My Life as a Quant (39 page)

Now we knew how to find every local volatility, step by step. We could select any node on the implied tree, read off the implied volatilities of the three options adjacent to it from the market's implied volatility surface, and then extract the local volatility at that node via our algorithm. In this way, one node at a time, we could find all the local volatilities. With those local volatilities and the implied binomial tree on which they sat, we could value and hedge any option on the index in a manner consistent with the smile. We were immensely elated, believing we had made the next big breakthrough in options pricing, an extension of the Black-Scholes model to make it consistent with reality.

We weren't the only excited ones. Mark Rubinstein and Bruno Dupire had also spent the past year developing similar extensions of Black-Scholes. A few weeks later Mark delivered his presidential address at the January 1994 meeting of the American Finance Association on
Implied Binomial Trees
. Speaking to him over subsequent years, I learned that he too imagined he had achieved a great breakthrough.

At about the same time, John Hull mailed me a copy of a talk Bruno had given on his version of implied trees at a meeting of the International Association of Financial Engineers (IAFE) in New York a few weeks earlier. In it, Bruno also claimed to have found a unique way to extract local volatilities from implied ones.

Each of us—Iraj and I together, Mark and Bruno separately—had tackled the inverse scattering problem in characteristically different styles. Iraj and I were used to dealing with the denizens of the investment banking world, the less numerate traders, salespeople and clients, and so we wrote our paper as simply and clearly as possible. We wanted anyone reading it to know exactly how to build their own implied tree. We explained, step by step, exactly how to build the tree and calibrate it to a given volatility surface, and we illustrated it with a fully worked-out numerical example involving a five-period tree that anyone could check.

Mark's presidential address was more discursive and academic, and covered the history of the problem. His initial solution to the inverse scattering problem focused on matching the market's implied volatilities at a single expiration, thereby ignoring some of the additional information embedded in the implied volatility surface.

Bruno's IAFE talk was the most tantalizing. As a Frenchman, he had a taste for formal mathematics, and his very brief report proposed an elegant formula for the local volatilities in terms of the slope and curvature of the implied volatility surface at the same strike and expiration. His paper wasn't easy to understand and people I spoke to were not sure it was correct. Sometimes I thought he was being intentionally obscure, staking out his claim without revealing exactly how he had derived it.

We studied Bruno's report and soon realized that his concise formula was exactly equivalent to what Iraj and I had developed on a discrete tree. Where he had used calculus, we had used algebra. Though it wasn't anywhere near as important or grand, we were playing Feynman to his Schwinger or Black and Scholes to his Merton. In an appendix to our paper we rederived Bruno's results and included a more transparent proof and a reference to his work.

In late December 1993 both Bruno and Iraj and I submitted our respective papers to Graham Cooper at
Risk
. Bruno's appeared in the January 1994 issue of
Risk
, together with an editorial explaining that ours would follow in February. The January issue also contained a page-long news report by Graham Cooper about the work that Mark, Bruno, and Iraj and I had done, referring to it as the new “supermodel,” and rather accurately describing the relative merits of our individual approaches. We were all suitably flattered.

For the next several years we toured the seminar and conference circuit. I spoke at dozens of university finance departments and business schools, at the Vienna options exchange, and at countless industry conferences. Salespeople in Japan, France, Switzerland, Spain and Italy, countries where the more sophisticated traders loved learning about quantitative theories, took me to see client after client. We gave day-long seminars to large groups in Zurich and London, Bilbao and Paris, Milan and Munich. It was exhilarating.

Iraj and I, together with Mike Kamal and Joe Zou, two more ex-physicists who joined our group in 1994, continued to enhance the model and to use it to value exotic options. We also worked with two software engineers in QS, Deniz Ergener and Alan Buckwalter, to embed the model in the desk's trading software. Most of all, we struggled to abstract the model's mathematical features into a visceral understanding that would make it accessible to traders. Since all their intuition was based on the Black-Scholes model, we developed a series of simple approximate corrections, rules of thumb that traders could use to nudge the Black-Scholes model in the direction of implied trees.

It turned out to be much harder to explain the model to our traders than to our clients. The traders were busy, their lives dominated by watching screens, servicing salespeople, and entering the terms of their transactions into computer systems. Each day they had to book hundreds or thousands of new trades, often staying late into the night to make sure all the details were correctly reconciled between the front-office risk management system on their desks and the back-office mainframe that held the firm's books and records. They were more interested in automating the bookkeeping aspects of their life than in improving their pricing.

The traders were in charge, and used whatever model they wanted without reason or justification. If you wanted them to hedge differently, the burden of proof was on you. They weren't stupid, just sensibly philistine, averse to using new models they didn't understand. Unfortunately, they were also averse to spending time on gaining understanding. When traders have no model at all, it's easy to get them to use the very first model available. Once they have something they rely on, it's much harder to get them to accept an improvement.

So, they simply stuck to using the Black-Scholes single-volatility framework for valuing exotics, even though it produced a flat volatility surface. To compensate, they put all their inventive energy and intuition into picking the “right” single volatility to use in the wrong model. One senior trader insisted that even if the implied tree model were correct and the Black-Scholes model were wrong, he would always be able to think his way through to the correspondingly appropriate single volatility that, when inserted into the wrong model, would nevertheless produce the correct value for an option. I was therefore very pleased one day to discover that there were certain exotic options whose correct value in the implied tree model lay completely outside the range of values we could obtain from the Black-Scholes model, no matter what single volatility we entered into it. For these options there was simply no appropriate single volatility that gave the correct answer from the wrong model. I showed these examples to the traders with great glee and some vindictiveness. It was merely a minor victory, as it failed to alter anyone's behavior.

By 2000, when I worked on approving the use of all models at Goldman, the tide had begun to turn. Increasingly, at Goldman and its competitors, trading desks were obligated to provide some justification that the models they used were appropriate for their market. Nevertheless, in the tug of war between traders and risk managers, traders usually have greater pull.

One day in late 1994 the tension between traders and quants led me to do something particularly stupid. I had spent all day dealing with impatient demands for systems support that I couldn't provide. That evening I took a limo out to Kennedy airport to catch a flight to Vienna, where I was scheduled to give a talk on implied trees at a conference at the OTÖB, the Austrian options exchange. Boarding my plane, I sat down in my business-class aisle seat and finally relaxed. I was frustrated by the constant battles at work, and swore that I would be pushed around no longer.

As I unwound before takeoff, a family of three boarded the plane at the last minute and began taking their seats, none of which were contiguous. The father, a gentleman of about fifty, took the window seat on my right, his son sat across the aisle from me, and his wife sat farther towards the front of the plane. As I flipped through the pages of the OTÖB conference program, the father asked the flight attendant if there were a set of three contiguous seats for his family. Finally, after about ten minutes of unsuccessful agitation, he turned to me and asked if I would switch aisle seats with his son. Still smarting from being pushed around all day, I remembered my “no more Mr. Nice Guy” promise to myself. I was seated in the aisle seat I had requested and I wasn't about to give it up for anyone. Turning to him with misplaced firmness, I said, “I'm sorry, but I'd rather stay where I am.”

As we took off for Vienna, I was appalled at my pointless recalcitrance. Both my seat and his son's seat were on the aisle. I was gaining nothing by being stubborn. And worse, I had now condemned myself to sit for ten hours next to someone to whom I had been unnecessarily objectionable. Guilt began to overwhelm me as I debated with myself how to undo what I had done.

While I writhed, I continued looking through the conference schedule. Then, I noticed, the man on my right extracted a similar schedule from his briefcase and began to look at it, too. I glanced at his face again, and suddenly realized I was sitting next to Bob Merton himself, the developer of continuous-time financial modeling, a Harvard professor and also a partner at Long-Term Capital Management. I had seen his picture just that morning on the inside flap of the cover of his famous book
Continuous Time Finance
when I had been reading about jump-diffusion models.

I turned to him shamefacedly and apologized for my rudeness. By now his son had fallen asleep, and there was no point in exchanging seats, so we talked for a couple of hours about the history of options pricing models. Though Bob was very gracious, I felt mortified at having displayed this unpleasant part of my character, and for the duration of the conference I found myself instinctively avoiding him and his family. Next time, I promised myself, I would be tough on the people who deserved it.

Though Mark, Bruno, and Iraj and I were among the first theorists to use local volatility to tackle the smile, various other people had similar ideas. In particular, I came across a closely connected paper by Avi Bick, an Israeli finance professor at Simon Fraser University in Canada. In November 1994, speaking at an NYU conference
Derivatives: The State of the Art
, I was buoyed to hear Gary Gastineau, himself the author of an options text, comment that our model would make markets more liquid by making options pricing more accurate. Although we did live to see local volatility become a household word and a textbook topic, I discovered that it was much more difficult than I had imagined to create a truly successful financial model.

Local volatility was an improvement on Black-Scholes in that it could account for the smile, but it had three genuine failings. First, our new model excluded the possibility that an index or stock could jump, and most market participants nowadays regard that possibility as the main factor determining the shape of the very short-term volatility smile. Our very first attempt to model the smile had indeed involved such jumps. We were never fond of jump models—since jumps are too violent and discontinuous to be hedged, when you include them you lose much of the coherence of the Black-Scholes model. But jumps are real, and omitting them made our model less realistic.

Second, implied trees were difficult to calibrate. Often, as you tried to build progressively finer-meshed trees for better computational accuracy, the local volatility surface grew wild, displaying unrealistic peaks and troughs as it varied from point to point. Over time we developed methods for smoothing these fluctuations, but the need to smooth them made it difficult to automate the production of implied trees for the desk. These waves and troughs were themselves a partial consequence of having excluded jumps; we were trying to model a violent phenomenon with a tranquil diffusion, something that was bound to make calibration unstable.

Finally, our model ignored the random nature of volatility itself. Iraj and I tried to enhance our implied tree model a few years later by adding a random component to the local volatilities, but it made calibration and calculation even more complex and unwieldy.

Over time we discovered that it is possible to obtain a smile with local volatility, jumps, random volatility, or some mixture of all three. So, local volatility and implied trees didn't become “the” single model of the smile. Years later, at a dinner in Manhattan discussing the past, Mark Rubinstein and I both laughed ruefully at the mismatch between what we had expected and the way things turned out.

Nevertheless, I was ultimately satisfied. Iraj and I had been among the first to propose a consistent model of a new and strange phenomenon; we had created a new framework and vocabulary, and we had done it from the front lines, on Wall Street, not from a leisurely research position in academia. The model was somewhat simplistic, as all models are, and it did not tell the whole story, but it was a plausible, self-consistent little world that captured one true and essential feature of equity volatility markets: Volatility tends to increase when markets fall. Local volatility models have become part of the standard arsenal of tools used by academics and practitioners.

During the 1990s, volatility smiles spread to almost all other options markets. Wherever there was fear of large market moves that could hurt investors—downward for equities, upwards for gold, in either direction for many currencies and interest rates—there smiles and skews appeared, and our model became a key tool in explaining at least some of these features. By the new millenium, smile models were ubiquitous, with
Risk
magazine conferences devoting entire sessions, year after year, to these issues. In the Firmwide Risk area at Goldman Sachs, I ran a team of about twelve PhDs in the Derivatives Analysis group that had to approve the prices set and the models used throughout the firm's derivatives businesses, and we soon found that every desk had its own smile model (all of them different), and that most of our work involved the verification of these models. The models differed from desk to desk because their markets differed; each market had its own characteristic smile for its own idiosyncratic reasons. Equity markets feared a crash; gold markets, after years of low prices, feared a sudden upward move; in interest-rate markets, bond investors feared high rates that would devalue their assets, while insurance companies, who often guaranteed their clients a minimum rate of interest, feared low rates that would diminish their incoming cash flow; in currency markets, investors feared a move outside some stable band. Each fear, based on bitter experience, corresponded to a different pattern and required a different model. There was no single model for the smile.