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Emanuel Der man on Fischer BlackDiscussion in 'Quant Matters' started by QuantNet, 7/22/10.

BY EMANUEL DERMAN

Introduction

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 

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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

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 

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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

ostponed.

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 couponsare 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.

Theories

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

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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:

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“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 thephysical 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

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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 thoughtknows 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.

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QuantNet, 7/22/10 #1

dstefan, 7/22/10 #2

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 wasreading 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 theadministration alike. He always had a peculiarly logical way of looking at things. Like the

positron of Dirac’s, his absence is a presence.

Excellent read, thanks a lot!

dstefanBaruch MFE Director 

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Sharat, 11/28/10 #3

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 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"

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