Emanuel Derman on Fischer Black


Staff member

I have been working for the last few months on a book about the way people are compelled to theorize and to build models of the natural and the social world, and somehow this led me to think about Fischer Black again, even before I received an invitation to speak here tonight. Fischer is, of course, most famous for the Black-Scholes model. But Fischer understood very well the qualities of models and their limitations. Among my three favorite sentences of Fischer’s are
"My job, I believe, is to persuade others that my conclusions are sound. I will use an array of devices to do this: theory, stylized facts, time-series data, surveys, appeals to introspection and so on."

"It’s better to ‘estimate’ a model than to test it. Best of all, though, is to ‘explore’ a model."

"In the real world of research, conventional tests of [statistical] significance seem almost worthless."

You can see that Fischer understood how models work and their limitations, that models are not gospel, not the world itself, but idols that we make to try to mimic it.
I want to begin this evening by speaking about models and theories and the people who make them, and end up with some related thoughts about Fischer.


Models & Theories
When you try to understand the laws that may drive the world, in my view, it’s especially important to distinguish between models and theories.

Models are fundamentally metaphors. They compare something you don’t understand very well to something you understand better, in order to add insight. Calling a computer an electronic brain, for example, is a metaphor that once cast light on the function of computers. Nevertheless, a computer is not an electronic brain. Conversely, calling the brain a computer is a model too – the brain is a long way from a computer. In tackling the mysterious world with models we do our best to explain the thus-far incomprehensible by describing it in terms of the things we already partially comprehend. Models, like metaphors, take the properties of something rich and project them onto something strange.
My favorite metaphor is Schopenhauer’s on sleep:
Sleep is the interest we have to pay on the capital which is called in at death; and the higher the rate of interest and the more regularly it is paid, the further the date of redemption is postponed.
By focusing on the periodic nature of sleep, a periodicity it shares with coupon payments, Schopenhauer takes the metaphor of a loan and extends it to life. Thus, since sleep and coupons are both periodic, and coupons are the result of a loan of principal that must be repaid, he depicts life as a loan from the void that leaves behind a hole in the darkness that must eventually be refilled. The loan of principal is life and consciousness, death is the final repayment, and sleep is la petite mort, a periodic little death.
This focus on the common periodicity and then extending it from one thing to another is a kind of analytic continuation, and analytic continuation in mathematics is a kind of metaphorical extension too. The factorial function for integer arguments n satisfies the recursive relation n! = n x (n-1)! Euler interpolated the recursive property of factorials to numbers that lie between the integers, and extrapolated it away from the integers to numbers in the complex plane, and so created the even richer Gamma function.
Good metaphors are expansive; they let you see in a new light both the object of interest and the substrate it rests on. Good metaphors enlighten upwards and downwards. But a models is still a toy, -- in this case comparing life, something we don’t understand, to finance, something we think we do.

If models are metaphors, shedding light by analogy, then theories are the real thing. They don’t compare, they describe and explain. Dirac created his theory of the electron, the Dirac equation, in 1928. He sought an equation that satisfied both quantum mechanics and special relativity. The one he found had four solutions. Two of them described the electron that physicists already knew about, a particle with negative charge and the two spin states that had already been postulated by Goudsmit and Uhlenbeck a few years earlier, based on the need to understand the details of atomic spectra. But Dirac’s equation had two additional solutions, similar to the ones he’d already found, except that they had unpleasantly negative energy. The positive-energy solutions described the electron so well that Dirac felt obliged to make sense of the negative-energy ones too.

Dirac postulated that the void, or the medium that we call empty space, what physicists call the vacuum is in fact filled to the rim with negative-energy electrons, and they constitute an infinite sea. This Dirac sea is the vacuum we inhabit, and, accustomed to it, we don’t notice the infinite number of negative charges surrounding us. (We smell only pollutants, not air itself.) If this is true, argued Dirac, then a sufficiently energetic photon of light can impart enough energy to an electron in the sea to make its energy positive and thereby emerge from the Dirac sea. What’s left behind is a hole. This absence of negative charge and absence of negative energy, this hole in the sea, behaves like an electron, but one with positive charge. Anderson discovered the positron in 1932, and astounded all the physicists uncomfortable with what had been a metaphorical stretch. Just as life is a temporary hole in the darkness, so here too absence too becomes a presence.

This was the first time a theory successfully predicted the existence of a previously unobserved particle, and it set the tone for particle physics thereafter. But, equally interestingly, Dirac’s equation transcended its metaphorical underpinning in the sea and became an accurate description of reality in quantum field theory. A brain may be like a computer, an atom may be like a miniature solar system, but an electron is the Dirac equation. Dirac’s theory of the electron stands on its own two feet, beyond metaphor, the thing itself. Like God in the burning bush identifying himself to Moses, the theory of the electron pronounces, “I am that which I am.”
Theories tell you what something is. Models tell you only what something is more or less like. Unless you constantly remember that, therein lies their danger.

Spinoza’s Theory of Emotions as Derivatives
A few months ago I reread parts of The Ethics, Spinoza’s attempt to derive the laws of appropriate human behavior from primitives, axioms and logic. Spinoza’s ideal style of theorizing was that of Euclid’s geometry, but applied to people.
Spinoza’s theory in The Ethics deals only in concepts transmitted through words and logic. Everything begins with pain, pleasure and desire, feelings so recognizable to inhabitants of bodies that their definition, though Spinoza provides it, is superfluous and even misleading. The emotions we feel, Spinoza claims, are derivatives of these underlyers. Love is pleasure associated with an external object. Hate is pain associated with an external object Envy is pain at another's pleasure. Cruelty involves all three primitives: it is our perception of the desire to inflict pain on someone we love.

I have illustrated the derivative structure of Spinoza’s theory of emotions in the figure below (click to expand). I call The Ethics a theory rather than a model, because, though he follows Euclid’s axiomatic method, Spinoza doesn’t make analogies; he doesn’t attempt to explain how humans should behave by comparing them to some other system. He begins with introspection and observation, what he sees about human beings as human beings, both others and himself.


Fischer, Spinoza and Newton
There were two qualities in particular that I admired about Fischer. The first was his intuition, the second his acceptance of the world.
About his intuition I wrote in my book six years ago:
“At bottom, he simply liked to think through everything for himself. His approach seemed to me to consist of unafraid hard thinking, intuition, and no great reliance on advanced mathematics.”

How do you get intuition about the world or about people?
Spinoza uses his theory of emotions as derivatives to deduce through logic how humans should live. I don’t have time to go into the details.
The highest endeavor of the mind, Spinoza concluded, and the highest virtue, is to understand things by the intuitive kind of knowledge. Intuition may sound casual and unfocused, but actually it takes intimate knowledge of the world that can be acquired only by careful observation and painstaking effort. When you struggle with a field of inquiry and a model for a long long time and you eventually master and incorporate not only its formalism but its content, you can make use of it to build things one level higher. Intuition is a merging of the understander with the understood.
Spinoza would argue that God’s understanding of the world is intuitive, an intuition so thorough that there is no remaining boundary between the creator and the created.

The other day someone sent me a speech of John Maynard Keynes delivered by his brother Geoffrey at the Newton Tercentenary in 1946. Keynes had written the speech but died three months earlier, and his brother read it there. It was based on Keynes’s reading of a box of Newton’s notes, many of them cryptic and mystical, on his attempts to understand not just the physical but the entire world. In his speech, Keynes wrote:
“Newton came to be thought of as the first and greatest of the modern age of scientists, a rationalist, one who taught us to think on the lines of cold and untinctured reason. I do not see him in this light. Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago …

I believe that the clue to his mind is to be found in his unusual powers of continuous concentrated introspection … His peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. I fancy his pre-eminence is due to his muscles of intuition being the strongest and most enduring with which a man has ever been gifted. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one's mind and apply all one's powers of concentration to piercing through it, and how it will dissolve and escape and you find that what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret. Then being a supreme mathematical technician he could dress it up, how you will, for purposes of exposition, but it was his intuition which was pre-eminently extraordinary - 'so happy in his conjectures', said De Morgan, 'as to seem to know more than he could possibly have any means of proving'.

There is the story of how he informed Halley of one of his most fundamental discoveries of planetary motion. 'Yes,' replied Halley, 'but how do you know that? Have you proved it?' Newton was taken aback - Why, I've known it for years', he replied. 'If you'll give me a few days, I'll certainly find you a proof of it' - as in due course he did.

I don’t mean to compare Fischer to Newton or Spinoza, but these passages remind me of him.
About Fischer’s attitude to the world, I wrote:
Whenever I think of Fischer I think of him as a consummately unsentimental realist, unafraid to see and take the world for what it is.”

Spinoza believed that his God didn’t make the world for humans, and hence he (i.e. Spinoza) had no obligation to explain away the unarguable fact that bad things happen to good people. He had no need to say that God acts in mysterious ways we cannot understand. Instead, he recognized that what offends humans is not what offends God. What he said, I think, is that the God he envisaged made everything you can think of. Every thing you can think of exists, and everything that exists can be thought of. Stuart Hampshire in his book on Spinoza referred to this principle as “The possibility cannot be greater than the actual.”

Spinoza argued that this is the universe we’re in and we should embrace it. When I was reading this a few months ago I thought of Fischer. One of the things that I learned from him, both in his attitude to company politics and to the world and its indignities, is that he seems to have had Spinoza’s attitude before I recognized it for what it was He seemed to me to take the world the way it was, not as the best of all possible worlds, but as the only one we have, like it or not, and make the best of it.
Fischer was both a man of principle and at the same time a pragmatist par excellence, and so I wonder what he would think of the desperation moves of the past few years by banks and the administration alike. He always had a peculiarly logical way of looking at things. Like the positron of Dirac’s, his absence is a presence.
All this reminds me of the great mathematician, Gian-Carlo Rota: "Research is not as much discovering something new as becoming aware of the prejudices that stop us from seeing what is in front of us"