My Life as a Quant (36 page)

Early in the 1990s, most options were written on simple global equity indexes, like the S&P 500 in the United States or the FTSE 100 in Great Britain; the exotic nature of the option lay not in the nature of the underlyer, but rather in the definition of the way in which the payout of the option depended upon the level of the underlyer. During the technology and biotech bubbles of the late 1990s, investors became more interested in options on complex underlyers, baskets of technology or pharmaceutical stocks, for example. As firms tried to outdo each other, options payouts became more refined and harder to apprehend. By the end of the decade, the most popular options were being created at French banks by elite École Normale Supérieure–trained mathematicians with a characteristically French taste for formal mathematics. The exotics they marketed were options on baskets of stocks whose actual composition kept changing over time. In one, for example, the number of constituent stocks in the basket each year decreased as the best-performing stock of the previous year was eliminated. As in any business, the sales-people liked tailoring and complexity because not only could you charge more for it, but it was also more difficult for a client to assess the value of its individual features. Complexity was also harder for competitors to copy.

Variations flourished. Riding the crest of this wave of innovation, and simultaneously propelling it, was Peter Field's inspired venture,
Risk
magazine, to which we turned each month for its mix of industry news, gossip, and quantitative articles. It was the first glossy in the options world, replete with advertising and cover art, a magazine aimed at quantitative practitioners rather than academics or money managers. It spurned the suffocatingly rigid format of academic periodicals like the
Journal of Finance
, and both academics and practitioners loved it. Fischer, too, commented admiringly on it. For several years in the early 1990s each issue contained topical articles on the newest exotic structures and how to value them. Soon
Risk
was organizing expensive courses on exotic options, a clever arbitrage by which they charged quants from one set of investment banks to listen to the lectures of quants from another set, while
Risk
pocketed the fee.

Other journals sprang up, too. The International Association of Financial Engineers, a new professional organization of quants, began to specify an educational syllabus suitable for the training of a quant. New textbooks and Master's of Financial Engineering programs sprouted and flourished at the increasing numbers of universities that, sensing a need, began to cater to those willing to pay to learn quantitative finance. In 1985 when I entered Wall Street it was amateur heaven, a fluidly makeshift field filled with retreads from other disciplines who could learn quickly, solve equations, and write their own programs. You had to learn options theory on your own and there were only a few textbooks to help you—those written by Jarrow and Rudd and Cox and Rubinstein were the only ones I could obtain. The only derivatives meeting I went to each year was the annual spring meeting of the Amex. By the late 1990s, there were scores of master's programs, hundreds of conferences, and thousands of books. Physicists and mathematicians, either unable to find academic positions or, grown weary of academic politics and pay, increasingly sought out Wall Street jobs.

The quantitative life of practitioners, formerly the happily casual domain of self-taught amateurs, was becoming a discipline, a business, and a profession. It was simultaneously becoming a little less fun, too.

The puzzle of the volatility smile

Beyond Black-Scholes: the race to develop local-volatility models of options

The right model is hard to find

I first heard about the smile in December 1990 from Dave Rogers, our head options trader in Tokyo. I had begun traveling to Japan regularly to bring our traders the latest releases of our risk management tools and to learn what new models and software they needed. Unlike the New York Stock Exchange, the Tokyo market closed at midday; traders grew less frenetic and went out for lunch, salespeople departed to meet clients, and there was time for leisurely conversations. While we chatted, Dave showed me the computer screen he used to watch the prices of options on the Nikkei 225 index. He pointed out a peculiar asymmetry in the Nikkei options prices: The prices of out-of-the-money puts were unexpectedly larger than those of other options.

Everyone referred to this asymmetry as “the smile,” or “the skew.” At first it looked only mildly interesting, a peculiar anomaly we could tolerate. Then, when I thought about it a little more, I realized that the existence of the smile was completely at odds with Black and Scholes's twenty-year-old foundation of options theory. And, if the Black-Scholes formula was wrong, then so was the predicted sensitivity of an option's price to movements in its underlying index, its so-called “delta.” In this case, all traders using the Black-Scholes model's delta were hedging their option books incorrectly. But the very essence of Black-Scholes was its prescription for replicating and hedging. The smile, therefore, poked a small hole deep into the dike of theory that sheltered options trading. If Black-Scholes were wrong, what
was
the right delta to use for hedging an option?

During the 1990s, the smile, initially a peculiarity of equity options, infected other markets, taking a slightly different form in each one. Understanding it became a dominating obsession for me and many of my quant contemporaries. It was an anomaly that sat right at the intersection of options trading and options theory, and I spent much of my intellectual energy trying to model it.

Our work started in the typical heady rush of energy and ambition that made me feel as though I were in physics again, racing to be the first to find the “right” model for something important and interesting. I fantasized about building a model that, embraced by everyone, would replace Black-Scholes. It wasn't as simple as I thought. During the next ten years I learned that “rightness” in financial modeling is a much fuzzier concept than I had imagined.

One of the things you learn repeatedly in a career in financial modeling is the importance of units. You always want the prices of securities to be quoted in a way that make it easy to compare their relative values.

When you need to compare the values of bonds, for example, their prices are insufficient, because each bond can have a different maturity and coupon. Instead, you quote their yields. A bond yield provides an estimate of the return the bond will generate for you irrespective of its coupon and maturity. You may not know whether a discount bond at 98 is better than a premium one at 105, but you do know that, all other things being equal, a yield of 5.3 percent is less attractive than 5.6 percent. This conversion of prices to yields is itself a model, albeit a simple one. It is a convenient way to communicate prices, as well as a good first step towards estimating value.

In the options world as well, price alone is an insufficient measure of value; it's impossible to tell whether ¥300 for an at-the-money put is more attractive than ¥40 for a deep out-of-the-money put. A better measure of value is the option's implied volatility. The Black-Scholes model views a stock option as a kind of bet on the future volatility of a stock's returns. The more volatile the stock, the more likely the bet will pay off, and therefore the more you should pay for it. You can use the model to convert an option price into the future volatility the stock must have in order for the option price to be fair. This measure is called the option's implied volatility. It is, so to speak, an option's view of the stock's future volatility.

The Black-Scholes model was the market standard. When I sat next to Dave in Tokyo that day, his computer screen showed the prices quoted in Black-Scholes implied volatilities. Even today, when no one believes that the Black-Scholes model is absolutely the best way to estimate option value, and even though more sophisticated traders sometimes use more complex models, the Black-Scholes model's implied volatilities are still the market convention for quoting prices.

Options are generally less liquid than stocks, and implied volatility market data is consequently coarse and approximate. Nevertheless, Dave pointed out to me what I was already dimly aware of: There was a severe skew in the implied volatilities, so that three-month options of low strike had much greater implied volatilities than three-month options of higher strikes. You can see a sketch of this asymmetry in
Figure 14.1
. This lopsided shape, though it's commonly called “the smile,” is more of a smirk.

Figure 14.1
A typical implied volatility smile for three-month options on the Nikkei index in late 1994. The dashed line shows the lack of skew that was common prior to the 1987 crash.

With implied volatility as your measure of value, low-strike puts are the most expensive Nikkei options. Anyone who was around on October 19, 1987 could easily guess why. Ever since that day when equity markets around the world plunged, investors remained constantly aware of the possibility of an instantaneous large jump down in the market, and were willing to pay up for protection. Out-of-the-money puts were the best and cheapest insurance. Like stableboys who shut the barn door after the horse has bolted, investors who lived through the 1987 crash were now willing to pay up for future insurance against the risks they had previously suffered. By 1990 there were similar smiles or skews in all equity markets. Before 1987, in contrast, more light-heartedly naive options markets were happy to charge about the same implied volatility for all strikes, as illustrated by the dashed line in
Figure 14.1
.

It was not only three-month implied volatilities that were skewed. A similar effect was visible for options of any expiration, so that implied volatility varied not only with strike but also with expiration. We began to plot this double variation of implied volatility in both the time and strike dimension as a two-dimensional implied volatility surface. A picture of the surface for options on the Standard & Poor's (S&P) 500 index is illustrated in
Figure 14.2
. Like the yield curve, it changes continually from minute to minute and day to day.

Figure 14.2
A typical implied volatility surface for the S&P 500 in mid-1995.

This tentlike surface was a challenge to theorists everywhere. The Black-Scholes model couldn't account for it. Black-Scholes attributed a single volatility at all future times to an index or a stock, and therefore always produced the dull, flat, featureless surface of
Figure 14.3a
. The best you could do, if you modified the Black-Scholes model to allow future index volatility to be different from today's, was to obtain a surface that slanted in the time direction, as depicted in
Figure 14.3b
. But the variation in two perpendicular directions, time and strike, was a puzzle. What was wrong with the classic Black-Scholes model? And what new kind of model could possibly explain that surface?

Figure 14.3
Implied volatility surfaces. (a) In the standard Black-Scholes model. (b) In an enhanced Black-Scholes model where volatility varies with time to expiration.