Risk Latte - Working Miracles with the Differential Operator - II

Diffusion Equation in Finance and the Gaussian Function

Team Latte
26th July 2014

[This article is excerpted from CFE Lecture Notes. ]

Before we talk about the diffusion equation and its characteristics and solutions, we need to spend a bit of a time talking about the Gaussian function. Over the past decade or so, many well-known practitioners in the field of banking and finance (as well as scores of financial journalists) have been throwing rocks at the Gaussian (normal) probability distribution and pinning most of the ills of the financial markets and the world of financial derivatives on this probability distribution. The argument is that financial markets and the way the financial derivatives behave in the real world are markedly non-Gaussian and that the normal distribution does not work in the real world of finance. Yet, a large part of finance theory, including those models and mathematical constructs dealing with valuation of financial derivatives, is totally Gaussian. For example, Black-Scholes option pricing model is Gaussian in nature. This is where the problem lies, or so goes the thinking. And, in many ways this thinking is correct.

A Gaussian function is any function of the form where, is any arbitrary real constant, or any constant multiple of this expression. An example of a Gaussian function would be

Where, and are constants. These functions have an extremely fascinating history and they occur in a wide variety of settings in mathematics, mathematical physics and engineering. The most famous, and widely used, example of such a function is the Gaussian (normal) probability distribution. The "bell shaped" curve in probability theory is described by this kind of a function and defines a normal or the Gaussian distribution. The probability density function with mean, and a standard deviation of is given by

All of us are familiar with the above density function and the Gaussian distribution. It is the most widely used probability distribution in finance. One of the interesting features of the above function is that it is impossible to find its integral (the indefinite integral). If we do not specify any limits of integral such as, we'll never be able to compute the integral of the Gaussian function.

It is a remarkable property of nature that solution to one of the most famous partial differential equations in physics, the heat equation or the diffusion equation, involves the Gaussian function. We shall explore in more detail the relationship between the probabilistic random walk model and the diffusion equation in a separate lecture, but for the moment we need to appreciate that the most elementary solution of a diffusion equation generates the Gaussian function.

What is a Diffusion Equation?

Black-Scholes partial differential equation (PDE) is a diffusion equation, the solution of which is a the famous Black-Scholes formula. In the field of financial derivatives diffusion equation plays a very important role. In our FEM course we talk about diffusion equation in detail (and look at its solutions in detail as well). Here, we will try to get a broad overview of what a diffusion equation is and how is it relevant in quantitative finance.

In mathematical physics, one of the most famous diffusion equations is a partial differential equation that describes the flow of heat and is given by

Where, is a function of space and time and is an arbitrary real constant known as the diffusion coefficient. The above is called a heat equation. Black-Scholes PDE, even though explicitly not a heat equation, can be transformed into a heat equation like the above with suitable transformation of variables. To be exact, the heat equation is not the most generalized form of diffusion equation that arise in physics, but we shall currently limit our discussion to the heat equation because that will suffice for our understanding of dynamics of financial derivatives.

We can think of a call option (on a financial asset) being governed by an equation similar to the above, where will be the call option price as a function of a spatial variable, , the asset price (here, ) and time, . The diffusion coefficient, , can be thought as the volatility of the asset price.

To see how we can find an elementary solution of this diffusion equation that connects back to the Gaussian functions please refer to CFE Lecture Notes.

Reference:

  1. Sneddon, Ian, Elements of Partial Differential Equations, McGraw Hill Kogakusha
  2. Lecture Notes of Prof Kent G. Merryfield, Department of Mathematics and Statistics, California State University, Long Beach, California

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