Fokker-Planck Equation and the Flow of Probability
26th June 2014
[This article is excerpted from the upcoming book Quantitative Finance: The Long Road Through Physics by Rahul Bhattacharya. This article is re-formatted and edited by our editorial team.]
Does probability - the concept that we learnt in high school and which we, more or less, use every day to describe events in our lives - behave like water or fluid? Just as water flows from a region of higher elevation to a region of lower elevation, does probability also display similar flow?
Many of you may think that it is silly to even contemplate such a comparison. In any case, we are not trained to think like this because "water" or "orange juice" or "plasma" is a fluid and probability is an abstract mathematical concept. Further, many of you may be surprised if I tell you that just as in high school physics we've learnt that total energy is conserved, total probability too is conserved and that you already know this in your head even though you may not think it in such terms.
When I was a trader, some 65 million years ago, my boss used to drill into our heads that the laws of probability are counter-intuitive. Be that as it may, over many years of backpacking through the financial markets I have realized that laws of probability also display a great degree of symmetry, just like the laws of physics. And it does not take great introspection or philosophical effort to see this. Every day we come to terms with this symmetry in the laws of probability.
What has really fascinated me about the concept of probability is that just like matter and energy in physics it too has a well-defined conservation law which is also why it displays symmetry. The most startling revelation for me has been that probability behaves like a fluid flowing from one region of the space to another over a period of time; and, during this flow while the distribution (probability distribution) changes over time the overall quantity is conserved.
Every high school kid learns about this conservation law of probability (though, they are seldom provided with an explanation) when they are told by their teachers that the total probability (sometimes, also called the "net probability") of an event is one. It will either rain today or not rain and the probability of rain plus the probability of not-rain is equal to one. The probability of getting the number 2 in a roll of a die is because there are six numbers (marked 1, 2, 3,4, 5 and 6) in an die. However, the probability of getting a number between 1 and 6 in a roll of a die is one because some number between 1 and 6 will definitely show up in a roll of a die and when we add the probabilities of all numbers in the die we get one . Similarly, a particle may exist in a certain region of the space and there is a probability associated with finding a particle in a certain region of space at a particular time. However, the total probability of finding the particle in the entire universe is equal to one because if we add up all the probabilities of finding the particle in every infinitesimally small region of the space over the volume of the entire universe then the sum will add up to one. The particle exists somewhere for sure.
This fact that total probability of an event always adds up to one essentially tells us that there is a conservation law at work. This simple and obvious fact about probability, which expresses the conservation of probability, has a deeper connection with the symmetry that found in nature.
But if probability displays conservation law, just like matter and energy, then we should be able to deduce an equation of continuity that expresses this conservation law of probability from the fundamental equation of motion. When we do such an exercise we see that this equation of continuity of probability takes a familiar form that arises in theory of fluid dynamics. Probability not only follows a well-defined conservation law, but it indeed, flows like water or orange juice or plasma. This is a remarkable result and has great implications in the study of financial derivatives as well.
This result goes by the general name of Fokker-Planck equation and is an equation that contains "probability density" and "probability current density vector".
Fokker Planck equation is fundamental to understanding of the Brownian motion of a particle and is used to model a wide variety of stochastic processed in physics. Fokker Planck equation is one of the key equations used to analyze financial derivatives. If is probability density and is the probability current density vector then we can write an equation of continuity for the probability as*:
There are many ways how we can derive the above equation. Most mathematicians and finance academics take the route of stochastic calculus and probability distributions to arrive at the above equation. However, there is another, very elegant and a very intuitive mathematical derivation of the Fokker-Planck using the Schrodinger's wave equation (this is the approach taken by many physicists).
There is a very nice interpretation to the Fokker-Planck equation, one that even finance professionals will find quite appealing. If we take a closed space , in the space, then the above equation of the flow of probability shows that the outward flux of across is equal to the rate at which the probability of locating the particle inside is decreasing.
If stocks and financial assets follow a Brownian path like a Brownian particle, then there is great similarity between such a mathematical framework and the framework that is employed while pricing financial derivatives. When a call option is at the money at time, the option has highest value and as the spot moves away from the strike over a period of time (going closer to maturity) the probability of the option finishing in the money - similar to the problem of locating a particle inside the space, - decreases.
*For more details on the derivation of Fokker-Planck equation, its explanation and its application please refer to CFE Lecture Notes.
- Lawden F., Derek, The Mathematical Principles of Quantum Mechanics, Dover Publications, New York, 1995.
- Bhattacharya, Rahul, Quantitative Finance: The Long Road Through Physics, CFE School Publishing, 2014.
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