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# Negative Probabilities, from Rocky Humbert

March 17, 2010 |

Most of us view the probability of an event as being between zero and one. But this is a simplification. Negative probabilities exist in physics, and they "probably" exist in the markets.

Additionally, probabilities greater than 1 exist too. Probabilities which are less than zero or greater than one are called "extended probabilities."

This is the first paper that I've seen which builds a mathematical model of interest rate options for negative probabilities. Previous papers have dealt with "risk-neutral" and "psedo-probabilities." The authors also promise an upcoming paper that describes financial models for events with a probability. The paper makes good reading for those who enjoy dividing by zero, and taking the square root of a negative number.

For the less geeky, besides negative interest rates, can anyone think of some real world examples of negative probabilities? Or probabilities greater than one?Is "Hell freezing over" an extended probability?

*Rocky Humbert, quantitative analyst, speculator and master chef, blogs as OneHonestMan.*

## Bruno Ombreux comments:

If it is less than 0 or more than 1, then by definition, it is not a probability. It is not even a measure. They could call them anything, for instance "tiger-striped ferrraris", but they should not call them "negative probabilities".

The reason for a semantic discipline requirement, is that this tongue-in-cheek article, is targeted at finance people. People in the finance industry are generally clueless and take this kind of joke at face value.

Right now, I am studying Bayesian statistics, where they make ample use of calculation hacks and gimmicks. For instance they use Dirac masses as probability densities (height is infinite, width is zero, and area 1 ). But they know exactly what they are doing and nobody is fooled by the vocabulary.

That's different when the public is the banks or HF unwashed masses. For these, a dog should be called a dog, a cat a cat…

## Rocky Humbert responds:

While I have often have my tongue planted in cheek (as well as foot planted in mouth), that is actually not the case here.

Mr. Bruno's first sentence is entirely correct with respect to "classical" probability theory. However, he might consider the possibility that Extended Probability Theory is analogous to Einstein's Relativity Theory extending Classical Newtonian Physics. (i.e. there are practical applications of atomic physics in our mundane lives — one doesn't need to travel at the speed of light to see this.)

I'm not a mathematician (and I don't play one on TV either) but I'm told that negative probabilities have a long history for people thinking about the foundations of quantum theory. Feynman wrote about them and the concepts led to his initial work on quantum computing. (Basic quantum computers now exist.)

Additionally, in markets, I believe that "Dutch Books" may give rise to extended probabilities.

I can't quarrel that some folks in the finance industry are clueless. (C'est moi, Monsieur!) but I don't find Brownian Motion-based options pricing models entirely satisfying either… hence I try to keep an open mind.

## Jon Longtin writes:

I would caution against mixing mathematics with physics. Math is an (actually the only) absolute science, who's existence is defined completely in terms of stated rules and relationships. It is, at the end of the day, a very large body of definitions.

Probability, in the mathematical sense, is the chance that a particular outcome will happen, with the assumption that that outcome can at least happen. If an event can never happen its probably is zero, and if it always happen it is one. *Mathematically* to speak of events outside of this context is meaningless.

Physics is, well, physics. The world is the way it is and it's our job to describe it to the best of our ability. A tool that does this remarkably well is mathematics. Often, though, as we learn more the physical models have to be revised, expanded, and reinterpreted, given new information and insights. When we look at our new and improved models through the lens of the old model, thought, strange things happen. This is true with relatively, quantum mechanics, and when they discovered that the sun went around the earth, to name a few.

There are, for example, new materials that are characterized as having a negative index of refraction, n (a measure of how strongly materials bend light, and is the reason a pencil looks bent when in a glass of water); classically vacuum has n = 1 and air is about 1.0001 or so, water = 1.33, with values less than 1 physically impossible. There are, however, new materials being developed that do not naturally exist in nature, but rather are engineered structures that give the illusion of having a negative index. The point is no physics are being violated here; only that the model needs to be revised, and fitting the new material into the old model will result in surprising and sometimes counterintuitive understanding.

It is sometimes tempting to introduce an extension to the old model, such as e.g., negative refractive index and negative probability, but the more rigorous approach is to redefine the physical model from the ground up to capture the new phenomenon in a rigorous way.

*Jon Longtin, Ph.D, is Associate Professor and Undergraduate Program Director, Department of Mechanical Engineering, State University of New York at Stony Brook*

## Bruno Ombreux replies:

In the case of the negative probability article, they are quick to dismiss the obvious and parsimonious solution that everyone has been using, and replace it with some harebrained theory.

Negative interest rates are nothing new and not a problem. We have had plenty of trade-ables that have always been able to get negative, with active OTC option markets in them, eg crack spreads…

The simple solution is to use a normal law instead of log-normal, and if you are still not happy, to use the empirical distribution.

## Tom Marks comments:

There is a fine volume recently out by the wonderfully polymathic Clifford Pickover called *The Math Book*. "From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics.

Reading each of these 250 entries is like doing a set of 25 push-ups for one's mind. And not just the left side, as the methodology behind fractal artwork indicates.

On the subject of squares of a negative number of which Rocky wrote, Dr. Pickover touches on the formerly ridiculed notion of imaginary numbers, the contributions of Bombelli, et al., and writes:

An imaginary number is one whose square has a negative value. The great mathematician Gottfried Leibniz called imaginary numbers 'a wonderful flight of God's Spirit; they are almost an amphibian between being and not being.' Because the square of any real number is positive, for centuries many mathematicians declared it impossible for a negative number to have a square root. Although various mathematicians had inklings of imaginary numbers, the history of imaginary numbers started to blossom in sixteenth-century Europe. The Italian engineer Rafael Bombelli, well known during his time for draining swamps, is today famous for his Algebra, published in 1572, that introduced a notation for √-1, which would be a valid solution for the equation x² + 1 = 0. He wrote, 'It was a wild thought in the judgment of many.' Numerous mathematicians were hesitant to 'believe' in imaginary numbers, including Descartes, who actually introduced the term imaginary as a kind of insult."

Leonard Euler in the eighteenth century introduced the symbol i for √-1 — for the first letter of the Latin word imaginarius — and we still use Euler's symbol today. Key advances in modern physics would not have been possible without the use of imaginary numbers, which ave aided physicists in a vast range of computations, including efficient calculations involving alternating currents, relativity theory, signal processing, fluid dynamics, and quantum mechanics. Imaginary numbers even play a role in the production of gorgeous fractal artworks that show a wealth of detail with increasing magnifications."From string theory to quantum theory, the deeper one studies physics, the closer one moves to pure mathematics. Some might even say that mathematics 'runs' reality in the same way that Microsoft's operating system runs a computer. Schrödinger's wave equation — which describes basic reality and events in terms of wave functions and probabilities — may be thought of as the evanescent substrate on which we all exist, and it relies on imaginary numbers.

Here is Dr. Pickover's website (well worth a look).

## Rocky Humbert replies:

With your reference to Dr. Pickover, you tied together many loose ends:

Mr. Maner alluded to the fact that there is a probability greater than one that "…vampires will invade the literary world and be a profitable genre." He referenced the epic drama: "Abraham Lincoln: Vampire Hunter " Low and behold, it was Dr. Pickover who invented vampire numbers– I would wager that this is the first time that negative probability, markets, Abraham Lincoln and vampires were all discussed in the same thread on Dailyspec. (The probability of this happening is comparable to the odds that the S&P will rise five more days in a row. I therefore conclude that this must be an omen, and I just bought ONE March S&P 116 call as a homage to negative probability and vampires.)

## Sushil Kedia writes:

The first and the simplest example of negative probability at work in the markets comes to my mind from the Chair's oft emphasis on deception in the markets.

Let me use an example:

A street conman's game very much prevalent in India, near the smaller train stations and ports, where the hustler holds a heavy bag of muslin with two hands and offers a wide peep inside for you to see a nicely mixed hoarde of coloured and natural peanuts. The odds offered are 3:1 for you to multiply your money if you lift up a coloured peanut. You rush in playing an unfair game apparently to your advantage. When you shove your hand in, the mouth of the bag is held much closer around your wrist than when you were inticed to take a game loaded in your favour.

You pull out the peanut and it is not coloured.

The trick deployed is that there is another bag within the bag containing only uncoloured peanuts.

In the awareness of the hustler, probability of the outcome is certain due to his ability at deception. In the awareness of the player the probability of the outcome is somewhere close to 50:50. In the awareness of an analyst like me who has burnt the hand that tried picking the coloured peanut many times 50/(50+50+100) or 0.25 assuming the hustler fails at closing out the bag with mixed peanuts forcing your hand into the hidden bag with only plain ones.

The difference in the probabilities known to the newbies and those who have burnt their hand and become analysts is the 0.75 gap which is really the negative probability on which the hustler is peddling his skill.

Replace the peanuts - plain and the mixed ones with earnings guidance and announcements and you realize the negative probability the masses face vis-a-vis the insiders assuming all else being equal.

Replace the peanuts - plain and the mixed ones with counted stats the pros are playing with and the bags of code (not only computational but simply deal flow informational) the glittering big firms can have.

So on and so forth.

With this perspective gaps in perception, information, imagination, awareness, model specification, ability to loot, peddle, hustle etc. etc. a concept of negative probability fits in well with comprehension. The Oracles of Delphi as explained in the Education of a Speculator played really on the negative probability the masses believe are non-existent and happy to live with such belief. Despair, disdain, pursuit of short-cuts, road to quick riches have all been built with bricks of negative probability.

The first and the simplest example of negative probability at work in the markets comes to my mind from the Chair's oft emphasis on deception in the markets.

Let me use an example:

A street conman's game very much prevalent in India, near the smaller train stations and ports, where the hustler holds a heavy bag of muslin with two hands and offers a wide peep inside for you to see a nicely mixed hoarde of coloured and natural peanuts. The odds offered are 3:1 for you to multiply your money if you lift up a coloured peanut. You rush in playing an unfair game apparently to your advantage. When you shove your hand in, the mouth of the bag is held much closer around your wrist than when you were inticed to take a game loaded in your favour.

You pull out the peanut and it is not coloured.

The trick deployed is that there is another bag within the bag containing only uncoloured peanuts.

In the awareness of the hustler, probability of the outcome is certain due to his ability at deception. In the awareness of the player the probability of the outcome is somewhere close to 50:50. In the awareness of an analyst like me who has burnt the hand that tried picking the coloured peanut many times 50/(50+50+100) or 0.25 assuming the hustler fails at closing out the bag with mixed peanuts forcing your hand into the hidden bag with only plain ones.

The difference in probabilities known to the newbies and the analysts is 0.5-0.25 = 0.25 the awareness advantage.

The difference in probabilities known to the analysts and the hustler is 1-0.25 = 0.75 the hustling advantage.

The difference in probabilities known to the hustler and the newbies is0.5-1.0 = -0.5 the ignorants negative probability

Awareness Advantage MINUS Ignorants negative probability = 0.25-(-0.5) = 0.75 = The Hustling Advantage or in other words, the negative probability is Awareness Advantage - Hustling Advantage.

Replace the peanuts - plain and the mixed ones with earnings guidance and announcements and you realize the negative probability the masses face vis-a-vis the insiders assuming all else being equal.

Replace the peanuts - plain and the mixed ones with counted stats the pros are playing with and the bags of code (not only computational but simply deal flow informational) the glittering big firms can have.

So on and so forth.

With this perspective gaps in perception, information, imagination, awareness, model specification, ability to loot, peddle, hustle etc. etc. a concept of negative probability fits in well with comprehension. The Oracles of Delphi as explained in the Education of a Speculator played really on the negative probability the masses believe are non-existent and happy to live with such belief. Despair, disdain, pursuit of short-cuts, road to quick riches have all been built with bricks of negative probability.

## T.K Marks writes:

On the subject of meals, I see today a poignant portrait of the food chain. These photos from Colorado are spectacular.

"…The starling seems to be completely unaware it is on the lunch menu as the bald eagle makes it attack at high speed…" On some level we've all been starlings at one point or another.

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