My Life as a Quant (37 page)

We knew that the Black-Scholes model oversimplifies the behavior of stock prices. It assumes that a stock price
diffuses
away into the future from its current value in a slow, random, continuous fashion, much like a cloud of smoke from the smoldering tip of a cigarette spreads through a room. Dense near the tip of the cigarette and sparse farther out, the intensity of the smoke cloud at a point represents the probability that a particle of smoke will diffuse to that location. In the Black-Scholes model, a similar cloud depicts the probability that the stock price will reach some particular future value at some future time.
Figure 14.4
illustrates that cloud of probabilities for a stock in the Black-Scholes model. The more dispersed the cloud, the more uncertain the future stock price. A single parameter, the stock's volatility traditionally denoted by the Greek letter sigma, s, determines the rate of diffusion and the width of the cloud. The greater the stock's volatility, the wider the cloud.

Figure 14.4
Simple diffusion in the Black-Scholes model. The shaded regions illustrate the range of possible future prices for a stock whose price is $100 today. The more time passes, the greater the uncertainty in the future price. The darker the shading, the more likely the price will be in that region.

Though simplification is the essence of modeling, the Black-Scholes picture of smoke-cloud diffusion is too restrictive. First, stock prices don't necessarily diffuse with a constant volatility; at some times a stock diffuses more rapidly than at others. Second, and more gravely, sometimes stocks
don't diffuse at all
. Diffusion, as shown in
Figure 14.4
, is a slow, continuous process; in diffusion, a stock price that moves from $100 to $99 passes through every possible price between them. This is not what happened during the 1987 crash, however; on that day the Dow-Jones index jumped its way downward through 500 points like an excited kid on a pogo stick.

Returning to New York from Tokyo, I began working with my QS colleagues Iraj Kani and Alex Bergier. I wanted to extend the Black-Scholes model just enough so that it could incorporate the smile. “Just enough” was always the aim. A model is only a model; you want to capture the essence of the phenomenon, not the thing itself. It is far too easy, in the name of realism, to add complexity to the simple evolution of stock prices assumed by Black and Scholes, but complexity without calibration is pointless.

The overwhelming fear of equity investors was another 1987-style crash, so we added this possibility to Black-Scholes. This was not something new; Merton had done it in his so-called jump-diffusion model in the mid-1970s, and to begin with, we did it even more crudely than he had. To the constant diffusion of stock prices we added just one new feature, the small probability
p
that the stock price might take one sizeable jump
J
downward. The probability cloud for this process is depicted in
Figure 14.5
. It shows the two scenarios the stock can now take: a jump J downward and subsequent diffusion with a volatility σ
H
, which is likely to be high because of the after-excitement of the crash; or, more likely, continued diffusion with the normal, lower volatility σ
L
.

Figure 14.5
The range of possible future prices for a stock that can jump once and then diffuse. The darker the shading, the more likely the price will be in that region.

Typically, we assumed the probability
p
to be of order one percent, implying that the market assigned about a one-in-a-hundred chance of a crash during the option's lifetime. We chose the σ
H
to be about 40 percent greater than σ
L
, based on a combination of intuition and experience about the after-effects of stock crashes. Our model now had only two unknown parameters, the jump size
J
of the crash scenario and the volatility s
L
associated with normal behavior. This was just one parameter richer than Black-Scholes, which contained only a single volatility. We calibrated these parameters by matching the model's option prices to the two implied volatilities that defined the shape of the three-month smile, those of an at-the-money option and a put option struck 5 percent out of the money. With a normal volatility s
L
in the vicinity of ten percent and a downward jump
J
of about 25 percent of the current stock price, we found we could produce smiles like those in
Figure 14.1
.

Our model looked at the world like this: During the option's lifetime the Nikkei had a roughly one-in-a-hundred chance of dropping about 25 percent. This is why you paid so much more for an out-of-the-money put. We then used the model to estimate the option's delta, the hedge ratio necessary to cancel its index risk. We also used it to value the more illiquid or exotic options that were becoming increasingly fashionable—barrier options, for example—whose prices were highly sensitive to the probability and size of the jump. We wanted our traders to search the market for options whose prices differed significantly from those produced by our model. They could then buy the apparently cheap ones and sell the rich ones in the hope that these outliers would eventually revert to our model's prices and generate a profit.

Though the jump model captured one essential sentiment responsible for the smile, it was ultimately too crude. Its view of a future in which the Nikkei awoke each morning and decided either to make one large instantaneous excited jump downward or else diffuse calmly was still too simplistic. Looking back, perhaps we should have added a distribution of possible jump sizes and jump times. But jumps occur rarely, and since there was little data on their distribution, we would have had to make many unverified assumptions, something that felt unaesthetic. Rightly or wrongly, we preferred a more constrained model whose parameters were totally fixed by calibration to observed options prices. Ten years later, though, more detailed jump-diffusion models of the smile became popular again.

Our initial model of the smile did find users in Goldman's risk arbitrage group, where savvy traders combined worldly knowledge with quantitative methods to take educated bets. Some arbitrageurs focused on acquisitions in which the acquiring company tenders for the stock of a target company at a public offer price that substantially exceeds its current level. If and when regulators approve the acquisition, the target's price will jump to the offer price. Until then, its price reflects an estimate of the probability of completion of the deal. For these situations, our jump model was a theoretically accurate picture, and the risk arbitrageurs made occasional use of it to see whether their estimate of the likelihood of the acquisition's approval matched the jump probability implied by the current price of the target.

Meanwhile, from mid-1991 through early 1993, Iraj and I and the rest of QS turned temporarily to the more pressing problem of enhancing our risk systems to handle the increasing number of exotic options we traded.

Unfortunately, the more we worked on exotics, the more we ran into the problems of the smile: Whenever we used the Black-Scholes framework to value the exotic options in the desk's book, we were using a model that produced the wrong value for much simpler standard options, a model inconsistent with the smile. This wasn't good—you can't trust a model for complex phenomena if it gets the simple stuff wrong. You wouldn't trust a NASA computer program that predicted the trajectory of an interplanetary probe from Earth to Mars if it couldn't first correctly predict the orbits of Earth and Mars around the sun.

The right place to start was with a model that could match the market prices of all standard options, the entire implied volatility surface. Only then, when it was correctly calibrated, could you sensibly use it to calculate the value of an exotic. How could we find a model that matched any surface?

I thought back to our development of BDT. In the mid-1980s, the fixed-income options world had undergone a similar crisis: Practitioners used a yield diffusion model like Ravi's to value an option on any single bond, but felt uncomfortable because it couldn't simultaneously match the prices of all Treasury bonds on the yield curve. BDT was one of the possible solutions to this dilemma.

We had a tremendous advantage in having come to the equity derivatives world with a background in fixed income. Iraj and I perceived the following analogy between bonds and their yields and options and their volatilities:

• Bond prices are quoted using current long-term yields, which reflect the market's expectation of future short-term rates.
  
• Options prices are quoted using current long-term implied volatilities, which reflect the market's expectation of future short-term volatilities.
  

Our ambition was to build a post-Black-Scholes model that allowed us to back out the market's expectation of future short-term volatilities from the current volatility surface. We weren't sure how to do it, but we knew that the world needed a better model and would reward its discoverers. Throughout 1993 we felt as though we were in a race with unnamed competitors to find it.

Iraj and I were great admirers of the binomial options model, a simple, picturesque, and yet accurate way of performing options-theoretic calculations on a gridlike tree of future stock prices. On a binomial tree, prices move like knights on a chessboard, one discrete step forward in time and up or down a notch in price. Binomial trees are easy to draw and, in a jerky way, mimic the behavior of real prices or indexes. As the grid of the chess board becomes progressively finer, prices move more and more continuously—they start to diffuse, in fact—and the binomial model becomes equivalent to the Black-Scholes model. Binomial trees were the Feynman diagrams of options theory, easy to picture and use, wonderful for simulating simple trading strategies or developing valuation models. Even the innumerate traders with whom we often dealt could understand them. Initially invented by William Sharpe soon after Black and Scholes wrote their paper, bimonial trees were then cleverly elaborated upon by John Cox, Mark Rubinstein and Steve Ross. As options theorists grew increasingly professional and better educated, binomial models fell into a low-tech disrepute, but we still found them immensely useful.

Therefore, we tried to use a binomial tree of index options prices like the one shown in
Figure 14.6
as a guide to extracting the market's view of future short-term volatilities. The left edge of the tree denotes the current index level. Each step up or down from there illustrates a potential future index move. Traditional binomial trees make the key assumption that all moves on the tree are of equal percentage magnitude; at any future time, at any future level, the index, whether it moves up or down, grows or shrinks by an identical percentage. In technical terms, the index has a constant volatility of returns, globally the same across the entire tree, identical at each future instant of time and index level. This constancy of the index's volatility in the Black-Scholes model leads to the associated flat implied volatility surface that is inconsistent with actual options markets.

Figure 14.6
At left, a binomial tree of future index moves. Each future percentage move is identical and represents a constant index volatility. At right, the shape of the corresponding implied volatility surface.