Negative Probabilities

Ireland is at the center of most maps of the world. In China however, you'll find the 'middle land' exactly where every Chinese person would expect.

If only Australia could shake off its colonial status once and for all and produce an 'upside down' map.

The mercator projection of the globe underestimates the land areas near the poles at the expense of those closer to the equator.

Europe looks far bigger and Africa far smaller than they actually are. It comforted many a colonialist.

When designing a two dimensional map of a three dimensional world, compromises needed to be made.

A map shows the world how their makers think it ought to be shown rather than how it actually is.

Death of a Model

Mathematical models used by quants are the same. Compromises need to be made in order to fit a complicated world into an equation.

During September 2008 our Andrew Davidson model for mortgage backed securities (MBSes) broke.

It used a trinomial tree model. As prices change from one period to another they can move in three different ways. Up, down or stable.

The problem is, rates were negative which caused the probability of moving up and down to sum up to more than one.

Which meant that the probability of staying stable was... that's right... negative. Probabilities need to add up to one, but should never be negative!

The model didn't describe how the world actually is, but showed us how it 'ought' to be. And then crashed. Over and over again.

Dipping A Toe Into Absurdity

When events defy old boundaries of reasonableness, the model maker updates what's possible. Before we lived in a world where negative interest rates were a fantasy, now we're living in a world where that's possible and we fix the model to accomodate these new realities.

There is another possibility however.

Espen Haug explores how negative probabilities can let models themselves open the door to such absurd worlds.

Common garden variety probabilities should be between one and zero, and the sum of outcomes sum up to one.

Summing up events with negative probabilities also sum up to one. Allowing for negative probabilities also results in outcomes which have a probability of more than one.

Negative probabilities give us an expanded canvas as new unforeseen events occur, without having to manually open up the model, take out a monkey wrench to expand the sample space.

You may retort, but what does it mean to have a minus ten percent chance of seeing negative interest rates?

Don't know for sure.

Perhaps it means our original canvas of event possibilities was 100% and now we have found 20% more canvas. Our 'model world' has grown by a fifth from our initial starting point.

The Half-Coin

A rather more prosaic example for those of you not familiar or interested with financial modelling is based on a flip of a coin.

Take a coin and toss it twice, add the result.

We have a one in four chance of seeing two zeroes.

A fifty fifty chance of seeing a one. I.e. we can 'expect'

And another one in four chance of seeing two ones.

We can described flipping two coins as:

0.25*0+0.5*1+0.25*2 = 1

This is called a probability generating function (PGF) and it adds up to 1.

The PGF of a single coin flip is:

0.5*0 + 0.5*1 = 1

which coincidentally is the square root of the previous two coin case.

Now take the root of a single coin and split it into a half-coin. The result is an infinite series which still adds up to one.

The infinite series includes negative probabilities.

Our half-coin is a very strange and very mathematical object. It has infinitely many sides!

As with the two coin sum example, two half-coin flips sum up to one normal coin flip.

Splitting Infinitives

Breaking a coin apart is like splitting the atom and uncovering a strange new quantum world. However fanciful, negative probabilities are completely natural in this unreal landscape.

Perhaps incorporating this open minded way of thinking about future events will help build more robust models which can seemlessly adjust as previously undreamt events become reality.