My Life as a Quant (34 page)

Just as a clothing designer must factor the cost of labor, cloth, and trimmings into a garment's price, so we had to reflect the cost of hedging in the price we charged for our GER option. The hedging strategy involved daily trading in the Nikkei and the yen, a strategy whose estimated cost we had to include in the price we charged for the Kingdom of Denmark puts.

Black and Scholes had shown in 1973 that the fair value of a standard stock option was the price of hedging it over its lifetime. They derived a partial differential equation for the option's value and had shown how to solve it. Ever since, academics and practitioners had kept busy by extending that insight to all sorts of other options. Now, in late 1989, Piotr Karasinski had found an analogous partial differential equation for the fair value of the Kingdom of Denmark GER put. Much to the surprise and near disbelief of everyone on the desk, he had shown that its value depends on the correlation between moves in the dollar-yen exchange rate and moves in the Nikkei index. It was counterintuitive, almost paradoxical that the degree of correlation between the Nikkei level and the yen should influence the value of a GER Nikkei put whose payoff was designed to be independent of the value of the yen.

When I returned to Goldman in that first month of 1990, everyone was focused on the Nikkei puts, in particular on how to value and hedge them. I soon met Dan O'Rourke, an options trader newly hired by Granny a few months earlier, and now responsible for the daily hedging of the Nikkei book. Dan and I saw the world similarly. First, we both understood that models alone were inadequate, no matter how good they were. What traders need is standardized systems that contain the models, systems that force them to use the models in disciplined ways. Our traders were managing their books on an inadequate and unreliable Lotus spreadsheet that any trader could modify at his whim. It was not the right way to run a business. We felt it was critical to build a risk management system tailored to Nikkei options, analogous to the sort of trading system I had helped build in FSG for the bond options desk, similar to the sort of system Garbasz had tried to create in Chicago. You needed a dedicated computer program to keep track of the hedging of the hundreds of options, futures, and currency positions that constituted our desk's Nikkei volatility book. Second, and more importantly in the short run, Dan and Bill Toy convinced me of the importance of demonstrating to the traders that Piotr's counterintuitive formula for the value of a GER put was correct.

I began by trying to penetrate to the essence of Piotr's formula for the value of a GER option. If you wanted to explain an options formula to a trader, you couldn't resort to stochastic calculus and partial differential equations. Even now, when traders are more likely to have a mathematical background, you still need to have an intuitive way of showing them that a formula makes sense. I'm never satisfied with myself until I understand equations in a low-tech, visceral way. Therefore I decided to forget about deriving the partial differential equation for the Kingdom of Denmark GER option, and instead tried to obtain a more easily understandable explanation of its fair value. Feynman's rules for quantum electrodynamics are a poor man's tool for calculating the probability of complex scattering events correctly. In a lesser but similar way, I wanted to find a set of consistent rules that would let you convince a poor trader that an option formula was right without resorting to the advanced mathematics behind dynamic options replication.

I thought about what the Black-Scholes formula really tells you. In principle, you can derive the formula from the Merton strategy of dynamic replication; from this point of view, the formula dictates in exquisite detail exactly how to synthesize a stock option out of a changing mixture of stock and riskless bonds. But looked at more naively, the formula gives you the fair price of the option in terms of the current price of the stock and the current price of a riskless bond. Its key insight is that the option is a mixture. Like the ancient Greeks' mythological centaur, part horse and part man, a call option is a hybrid, too—part stock and part bond. From this point of view, I came to regard the Black-Scholes formula as a simple and sensible way of interpolating from the known market prices of a stock and a bond to the fair value of the hybrid. Several economists, Paul Samuelson among them, had come almost imperceptibly close to obtaining the Black-Scholes formula before Black and Scholes by means of this kind of reasoning.

When you want to estimate the price of fruit salad, you average the prices of the fruit the mixture contains. In a similar vein, I thought of the option formula as a prescription for estimating the price of a hybrid by averaging the known market prices of its ingredients. So I experimented with making myself a set of rules that a poor man could use for valuing options as mixtures; I tried to see if I could use them to get the right formula for the GER option. I eventually came up with the following rationale.

First, as in any mathematical problem, you have to choose your units, the
currency
in which to quote all security values. You can choose any currency you like—yen, dollars, or even shares of IBM stock as the unit measure of value; it's like deciding on inches or centimeters in reporting heights. The cost of creating an option or an apartment building should not depend on which currency you choose to perform your estimates.

In practice, a little thought often suggests a natural choice of currency that can immensely simplify the problem. To take an example outside the options world, stock market analysts often cite a stock's P/E ratio—its price divided by its annual earnings—as a measure of a stock's value. This is equivalent to using the stock's annual earnings in dollars, rather than the dollar itself, as the unit in which to report a company's stock prices. The stock price, quoted this way, automatically tells you how many years of presumed earnings you are paying for when you buy the stock at its current price.

Many of the advances in options theory in the 1990s involved nothing more than Black, Scholes, and Merton's original insight augmented by the clever choice of a more subtle unit of value than the dollar, a trick first exploited by Bill Margrabe soon after Black and Scholes invented their model. Options theorists of a mathematical bent call this trick “the choice of numeraire.”

Since risk is what you are dealing with, you must then specify the
risky ingredients
of the option, those elements whose future values are unknown. Here, you aim to think of the complex option as a mix of the simplest underlying securities whose risks you understand. Options modelers call these the risk factors. For a standard stock option, for example, the stock price is the major risk factor. For the Kingdom of Denmark Nikkei put, the most important risk factors were the level of the Nikkei and the value of the dollar-yen exchange rate.

Next, you must describe
the range of future scenario
s that the values of the risky securities can take. The range is usually described by several model parameters that will have to be specified when the model is calibrated. Once you know the range of future scenarios for the risky ingredients, you can then estimate the value of any other risky security (an option, for example) as the average of its future payoffs over each of the scenarios, discounted to the present time. In the Black-Scholes model, for instance, one assumes that future stock returns lie on the common bell-shaped distribution familiar to all users of elementary statistics. This distribution is specified when you know the parameters that determine its center and width, or, put more mathematically, its mean and its standard deviation.

You must then
calibrate
the model, which means you must make its scenario parameters consistent with the current prices of the simpler, risky, underlying securities in the natural currency you've chosen. In the Black-Scholes model, calibration means ensuring that when you use the model to calculate the value of the stock itself, you reproduce the current stock price, and when you use the model to calculate the value of a riskless bond, you reproduce its price. This constraint is almost enough to pin down the Black-Scholes formula. Calibration is absolutely critical. Anytime you use your model to calculate the value of a simple risky security whose risk you understand and whose market price you know, your model must match that price—if it doesn't, you're starting off from the wrong place.

Once the model is calibrated, you can use it to calculate the value of the option as a discounted average of its future payouts over the distribution of scenarios. I like to call this interpolation, because the model is used to calculate the “middle” value of a mixture from the known prices of its ingredients at either end.

There is nothing especially original about my prescription, but it does demystify the method of options valuation while still preserving much of the economic logic. I found it a helpful way to explain models to traders, and a useful intuitive way of thinking myself. You can get the answer to many complex derivatives problems with less mathematics than you think.

I now proceeded to apply this logic, one step at a time, to the Kingdom of Denmark GER Nikkei put. Since it paid out in American dollars, I chose dollars as the natural currency. The relevant risk factors were the level of the Nikkei (in yen) and the value of the yen (in dollars). Piotr had assumed the common bell-shaped scenarios of future returns for both the Nikkei and the yen—good enough at that time when no one yet worried about the fat tails in the distributions and their effect on options prices.

To calibrate the model, I chose the values of the respective centers of the bell-shaped distributions of the Nikkei in yen and the yen in dollars so that both the prices for the Nikkei and the yen, quoted in dollars, matched their current market values. Now the model was fully specified and ready for interpolation.

Next I calculated the fair value of the put option by averaging its payout over all future Nikkei and yen scenarios, first converting that payout to dollars at the guaranteed exchange rate. To my great gratification, I immediately obtained Piotr's formula more directly. The value of the option did indeed depend on the correlation between the Nikkei in yen and the yen in dollars. My method was much easier to explain to traders; always on the search for pricing discrepancies, they instinctively grasped that a model had to be calibrated to agree with the market's dollar prices for the Nikkei and yen, securities they could buy or sell at any instant.

In physics I always used to do calculations at least two different ways and see if they agreed. I decided to do that here, too. If my rules were right and if I was careful to use them correctly, I should have obtained the same final answer for the value of the GER put even if I chose an unnatural currency. To confirm this, I perversely decided to choose the yen rather than the dollar as my currency, a more contrived choice of numeraire for evaluating a GER put that pays out in dollars. Now I calibrated the distributions to the market prices of both the Nikkei in yen and the dollar in yen. I worked out the fair value of the Nikkei put warrant in yen, even though it paid out in dollars. When I converted the warrant's final yen value to dollars at the current exchange rate, the answer was identical. Everything was consistent no matter which currency you used to solve the problem.

There was an element of paradox to the appearance of the Nikkeiyen correlation in the final formula, but over time we got to understand better why this correlation was so important. The Kingdom of Denmark Nikkei put's payout in dollars was independent of the dollar-yen exchange rate; it varied only with the level of the Nikkei. But, in order to hedge the put, you had to hedge its exposure to the Nikkei, which required taking a position in Nikkei futures whose value depended on the level of the Nikkei in yen. Owning these Nikkei futures induced a secondary exposure to the yen, which now had to be hedged, too. It was this secondary (yen) hedging of the primary (Nikkei) hedge that induced a dependence on the correlation. Once I included the cost of carrying out both hedging transactions over the lifetime of the option, I obtained the correct value for the GER put.

I was very pleased with my poor man's rederivation of Piotr's elegant result, and ran from trader to quant, excitedly explaining my understanding to anyone who would listen. Over the next few months Piotr, Jeff, and I wrote a paper,
Understanding Guaranteed Exchange-Rate Options
, intended to be the first of a new series of Quantitative Strategies Research Reports. Though it contained some mathematics, it was written in a relatively casual style, intended to educate clients and salespeople about these products. For a brief period I felt that we knew something interesting that few other people understood, and I looked forward to sending our report to academic friends and clients. I was happy to be in the publication business again.

It wasn't that easy, however. When you send clients a research report from an investment bank you have to be careful that nothing you write can be construed as a recommendation that might cause legal problems. Statements about hedging require disclaimers on the limited efficacy of theoretical models in the face of actual markets. Furthermore, you obviously don't want to reveal anything that could damage your business franchise. Therefore, after preparing our report, we took it to the new head of the trading desk for approval.

Several days later he asked us not to distribute the report outside the firm. Though I was frustrated at not being able to communicate what we knew to people in our field, I could see that he honestly felt that option pricing formulas were akin to trade secrets that neither competitors or clients should know about.

I thought he was wrong. The true commercial value lay in Granny's great idea—selling Nikkei insurance that was insensitive to a drop in the value of the yen to American investors who thought that the Nikkei was overpriced. But this idea was already in the public domain, since we had listed the warrants on the American Stock Exchange; sure enough, it was almost immediately copied by other banks.