more time passes, the greater the uncertainty of future rates.

In order to complete our model, we now had to determine the range of future one-year rates at every year in the future. In our model, the key principle was to think of longer-term bonds as being generated by successive investments in short-term bonds. From this point of view, two years of interest is obtained by two successive one-year investments, the first at a known rate, the second at an uncertain one. The market's price for a two-year bond today depends on its view of the distribution of future one-year rates. You can calculate the logical value of the current two-year bond yield, from the current one-year yield and
the distribution of one-year rates one year hence
. Similarly, you can calculate the volatility or uncertainty of the current two-year yield from the volatility of the distribution of one-year rates one year hence. Alternatively, working backwards, since the current two-year yield and its volatility is known from the market, you can deduce
the distribution of one-year rates one-year hence
, as shown in
Figure 10.5
.

Figure 10.5
How the distribution of future one-year rates is deduced from the current yield curve in the Black-Derman-Toy model. The two-year yield to maturity fixes the distribution of one-year rates after one year, the three-year yield to maturity fixes the distribution of one-year rates after two years, and so on.

In the same way, the value of the current three-year yield can be found from the current one-year rate, the known distribution of one-year rates one year hence (already deduced from the current two-year yield) and
the distribution of one-year rates two years hence
. But, since the value of the current three-year yield is known, you can use it to deduce
the distribution of one-year rates two years hence
. Continuing in this way, you can use the current yield curve at any instant to pin down the range of all future one-year rates, as illustrated in
Figure 10.5
.

This was the essence of our model. When Bill and I programmed it, it seemed to work—we could extract the market's expectation of the distributions of future one-year rates from the current yield curve and its volatility. There was nothing holy about the one-year time steps we started with. Once the model worked, we used monthly, weekly, or sometimes even daily steps on a lattice, determining the market's view of the distribution of future short-term rates at any instant from the current yield curve. A typical lattice (or tree, as we called it, because of the way an initial interest rate forked out into progressively wider branches) had hundreds or thousands of equally spaced short periods, as illustrated in
Figure 10.6
.

Figure 10.6
A schematic illustration of a multiperiod short-rate tree in the Black-Derman-Toy model with equal periods extending for many years.