My Life as a Quant (41 page)

Each day, like clockwork, we computed and reported the VaR for the firm and its parts. Several times a week senior members of our group met with each division's risk committee, and once a week, in an unpleasantly early 7:30 A.M. global conference call, we met with the central risk committee who fine-tuned the entire firm's risk by setting limits on VaR. During turbulent times they lowered the cap and during quiet times they raised it. The aim was to take the appropriate amount of risk for the economic environment, not to eliminate it. No risk, no return.

VaR is not a panacea, and there are many legitimate objections to its use. Most importantly, VaR by its nature uses statistical distributions to project the firm's value into the future, and statistics are inevitably based upon the past. But the past doesn't repeat itself verbatim; people learn enough from their past experiences to make new mistakes as they struggle between greed and the effort to avoid the old ones.

VaR is also too simplistic: it's not possible to capture the profit-and-loss potential of a complex distribution of future portfolio values in a single number, because very different distributions can have the same 99.6 percentile. We are ignorant of the true probabilities—the extreme tails of stock and bond price distributions, not to mention the overall distributions of complex securities like swaptions or weather derivatives, are poorly understood and may not even be stationary.

Even if you insist on representing risk with a single number, VaR isn't the best one. Percentiles don't reflect the psychology of risk perception very well—two different securities can each have their own small percentile losses that combine to produce a greater percentile loss for the portfolio. The VaR of a portfolio can therefore be greater than the VaR of its components, in counterintuitive contradiction with the idea that diversification diminishes risk.

As a result, though we used VaR, we didn't make it our religion. We were pantheists, praying to many parallel risk gods. For example, we had all lived through various market meltdowns—the 1987 stock market crash, the 1998 Russian default crisis—whose reoccurrence would cause great losses even though we had no idea of their probability. Thus, the firm imposed a bound on the positions on each desk so that they could lose no more than a tolerable amount in a repetition of such events. These and other stress-test limits based on similar experienced or imagined catastrophes were imposed over and above the VaR limits.

I think we did a pretty good job by taking a many-sided view of risk, but it was a never ending enterprise. Each year we collected more data to improve our statistics. Each year the simulations were enhanced to better reflect the behavior of familiar markets. And, whenever Goldman entered a new business—energy or weather derivatives, for example—that had no history, we strived to create an imagined history and statistics that would reflect the risk we perceived. So much of financial modeling is an exercise of the imagination.

Though most of Firmwide Risk focused on VaR, I didn't have much to do with it. I found it useful but inelegant work, aimed more at an audience of regulators rather than traders. Instead, I spent most of my time in Firmwide Risk as head of the Derivatives Analysis group, a collection of about twelve PhDs in New York, London, and Tokyo. Our mission was to ensure that every trading desk's derivative deals were “marked to market” accurately.

Marking to market is the act of assigning to each security in your portfolio a price that reflects its current market value. You have a very good idea of the value of a share of Microsoft, which trades millions of times a day at a publicly listed price. Marking it to market is easy. But the derivatives trading desks at Goldman (and most other investment banks) had developed increasingly large positions in long-term or exotic over-the-counter derivative securities, tailored to satisfy the needs of particular customers. It was harder to mark them to market. It's much simpler to price a new pair of Levi's than a secondhand, custom-tailored Christian Lacroix evening dress.

Our firm owned hosts of illiquid derivatives in the interest-rate, equity, currency, commodity, energy, and credit markets. Some structured products spanned many markets—we traded yen-denominated bonds whose coupons would increase if the S&P 500 index rose. To manage this position you had to hedge changes in Japanese and American interest rates, the yen, and the S&P 500.

There
were
no current market prices for such securities. Quants and traders “marked them to model,” which meant they figured out their value by means of carefully calibrated models, developed with mathematics and written in computer code that was then embedded in front-office risk management systems. And there was no secondary market for most of these exotics—you had to keep them on your books until they expired, and hedge them by means of the same model along the way.

What is the fair value of an illiquid security that you have to hold for several years? This was an immensely significant and practical question, because the value of these securities determines the company's earnings, its stock price, and the bonuses of the traders that manage them.

The trading desks typically valued their illiquid positions using their own models. But this involved a moral hazard; when a trader's year-end bonus depends upon the value of a security whose price is both obscure and under his control, he may be tempted to embellish his profit when payday approaches.

The Derivatives Analysis group was the firm's model police: Our job was to confirm that the billions of dollars of exotic or illiquid derivatives were being marked fairly. It's an honorable and important task, one that requires both knowledge and wisdom. That first year in Firmwide Risk I saw a plethora of idiosyncratic and illiquid securities on the firm's books in New York, London, and Tokyo. The Equities Division was long out-of-the-money, four-year calls and short out-of-the-money puts on individual technology stocks, sold to them by insiders who were hedging the stock they had received when their firm went public. The value of these options was somewhat uncertain because there was no listed market at those distant expirations and strikes. The fixed-income division had much more complex swaptions on their books, similar to those yen-denominated bonds with coupons indexed to the S&P 500. The commodities desks traded long-term barrier options on gold, structured to help mining companies hedge their future profits against gold price declines. All of these markets manifested volatility smiles that made the accurate valuation of these highly complicated securities difficult and uncertain.

Slowly it began to dawn on me that what we faced was not so much
risk
as
uncertainty
. Risk is what you bear when you own, for example, 100 shares of Microsoft—you know exactly what those shares are worth because you can sell them in a second at something very close to the last traded price. There is no uncertainty about their current value, only the risk that their value will change in the next instant. But when you own an exotic illiquid option, uncertainty precedes its risk—you don't even know exactly what the option is currently worth because you don't know whether the model you are using is right or wrong. Or, more accurately, you know that the model you are using is both naive and wrong—the only question is
how
naive and
how
wrong.

No one truly knows the precise value of a stock option that cannot be traded. Its value depends on what you can squeeze out of it by hedging its underlying stock through all its future random price changes to expiration. And this depends on the nature of the randomness and on the particular strategy employed to hedge against it, both of which cannot be totally determined in advance.

Faced with uncertainty, I took a many-worlds view of derivatives values. I assumed that the future could be one of several foreseeable worlds, and that there was consequently a range of possible values one could legitimately assign to the option. In one future world, for example, stock volatility might decrease as stock prices increased. In another, volatility might simply be random. In each of a host of plausible different worlds, all consistent with today's observations, you can work out the correspondingly appropriate theoretical model and then use it to value the option. You can then compare the desk's mark-to-model value with the spectrum of many-worlds values, and see whether it lies inside their range. If it does, then it's reasonable and can be allowed as a mark. But if it doesn't, you negotiate with the desk to either justify or amend its estimate. Furthermore, since there is a range of plausible values, we recommended that Goldman hold in reserve an amount equal to the average mismatch between the range of plausible values and the desk's actual mark. This reserve represented a portion of their expected profit to be held in escrow against the possibility that their model or hedging strategy was wrong. It would be released only when the trade was finally unwound and the true market value of the option would finally be realized and revealed.

This was the strategy we adopted for appraising the firm's exotic and illiquid derivatives, which we regarded in the same way that an art dealer might think about his inventory of impressionist painting. Suppose an art dealer's agent has acquired a valuable Renoir painting for $10 million when he believes it to be worth $13 million, and now asks to be paid a bonus of ten percent of the expected profit of $3 million, or $300,000. It would be foolish of the dealer to pay the bonus until the Renoir has been sold. Until then, the painting's value is not merely risky, but uncertain. Illiquid option or illiquid painting, you shouldn't count your chickens until they're hatched.

The more I thought about it, the more that appraisal seemed like the right metaphor for options valuation. When you want to estimate the value of an antique that hasn't changed hands in a long time, you compare it to analogous
objets d'art
that have been auctioned more recently. The more I dealt with estimating the value of the panoply of option structures and models we came across, the more I believed that options valuation was really valuation by analogy.

I recalled that as a child in bible school, I had learned the story of Hillel, a famous sage, who was asked to recite the essence of God's laws while standing on one leg. “Do not do unto others as you would not have them do unto you,” he is supposed to have said. “All the rest is commentary. Go and learn.” I believe that you can summarize the essence of quantitative finance on one leg, too: “If you want to know the value of a security, use the price of another security that's as similar to it as possible. All the rest is modeling. Go and build.”

Financial economists grandiosely refer to this law as the
law of one price
, which states that securities with identical future payouts, no matter how the future turns out, should have identical current prices. It's the essential—perhaps the only—principle of the field. To estimate the value of an illiquid security, you find a set of similar liquid securities, with known market prices, whose payouts match those of the illiquid security under all circumstances. The best estimate for the value of the illiquid security is then the value of the set of liquid securities with the same payouts.

Where do models enter? It takes a model to show that the illiquid security and the liquid portfolio have identical future payouts under all circumstances. Your model must specify what you mean by “all circumstances,” and you must show that the replicating portfolio, in every future circumstance, has the same payout. Most of the mathematical complexity in finance involves the elaboration of this single principle.

Models are only models, toylike descriptions of idealized worlds. Simple models envisage a simple future; more sophisticated models incorporate a more complex set of future scenarios that can more closely approximate actual markets. But no mathematical model can capture the intricacies of human psychology. Watching traders occasionally put too much faith in the power of formalism and mathematics, I saw that if you listen to the models' siren song for too long, you may end up on the rocks or in the whirlpool.

In September 2000 I was elected Financial Engineer of the Year by the International Association of Financial Engineers (IAFE). I was the first and, thus far, only practitioner to receive this award, lucky to be temporarily in the company of the previous winners: Robert Merton, Fischer Black, Mark Rubinstein, Stephen Ross, Robert Jarrow, John Cox, and John Hull, all renowned contributors to the field who had made their major contributions while in academic life. I considered myself fortunate to have been able to make a small mark after a late start.

As a practitioner I had always been hands-on. What I had enjoyed most was research, the primary work that no one has done before, carried out for a small number of people in trading who were genuinely interested in the result. Now, in Firmwide Risk at a large firm, I had to accustom myself to doing secondary research, merely helping to validate the results of other quants on the desk who had taken the first crack at a problem. I was also part of a large bureaucracy. Each week I went to two fixed-income risk meetings, one equities risk meeting, one firmwide risk committee meeting, at least two Derivatives Analysis group meetings, and a meeting of all the managers in Firmwide Risk. Then there were three different meetings with three different controllers for equities, fixed income, and currencies and commodities respectively, as well as the periodic meeting of all the VPs in Firmwide Risk.

That was when times were good. By mid-2000, after the bursting of the technology bubble and the subsequent decline in all stock markets, I had many more meetings with discouraged young quants in my group who foresaw a very limited upside. By early 2001 I was spending a large fraction of my time trying to cheer up disgruntled but talented people.

There was one real perquisite to being a senior person in Firmwide Risk—you got to participate in the firm's central risk meeting once a week and watch all the biggest big shots in action as we listened to the state of our business prospects and discussed current events and strategies. I was invited to hear Wesley Clark address us on Iraq, more than a year before the invasion. In the end, however, I was an outsider. Running the firm was their world, not mine. I liked smaller worlds and I preferred working on more specific and detailed topics. Soon, I knew, I would move on.