I first met Fischer in 1986, several months after I came to work in Fixed Income Research at Goldman, Sachs & Co. I knew little about options. But, as an ex-physicist who had spent a few years in software development at Bell Laboratories, I was considered to have the right mix of analytic and computer skills for working on these sorts of problems, so Ravi Dattatreya assigned me to work for the rapidly growing business in over-the-counter Treasury bond options.

The straightforward and simplistic way to model bond options then was to replace the stock price in a Cox-Ross-Rubinstein binomial tree with the bond price or, more strictly, the bond present value. This stochastic price model has many limitations, but the most disturbing is the discrepancy between the future uncertainty of the bond price in the model, which increased (like a stock price) without limit as time passed, and the certain and exact "convergence to par" at maturity of actual bond prices. Consequently, the model was inappropriate for valuing options that expire close to the maturity of the underlying bonds, and was reasonable only for short-term options on long-term bonds.

Various researchers had attempted to remedy this problem by constraining the bond's price volatility to decrease parametrically with time. More practically, earlier the previous year at Goldman Sachs, Ravi had created the then-current Goldman Sachs bond options model by ingeniously converting the so-called price-volatility model into a "yield volatility" model. To do this, he used the underlying stochastic state variable in the binomial tree to represent the yield to maturity of the bond on which the option was written. The uncertainty in yield also grew limitlessly larger with time, but, since any value for the yield at maturity corresponds to a price of par, the damage was contained, and the price distribution close to maturity was more realistic.

This picture worked well for making markets in individual bond options. Using the yield and its volatility as a fundamental modeling factor mirrored a trader's economic and financial intuition, and the computer implementation was simple and ran quickly. I believe several other groups on Wall Street came up with similar models around the same time.

When I started working at Goldman Sachs, the model still had several problems. The first was practical: The model had no graphical user interface, and running it repeatedly from the command line by entering all the necessary arguments made for slow market-making. Ravi had me rewrite the model in C and add a friendlier interface. From then on, it was hard to forget the difference a functional user interface and a flat-file data base for storing trades-in-progress makes to a desk's capacity to develop rapid quotes and business on a variety of coupons and maturities.

At this point, another problem became apparent. As we pushed the use of the model to longer maturities, a violation of put-call parity showed up above the background hum of the numerical methods. Upon careful inspection of the algorithm, we noticed a small oversight in the way the model used flat price plus accrued interest in calculating the risk-neutral probabilities in the binomial tree, and corrected it.

This careful inspection of put-call parity threw the spotlight on a more fundamental difficulty. The yield volatility model for bond options required two rates as input: the current yield to maturity of the bond, which specified its price, and the current zero-coupon discount rate for the expiration of the European-style option that determined the present value of the payoff. In the simple version of the model, only the long-term yield varied stochastically. In the more realistic version, Ravi had linked the evolution of the short rate to the evolution of the long rate by making them both evolve proportionately, which corresponds roughly to constraining all future yield curve movements to be parallel shifts.

While this added some semblance of realism to the model, it introduced arbitrage violations: The stochastically varying discount rate led to an average effective discount rate over the tree that didn't match the initially entered market discount rate that corresponded to the option's expiration. This meant that European-style put-call parity was violated.

I experimented with tuning the initial discount rate to force it to satisfy put-call parity, which actually turned out to work quite well. But this trick meant you had to retune the assumed discount rate to the option's expiration for each new underlying bond. Tuning was difficult if not impossible for American-style options, because of their uncertain exercise date.

It was then that I met Fischer. Although he was head of the Quantitative Strategies group in the Equities Division, Fischer was the expert for all options at Goldman Sachs. The fixed-income options traders had already consulted him about building a more advanced options model. I was sent to meet him.

He quizzed me lightly, in a sort of informal interview, as I showed him the model and the software we were using; Fischer wasn't the sort of person to ask you a series of pass-fail test questions. He and Bill Toy had already started investigating a model in which you calibrate the instantaneous short rates to the yield curve and its volatility, and I joined them.

When you knocked on Fischer's door and entered his office, it was usually quiet. Fischer didn't seem to allow his work life to be dominated by the multiplexing, interrupt-driven set of precedences that became a way of life for anyone who'd been at an investment bank for several years. He made time for you, and never seemed to be in the prototypical frantic rush.

Most often, you'd find him reading, or on the telephone, or sitting with his PC keyboard on his lap, swiveled in his chair to face the PC behind him, entering notes in Thinktank, a hierarchical outliner he used constantly. Fischer was organized. He filed his notes in Thinktank, and people I knew in his group at that time claimed that he conducted a continuing and eager correspondence with the designers of Thinktank to suggest the addition or modification of features he'd like.

If you said something he found useful, he'd write it down with his fine-pointed mechanical pencil on a fresh sheet of his ruled white pad, and then insert it into a newly labeled manila folder, which eventually went into one of his file drawers. He wore a Casio information-storing watch (very un-Wall Street then, when important people had desks without computers, or even offices without desks), and several people who worked for him followed his unfashionable lead.

It was Fischer's meticulous devotion to clarity and simplicity of presentation, in speaking as well as writing, that struck me most. (This is not to say that he wasn't equally dedicated to truth and accuracy in modeling; it's just that this wasn't so surprising.) Fischer's clarity nevertheless took a very shorthand form - if you didn't understand what he said, he often simply repeated it, as though he couldn't do any better than tell you the truth as he saw it, leaving you to come to recognize it. He didn't often try to persuade.

When I went to show Fischer the yield volatility model we were using, I got my first taste of his devotion to presentation style. I logged him into the VAX on a VT-100 terminal on his PC. Almost as soon as I started to demonstrate the interactive program, the VAX crashed, and we were left with a dead screen image of the interface to the bond options calculator. You couldn't toggle or change the values of any of the fields, but you could look at them carefully.

I offered to come back later when the VAX was up, but Fischer spent the next hour with me looking at the screen, commenting on the layout of the different areas that corresponded to input and output areas for the bond and the option. He pointed out to me the inadequacy of some of the abbreviations I'd concocted in order to fit long variable names into an eighty-character by twenty-five-line screen. (I had naively contracted "bond duration" to "durn." which he particularly disliked.)

I was surprised that he was willing to spend so much time on the interface, when he hadn't actually yet seen or used or had explained to him the mechanism behind the model. But clear expression was an invariant characteristic of Fischer's. I didn't yet appreciate this effort as much as I came to later, and I think I scoffed at it a little, at least inwardly.

He was even more insistent about writing clearly, and yet informally, about technical matters. When we wrote the first draft of our one-factor model of interest rates, he wanted no equations in it. Being used to writing physics articles, I took reader interest for granted. But on Wall Street we were writing for traders and salespeople as well. He knew his audience, and he made me write repeated drafts that first had to pass the inspection of his technical editor, Beverly Bell, who knew the style he liked. He wanted accuracy and honesty without the technical details.

I assumed that "leaving out the equations" meant writing down the verbal equivalent of a formula, which led to a sort of dense description that may have been accurate but was often impenetrable. It took me a while before I realized that one first had to struggle to apprehend the qualitative dynamic behavior of the model in a visceral, intuitive sort of way, and then convey that understanding.

I think the reason that our model became fairly popular in trading circles is that it involved no differential equations and was transparently explicit about implementation, at a time when many of the people who built and used these models were not yet comfortable with the formalism of continuous-time stochastic differential equations.

Fischer didn't appear to believe in keeping models secret, and thought the advantages of dissemination outweighed the arguments for secrecy in valuation. He wouldn't give away software, though, and knew that much of a model's value lies in its successful integration into the trading environment, and in gaining intuition about how to use it.

Sometimes, Fischer's insistence on clarity and style led him down some quirky paths, interesting to recount only because they seem to contrast so dramatically with his expertise in financial areas. In his first few years at Goldman Sachs, his strong tastes (and distastes) imposed a strict set of standards on the display of numbers on the computer screen. He intensely disliked a display of decimal numbers that include more digits after the decimal point than is warranted by the accuracy of the procedure that produced them.

This seemed sensible to me, although somewhat secondary in importance to calculating the number in the first place, but he extended his disapproval (always a powerful force) to the displaying of any excess zeros at the end of a decimal number, digits he called "trailing zeros." So, it was permissible to display a computed yield of 12.72% on the screen as "12.72." But if the computed value happened to be 12.00%, it was obligatory to display it as "12%," with the last two ("trailing") zeros omitted. This "Fischer notation" meant that the number of decimal points to be displayed depended on the number itself, and no standard routine in the C programming language did this.

Fischer's resolve about this led every programmer to write a routine that first printed numbers into a buffer in memory, and then stripped off any final zeros after the decimal point before displaying the character string on the screen or printing it on paper. We used to joke that you could tell if people worked with Fischer by searching their subroutine library for a routine named "removeTrailingZeros(char*string)."

Fischer took this to what seemed to me an extreme position when he insisted that the number zero itself have no trailing zeros. Someone in his group built a stock options valuation calculator that required the entry of the stock dividend yield. If the stock paid no dividend, you entered "0" into that field. When you did, the obligatory "removeTrailingZeros( )" subroutine kicked in and removed the single zero, leaving the field blank.

The next time you looked at the screen, you thought you'd forgotten to enter the dividend, and as soon as you re-entered the zero, it rapidly disappeared again. This gave you the uneasy sense that the program incorporated some sort of practical joke or party trick, like birthday candles that burst into flame again right after you blow them out. Fischer somehow refused to acknowledge the semantic difference between not yet having entered anything and entering an actual zero.

He had other strong feelings about computer interfaces. He disapproved of "mice" and other pointing devices, and thought keyboards were the best means of data entry. In his opinion, everything that could be done with a mouse could be done better using macros or other little ad hoc languages. He similarly disliked graphics, and thought that plain tables of numbers were more evocative. There was not much persuading him in these matters.

Enough of idiosyncrasies of style. Not idiosyncratic at all was his dedication to clear, but casual, unadorned English in writing and speech. His writing style was distinctively terse and conversational. Sometimes his prose seemed to me to lack the ands, buts, thuses, and therefores commonly used to link the flow of sentences and thoughts in technical articles. As I recall this now, his avoidance of conjunctions seems to me the literary analogue of his dislike of trailing zeros.

In the same spirit of dislike of superfluity, he also was uncomfortable with small talk on the telephone. When Fischer had nothing to say, he said nothing; this could be particularly disconcerting on the telephone, where Fischer often simply kept silent, sometimes for a minute or two, and then said an abrupt good-bye. In conversation or discussion, he was pleasant and precise. But you had to speak his language to him. He thought about things a certain way, and had earned the right to do so, and it was most effective if you could tell him your ideas in his style.

Fischer had great physical (or perhaps financial) intuition. His mathematics skills were good, but his insight and intuition, his instinctive sense of what ought to happen in a theory on which he was working, were very strong, and his perseverance generally kept him moving in the right direction through the copses and thickets that temporarily obstructed the view. I noticed his intuition at work when we were trying to speed up our yield curve model.

I had started experimenting with choosing time spacings that grew geometrically from level to level in our interest rate tree. At some point, we ran across certain initial yield and volatility term structures that couldn't be calibrated, and we began to realize that the computational method (varying the level spacing) might somehow be affecting the content of the theory itself. Fischer quickly realized, with geometric insight sans equations or real mathematics, that the variable level spacing had induced mean reversion in the evolution of the interest rates.

To me, Fischer's approach to modeling seemed to consist of unafraid hard thinking, intuition, and no great reliance on advanced mathematics. This was inspiring. He attacked puzzles in a direct way, with whatever skills he had at his command, and often it worked. I saw him try to solve differential equations by playing with series expansions rather than trying to find out if they had known solutions.

He gave you the sense (perhaps misguided) that you could discover deep things, or at least do useful work with whatever skills you had, if you were willing to think hard. I also liked the fact that he had a physical taste in models. He seemed to prefer describing the financial world with variables that represented observable phenomena, as opposed to hidden statistical factors identifiable only by decomposition or analysis.

He had a surprisingly pragmatic streak. Coming from a particle physics background, I initially thought of the work we were doing on yield curve modeling as a sort of attempt at a "grand unified theory" of interest rate pricing; we would henceforth use one model for all interest rate securities. But Fischer was more of a realist, and probably had experience with the vast limitations to predictability in finance compared with hard science. [*] He preferred not to play up the "one grand model for all instruments" angle that I, perhaps somewhat naively, found so appealing. One day he said he thought it was quite possible to prefer different models in different (overlapping) sectors that didn't have to be strictly consistent with each other.

Fischer always seemed more free of artifice than anyone I knew, although this sometimes made him less easy to deal with. He once said that one of the things that limited his influence was the fact that he always told people the truth, even if they didn't want to hear it.

This is true. He didn't soft-pedal in voicing his opinion of work you had done or actions you had taken, but just told you what he thought was important in the relevant arena. He had a clear view of what was important, and this generally meant important for the long run, rather than the short.

In finance, I deduce that he thought practical usefulness and accuracy were more important than elegance, despite the unquestionable elegance that lends so much appeal to the Black-Scholes-Merton options pricing framework. He wasn't a pure mathematician, and when he was awarded the Financial Engineer of the Year prize, he said he preferred applied research to pure research. He also argued that professors should be paid only for their teaching; in their efforts to become good teachers, they would naturally be led to do research.

When we worked on our yield curve model in 1986 and 1987, and started to understand how to implement the tree successfully, he got excited and wanted to drop the one-factor model in favor of embarking on a more realistic two-factor version that took better account of the independent volatilities of both the one-period and the two-period rates. Similarly, when he understood the effect of variable lattice spacing on mean reversion of interest rates, he pushed ahead with Piotr Karasinski on these developments, because they approximated reality better.

Conversely, when Iraj Kani and I were working on models of the volatility smile, Fischer always insisted that anything we did would be unsatisfactory as long as we ignored the effect of the expectation of jumps on the implied distribution of the index, no matter how difficult that might be. No excuse about the difficulty of doing so, or the fact that jump models were usually not preference-free, placated him, because jumps were there.

Fischer's sense of the important made him more interested in new content rather than new methods for solving models whose content he already knew. After he moved from the Equities Division at Goldman Sachs to Goldman Sachs Asset Management, and then later to Fixed Income, I sometimes invited him to come to occasional seminars in Quantitative Strategies. I noticed that he didn't attend if the seminars were on new or improved numerical solutions to problems that were already soluble; it's not that he was against numerical solutions; he was simply interested in content more than technique.

He didn't get carried away by the need to find analytical solutions, either; he just wanted to solve the right problem with the right model, and was quite happy to use numerical methods when fast computers were available. In this regard, I'm always slightly mystified when I hear conference presentations that specify one of the ideal objectives for a yield curve model as being the existence of a simple analytic solution.

Fischer grasped the importance of computing and systems to using models, despite his idiosyncratic computer taste. People sometimes ask why we published the article on our yield curve model in 1990, given that we worked at a profit-making investment bank.

First of all, when we published it, traders at Goldman had already been using it for several years. But, more important, Fischer distinguished between releasing the ideas behind models (which he thought legitimate) and releasing computer implementations or trading systems that incorporate the model (which I think he thought should be charged for).

Creating good software probably requires at least three to four times more resources than creating a model itself - models are ideas and principles; systems are expensive engineering. People unfamiliar with trading tend to either over- or underestimate the value of pure models. There are very few models that lead to direct riskless arbitrage profits or clear trading strategies; these are obviously kept internal.

A good modeling team that participates openly in the communal research world contributes not only to the traders in the firm, but also to the firm's relationship with clients, its general level of intellectual rigor, and its ability to attract and recruit other talented people.

Deep inside, Fischer seemed to rely on the equilibrium approach of the capital asset pricing model as the source for his intuition about options pricing. I believe this is the way the Black-Scholes equation was originally derived, although the first derivation of the options pricing formula in the Black-Scholes article is based on valuation by replication.

I had a humorously insightful glimpse of this several years ago, when a few of us in Quantitative Strategies tried to crudely estimate options prices in the presence of transaction costs. (I think Fischer often thought this to be a misspecified problem; he always stressed that the transaction costs are not a function of the option alone, but of the portfolio to which it belongs.)

We built a Monte Carlo simulator that, on each stock path, made the underlying stock price evolve through small increments of time, at each instant hedging it according to the Black-Scholes formula, and then adding in the assumed transaction fees. The mean cost of replication over all paths was taken as an estimate of an option's cost. When transaction costs are zero, and rehedging takes place continuously, the option cost on any path turns out to be the Black-Scholes value.

This is the essence of the Merton approach to options pricing: All paths in the Monte Carlo simulation yield the same options value, and there is no variance about the Black-Scholes mean. We intended to replace continuous with discrete rehedging, add transaction costs, and then explore the resulting tension between accurate replication, which requires frequent rehedging, and low cost, which dictates minimal rehedging.

One of my colleagues wrote a Monte Carlo simulation program. To test it, we ran it with zero transaction costs and frequent rehedging, expecting to reproduce the price of a Black-Scholes call with zero variance across the simulation. When we ran it, the variance decreased to zero so slowly as the rehedging frequency increased that even at 10,000 rehedgings on a one-year option, there was still appreciable variance, and we couldn't regain the exact Black-Scholes value on a single path.

This seemed strange, so I wrote my own version of the simulation and found the same apparently-too-slow decrease in the variance about the Black-Scholes value. This was puzzling, as it suggested that, even in a true Black-Scholes world with fixed volatility and no transaction costs, someone who owned a single option and rebalanced the hedge a hundred times a day would still have to live with a large uncertainty in the P&L. I went to see Fischer about this, since I thought I was misunderstanding something.

He became quite excited at the apparent inability of the replication to reproduce the Black-Scholes value over a single path, and said something like, "You know, I always thought there was something wrong with the replication method." In his heart, I think, he felt more comfortable with the notion of value being determined by market equilibrium than by replication. Sad to say, both my colleague's and my simulation programs had small errors, which, once corrected, allowed the rapid convergence of the option replication cost to the Black-Scholes value on any path to become clearly visible!

Fischer had an unbiased view of his contributions. Once, when I was going to give a talk at a conference where Merton was the keynote speaker, I called Fischer (already ill, but more than a year before his death) and left him voice mail asking the appropriate way to refer to "the model" - should I call it "Black-Scholes" or "Black-Scholes-Merton?" Fischer responded saying it was fine to call it the Black-Scholes-Merton model, and then added that Merton had come up with the replication argument for valuing the option, noting quite unperturbedly that "that's the part that many people think is the most important."

Finally, Fischer was an inspiration in dealing with the manifold structural problems involved in developing and distributing new models. Most modeling groups are multidisciplinary; we use financial theory, mathematics, and computing. Because of this, people in modeling groups seldom have real mentors - most other colleagues are either salespeople, traders, or programmers. We don't fit neatly into any of these slots, and have few guides to lead us, and no well-worn path to follow on the Street.

Fischer perhaps came the closest to a role model for people like this, although he was a little too exceptional in his historical contribution and reputation to fit that role perfectly. But he was inspiring to people who were struggling to define a place for quantitative skills in investment banking firms. Under these circumstances, Fischer encouraged you to refuse to be swayed by sheer political obstacles, and reinvigorated your sense that it is important to do the right thing and to concentrate on quality even if people around you sometimes don't appreciate it.

He was a quiet, always north-seeking source of sense when things didn't go smoothly, because he kept your eye on the goal, which was doing things the best way you could, and breaking new ground. On issues of turf, he put the welfare of the firm first. In political situations, he counseled avoiding the quagmire of turf issues by always looking for new modeling and systems-building opportunities to exploit.

In the same way that Fischer avoided meaningless zeros on the computer screen, pointless small talk in conversation, and unnecessary conjunctions in his written work, he didn't seem to be afraid of dying. I didn't know him well on a personal level, but he always seemed a consummate realist. At his memorial service in Cambridge, I heard a moving speech by Jack Treynor, who concluded by saying that as regards death, "Fischer wasn't afraid at all." That's the way I saw it too. He rarely seemed to delude himself about the way of the world or its standards.

When he became terminally ill, he neither hid it nor announced it, but told the necessary people. He didn't complain to anyone I knew at Goldman Sachs, and he spoke about it in a fairly detached, almost objective sort of way. I never heard him complain.

He had a massive operation, and was full of genuine praise for the surgeon, who he said was a genius. After the operation, he made a temporary recovery, and worked again, assiduously. For a while we sometimes spoke on the telephone about working on models of the volatility smile that included jumps in the underlying index - he always thought jumps were a critical part of anticipated index distributions at short expirations.

He was frank if you asked about his health, but never volunteered any information if you didn't. Later, when one could sense from occasional remarks and rumors that his condition had worsened, I summoned the courage to say that I always avoided asking how he was doing, but hoped things were going all right. He simply said that things "look pretty iffy right now."

When he finally stopped coming to work, he communicated with anyone who wrote to him via e-mail. I liked to keep in contact with him, and would send him comments or short bits of news from work. If they were purely superfluous, in the nature of small talk or complaint, then, true to his style, he seldom replied. But if you wrote to him about some genuine finance issue, you got a prompt answer.

I asked him once if these e-mail questions were bothering him, and he immediately replied to say no - he liked to receive these questions or puzzles.

ENDNOTES

The author is grateful to Beverly Bell, Fischer's editor for many years at Goldman Sachs, for some helpful comments on this article.

* I've often found that researchers who don't have practical experience with using financial models to make markets have much more faith in testing the predictability of models than is justified or practical. People working on Wall Street are more comfortable with the notion that a model provides a useful way of thinking. "In the end," as Fischer wrote, "a theory is accepted not because it is confirmed by conventional empirical tests, but because researchers persuade one another that the theory is correct and relevant."