Integrating mathematical finance into PDE

Integrating mathematical finance into a course on PDE

Many math students are interested in mathematical finance. The core of mathematical finance is a partial differential equation called Black-Scholes equation. It would be natural to integrate mathematical finance into a course on PDE. In this notes, I will discuss some background information that may help facilitate such an integration.

Many people think Black-Scholes equation is very technical and hence mathematical finance is a specialized subject. Think about Schrodinger equation. It is highly technical. But Schrodinger equation applies to all particle movements. Schrodinger equation covers very broad areas. Likewise, Black-Scholes equation applies to all financial derivatives. Mathematical finance is a very broad subject. 

In most cases, Schrodinger equation doesn’t have analytical solutions. But in the simple case of spherical symmetry, it does possess analytical solutions. They give simple descriptions on the structure of periodical table of the elements. That was a great triumph of quantum physics. Likewise, in most cases, Black-Scholes equation doesn’t have analytical solutions. But in the simple case of European options, it does possess analytical solutions called Black-Scholes formulas. They give simple descriptions how basic observable quantities in the market determine derivative prices. That was a great triumph in finance and social sciences. It shows that there exists precise and quantitative relations in social sciences, not just statistical regressions. 

 Black-Scholes equation is a parabolic partial differential equation. The equation and its solutions can be readily discussed in the first course on partial differential equations. Many people are concerned about the lack of applications relevant to the real world in math teaching. Black-Scholes equation, and its solutions, Black-Scholes formulas provide exciting applications to the highly active and highly visible financial market. Black-Scholes equation is much simpler than Schrodinger equation. Its applications are in finance and economics, which are much more familiar to most people than the quantum world.

Black-Scholes equation is a reverse differential equation. In financial contracts, payments at the end of contracts’ life (in the future) are specified. We want to solve for the contracts price today. That is why Black-Scholes equation is a reverse equation. In real world, we invest first, harvest later. Most economic activities are described by regular equations (forward in time direction), not reverse equations. If a reverse equation has so many applications, one might wonder if a regular equation  has even more applications. That is indeed the case. 

From the very beginning of mathematical finance, some pioneers, such as Fischer Black, has dreamed about a similar theory on real economic activities. We will show a regular equation, a thermodynamic equation, will provide an accurate description on a broad spectrum of problems in the economic, social and biological activities, very much like Schrodinger equation provides an accurate description of the physical world. 

John Galbraith once said, “Politics is not the art of the possible. It consists in choosing between the disastrous and the unpalatable.”  Since most policies are designed to be palatable, they often tend to be disastrous over long term. When we solve the equation, the long term consequences of monetary policies, fiscal policies and many other policies become very clear to us.  This will help us question those policies that sow the seeds of large scale financial crises and social decline.

With an integration of mathematical finance into a course on PDE, we can apply the techniques of PDE systematically to human decision making, which are the most relevant and most familiar to us.

More information on the PDE and its solutions on economic activities can be found at

An Analytical Theory of Major factors in Economics, 

http://web.unbc.ca/~chenj/papers/AnalyticalTheory.pdf