The Problem with the Probability measure
February 26, 2009
Probability is a very beautiful and a very dangerous concept when applied to financial markets and financial risk management.
The fundamental problem with financial risk, be it market, credit or operational risk, is that it is predicated on the notion of probability. Gaussian distribution, bell curve, Copulas, Poisson processes, VaR, default time, all these and in fact, each an every other concept and axiom in financial risk management is tied to the concept of probability or the "probability measure".
What is probability? Well, you have the objective probability measure and the subjective (Bayesian) probability measure, but in its most fundamental form probability can be simply defined as the chance of a particular event happening. But this chance, this probability measure, does not exist in nature.
Firstly, probability exists in our minds and is function of how much information we have with us. Secondly, we cannot catch probability in the act, just like we can watch the earth going round the sun or observe a baby being born.
Let's analyze the first pointi. Say, I am holding a coin in one of my folded palms and ask you what the probability is that the coin is in my right palm. You would immediately answer 50%. There is indeed a 50% chance that the coin is in my right palm. But that is so because you don't know in which of my palms I have hidden the coin and so you are making an estimate; it can either be in my right palm or left palm and since the coin has to be in one of my folded palms (there are only two possibilities and the total probability is one) you can immediately deduce that the probability of the coin being in my right palm is 50%. You estimate is based on lack of information on your part.
However, I know exactly in which of my palms I am holding the coin. If it is indeed in my right palm then, for me, the probability of the coin being held in my right palm is 100%. Otherwise, the probability, for me, is 0%. Thus the game is rigged. The person, and in this case it is me, who is holding the coin in his palm, always knows with certainty where the coin is (equivalent of saying he knows how the history will unfold). Everyone else is in the dark and guessing. You may say that well someone always knows the answer and that "someone" might as well be God.
This may sound like a philosophical statement and, as Einstein once famously quipped, God does not play dice. For Him there is no probability.
In the financial markets a Goldman Sachs, a Morgan Stanley or a Delhi-Bombay bank has so far tried to play the equivalent of God (of course, this time around they got stumped themselves!). They normally know in which of their palms they are holding the coin and you are the one who gets stuffed. Risk – at any rate, financial risk – has been a one sided game so far. It’s all a question of information. Whoever has information controls the game.
And ironical as it may seem or over simplistic in its analysis, in the world of financial risk it always comes down to someone holding a coin in one of his palms and rest of the people guessing which one is it.
Now, coming to the second point; you cannot catch probability in the act. The probability measure exists in our minds and is hidden inside a mathematical model. It can never be observed as reality unfolds.
In financial risk and derivatives models, you work with the likes of stochastic differential equations which model the future movement of a stock price – the evolution of the stock price in time – as a diffusion process. This concept is borrowed from physics. Note the word “stochastic” which means random. Stock prices, and indeed the prices of all financial assets, be it currencies, commodities, interest rates, etc. are assumed to follow a random walk. This randomness is all about probability. If you want to simulate stock prices going forward from today then you pick a random number from a probability distribution, multiply it with the volatility of the stock and the product gives you an estimate of where the stock will be in the next instant. And since this random number is a probability measure there are infinite possibilities as to where the stock will be tomorrow given where it is today (as long as the return of the stock follows a predefined distribution, such as say, a lognormal distribution).
Thus our mind is befuddled with all these infinite possibilities of where a stock can go tomorrow or in the next period from today’s starting price. No one knows precisely what is going to happen, where the stock would end up tomorrow. There is no mathematical formula or equation that deterministically generates an exact value of the stock price tomorrow given today’s stock price and certain observable input parameters. It’s all up in the air.
Physicists can tell you when Hailey’s comet will return, when the next solar eclipse will happen or when our earth will be gobbled up by the sun. Doctors can more a less tell a pregnant woman as to when precisely she can expect her baby or how healthy her baby will be. Their mathematical and scientific models produce precise dates and time for the happening of such events. But a quantitative modeler in a bank who forecasts the stock price does not have this luxury. He is at the mercy of a model that is totally dependent on “probability distributions” and the notion of randomness.
But this randomness is not there in nature. Nature does not display any randomness at all. Look around you, look to the stars and heaven, look into the depths of ocean and trees and foliage around us, and look at the wildlife and even at our own human society. Nowhere do we see any randomness. Nature may follow non-linear dynamics, it may be chaotic but there is not an iota of randomness in its fabric.
Probability does not exist in nature. It’s only hidden in our minds and financial models.
Therefore, as we sit in front of our computers trying to model the evolution of stock prices in future and value financial derivatives, we create this beautiful randomness around us which is nothing but an illusion. The math – stochastic differential equations, probability theory – the models, the computer code, all these combine to create millions and millions of different competing worlds in future one of which will actually exist. In fact, so clueless is the financial modeler and so pressed for time he is given the humongous task that faces him – to simulate a world out of infinite possibilities – that it all turns out him picking a random slice out of an infinite universe and calling it the reality.
However, tomorrow when the reality arrives, the stock actually takes on the value the randomness vanishes. The probability measure which was used to predict today’s stock price yesterday is no longer visible and has no relevance any more. We now need a new probability for predicting tomorrow’s price.
Bart Kosko, in his excellent text, Fuzzy Thinking likens this disappearance of randomness to a movie being watched on DVD. As you are watching a movie, events are randomly unfolding in front of your eyes and the future – the end of the movie – is uncertain. Only the director of the film knows how it will end. You don’t know and hence you are guessing. This randomness adds thrill and excitement to movie watching. Now after the movie finishes, rewind the movie and watch it backwards on your TV screen. As the movie runs backwards, you’d see all randomness has disappeared. The story – in reverse – has an exact order to it and fits in to the last detail. Where did the randomness go?
That is the question we must ask? As the stock price unfolds today, as the financial markets take shape and as the economy generates its output what happened to all that randomness that preoccupied our thinking yesterday.
Nassim Taleb is right; we are continuously being fooled by randomness.
i This is taken from Bart Kosko’s excellent text Fuzzy Thinking
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