My Life as a Quant (26 page)

We had aimed to make our model simple and consistent, and it was—we could match all current bond prices with one tree. We could then use the same calibrated tree to value
any
security whose payouts in the future had a known dependence on interest rates by averaging those payouts over the distribution. In particular, we could value the payout of an option of any expiration on a bond of any maturity.

It was particularly attractive that our new model satisfied the law of one price. Our tree functioned as a computational engine that produced the current value of a security by averaging its future payouts; you put future payouts onto the end of the tree, cranked the handle that averaged and then discounted them over the interest rate distribution, and ended up with the current price. The engine didn't care what name you gave to the security that produced the payout—bond, option, call it what you like. As long as the future payouts were identical, the engine produced the same price.

By late 1986, less than a year after I came to Goldman, we had most of the model implemented and running at reasonable speed. Now we began trying it out on actual options traded by the desk, cataloging the ways in which its prices differed from those of Ravi's model. Our traders' intuition had been honed on the old model, and they were sensibly conservative about switching—it is never wise to start using something new until you understand how well it glues on to what you used before. You need a feel for a model before you can begin to rely on it. Therefore, some of the sales assistants in Financial Strategies began to test it, and slowly convinced themselves that it satisfied the law of one price, something that was theoretically obvious to us but not yet practically clear to them.

I was tremendously excited by what we had done. Still at heart a physicist, and still philosophically naive about financial modeling, I half-thought of what we had built as a grand unified theory of interest rates, and imagined we could use it to value every interest rate-sensitive security in the universe.

Fischer, however, disliked this view of our model. More practical and much more experienced, he knew that there were financial forces that lay outside our model; it seemed perfectly possible to him that the model might be good (if it was good at all) for one sector—simple options on Treasury bonds, for example—but not for callable bonds, caps, or a host of other optionlike fixed-income instruments. He called what we had created an “as if” model, by which he meant that we were assuming that the world of bond market investors behaved
as if
only short rates mattered.

Despite my desire to have done something grand, I, too, began to recognize our model for what it was: a simple phenomenological model in the sense that physicists use the adjective, a useful but limited toy that we should take tentatively but seriously, trying to see how far we could push it. We had assumed that markets care only about a single factor, the short-term interest rate, and that all longer-term rates simply reflect the market's opinion of future short-term rates and their volatilities. Was that strictly true? Of course not! The world is indescribably more complex. But what we had was a good starting point from which to capture some of the rational linkages between long- and short-term rates. Our model may not have been
the
real world, but it was a
possible
real world, one of many, and that made its prices interesting.

Fischer wanted the paper we were writing to be clear, accurate and yet not overly mathematical. Over the next year I wrote multiple drafts which he read and then returned with comments, and with the help of his editor Beverly, the paper grew progressively shorter. In my previous life as a physicist I had been a somewhat careless writer with little patience for revision. That year of draft writing taught me the importance and pleasure of formulating concepts qualitatively but precisely, and from then on I was willing to struggle over small things to communicate them clearly.

Our paper,
A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options
, eventually appeared in the
Financial Analysts Journal
in 1990, almost four years after we had developed it. To my pleasure and surprise, it was widely adopted and rapidly became known by the acronym for our names, BDT. It was a pity we hadn't published earlier, but Fischer was a perfectionist, reluctant to release it until he was truly satisfied.

We weren't the only builders of consistent models of the yield curve. At Salomon Brothers, my friend Mark Koenigsberg and his boss, Bob Kopprasch, who later came to head up Financial Strategies at Goldman, had also built a fairly simple model of interest rates, though it wasn't quite as general as ours. Oldrich Vasiçek, always a pioneer, had published the first Black-Scholes-style model of interest rates a good ten years earlier, in 1977; Cox, Ingersoll, and Ross had published a more complex but related model in 1985. There were others, too. So why was our model so widely embraced?

I can think of three reasons. First, we were practitioners and therefore knew what a trading desk needed. While Vasiçek and the Cox-Ingersoll-Ross team had been focused on the general problem of modeling yield curves, we aimed squarely at bond options just when that market was expanding.

Second, our model was easily accessible to practitioners. It was written in a down-to-earth style using the language of binomial trees, a poor man's description of volatility that everyone on Wall Street could grasp. What we described was so close to an algorithm that anyone who took the effort could implement our model as a computer program. Most other contemporary models demanded a stronger grasp of stochastic calculus to understand them and a separate effort at numerical analysis to turn them into a useful computer program; in our model, the medium was the message: The description and the implementation were almost one.

Finally, unlike most prior models that produced analytical formulas that didn't match the shape of actual yield curves, ours could be calibrated to almost any curve, and therefore was ready for the trading world. Indeed, the description of the calibration was a critical part of our paper.

In subsequent years, hosts of new yield-curve models appeared, each known by their authors' acronym, and ingenious academics and quants keep finding new and perhaps more realistic ones. You can now pick from BK (Black and Karasinski), HW (Hull and White), and HJM (Heath, Jarrow, and Morton) or its extension BGM (Brace, Gatarek, and Musiela). The process continues, though which model to employ is still a matter of taste and compromise. Selecting a model becomes a question of finding one that is rich enough to represent most of the risks your product faces, efficient enough to run on a computer in a tolerable amount of time, and simple enough so that programming it is not too complex and burdensome a task.

BDT was simple and consistent, but, like any model, it was not entirely realistic, and its limitations became increasingly apparent over the years. At bottom, it had only a single factor of uncertainty, the future distribution of short-term rates, and so all rates, both short and long, tended to move together, preserving the curve's initial shape a little too well. Therefore, though the model was good for valuing options on bonds, it was not well suited to valuing options on more arcane properties of the yield curve, such as its slope or curvature.

Still, the model's simplicity made it an easy entry point into the art of yield-curve modeling for both practitioners and academics, and it left its mark. It continues to be used even after superior but more complex models have arrived on the scene.

Building the model had been tremendously absorbing, just like the old days in physics. For most of each day I had banged my head single-mindedly against the same problem, thinking about the trees, writing the computer program that embodied it, searching for ways to speed it up, speaking with Fischer and Bill, examining the computer's results, making further modifications. Sometimes I had difficulty sleeping, waking up spontaneously in the middle of the night, unable to return to bed until I had tried out a new scheme.

Unlike me, Fischer was patient. He took an efficient-market attitude towards building models: Each day you looked at what you knew and decided what was the best way to proceed next. On at least two occasions he thought we could do better if we stopped everything and started again from scratch. It was a trait I admired, but I didn't have the stomach for it. Bill and I were new to the field and eager to have our first contribution complete and disseminated.

The first time Fischer wanted to begin all over again was when we learned how to introduce the mean reversion of interest rates into our model, an insight that, not untypically, came from an accidental observation. Until then we had always run the model with equal time periods in the tree. In order to speed up the program, I experimented with varying the period length in the tree, letting it start out small in the present and then grow large in the future. In this way I aimed to make our valuation engine take a detailed look at distributions in the present but only a coarse look at their behavior in the far distant and uncertain future.

When I tried to calibrate these unequally spaced trees to the current yield curve, I ran into trouble. It often became mysteriously impossible to find any distribution of future short-term rates that were consistent with the yield curve. The mystery disappeared when we drew the trees and examined their topology.
Figure 10.7a
shows a tree with equal-length periods. We could now see that the increasing period lengths of
Figure 10.7b
caused the tree to grow visibly wider over time, reflecting a tendency for interest rates to move
away
from their mean, a runaway behavior that doesn't mesh too well with reality. This was what caused the difficulties with calibration. Similarly, if we decreased the future period lengths as shown in
Figure10.7c
, we observed that short-term rates reverted to the mean and the tree tended to grow ever narrower with time. This narrowing tree incorporated a restoring force that stabilized interest rates, akin to what happens in real markets when governments and central banks intervene to stabilize economies.

Figure 10.7
BDT short-rate trees with constant volatility but varying periods.

When Fischer realized that a tree with decreasing periods could describe mean reversion, he wanted us to drop our equal-period version and begin all over again. Bill Toy and I were more anxious to complete what we had started, and eventually we prevailed. The trees with variable periods eventually became part of the Black-Karasinski model.

Fischer was the first to notice the link between period length and mean reversion by examining the topology of the trees, and it made me aware of his intuitive powers. Looking at the tree, he could see the dynamics that its shape implied. It was only much later, after completing our paper, that we worked out the description of our model in the elegant language of stochastic calculus, the way it is now described in textbooks.

On a second occasion, sometime during early 1987, Fischer became entranced by the idea of adding a second stochastic factor to our model. The idea was sensible; our original model had simplistically assumed the entire yield curve was driven by short-term rates, and we were tempted to make the model truer to life by allowing longer-term rates to vary independently of the short-term rate. Adding a second factor in our framework meant working with two-dimensional trees. We played around with calibrating such trees to the yield curve. But trees with two or more dimensions are not only harder to visualize or draw, as you can see from
Figure 10.8
, but also appreciably more difficult to handle in a computer program, and I was again relieved when we postponed this elaboration in favor of completing the delivery of our simpler model to the trading desk.

Figure 10.8
A two-dimensional tree of future interest rates.

Several years later, Fischer, Iraj Kani, and I began investigating a two-dimensional version of the BDT model, but we never completed the work.