Partial Differential Equations in Finance
May 4, 2007
Recently we had a very lively (single sided) discussion with a group of French quants in a bank in Asia regarding partial differential equations (PDEs) and their applications in financial engineering and derivatives. The discussion was over lunch and specifically centered on the non-parabolic PDEs and their applications. It is difficult to suggest which was more appetizing, the lunch or talking to those quants. We must admit that the French are head and shoulders above a lot of others - certainly us - where quantitative modeling is concerned. Also, during this lunch we listened and they talked.
It would be a shame if we don't share that knowledge - in its most general form - with our readers. Two excellent books on PDEs are: one by Daniel J Duffy and the other by Paul Wimott*.
Partial differential equations, without which there would be no Black-Scholes, option trading and the world of exotics and structured products, are divided into three major categories:
The parabolic equation, of which diffusion and convection-diffusion equation is a sub class, is the most used and popular (read "well known") one in the world of derivatives. Black-Scholes equation, the solution of which gives the celebrated Black-Scholes option pricing formula, is a convection-diffusion equation. However, elliptic and hyperbolic (a 2 nd order hyperbolic equation is a wave equation) equations are also found in some derivatives applications.
Generally speaking, all partial differential equations, at the algebraic level are quadratic equations of the form:
Where a, b, and c are constants and is a generalized variable. As we have discussed elsewhere in this site this kind of quadratic equations are used quite often in math modelling of financial derivatives and risk. In the case of partial differential equations we can treat the variable as a partial derivate of two other variables. For example we could have a first or second order PDE with terms involving , and , and could be a transformed variable of these.
The roots of the above quadratic equation are:
The categorization of a PDE into elliptic, parabolic and hyberbolic is due the nature of the roots of the above quadratic equation as shown below:
This is how PDEs are typically classified. Time and resource permitting we shall try to discuss some more on PDEs and especially the non-parabolic PDEs which are not that common in financial engineering applications.
*Reference: Finite Difference Methods in Financial Engineering by Daniel J Duffy and Quantitative Finance by Paul Wilmott (Wiley Finance)
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