Register to reply

Is the Black-Scholes equation a differential equation?

by TheGhostInside
Tags: blackscholes, differential, equation
Share this thread:
Jan28-13, 04:30 PM
P: 1
Hi everyone, first post. To anyone who has had experience with the background of the Black-Scholes equation used in finance to price options based on underlying assets (wiki here), I have just one simple question to ask regarding a research paper I must write.

Is this equation a stochastic differential equation, or a PDE?
Phys.Org News Partner Science news on
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Jan28-13, 05:39 PM
HW Helper
P: 1,391
In a certain sense it's both. There is a stochastic differential equation which is equivalent to a PDE for the probability density.

Basically, if you have a stochastic differential equation with brownian noise,


$$dX_t = a(X_t,t)dt + b(X_t,t)dB_t,$$

then one can show that this is equivalent to a PDE called the Fokker-Planck equation:

$$\frac{\partial f(x,t)}{\partial t} = -\frac{\partial}{\partial x}\left[a(x,t)f(x,t)\right] + \frac{1}{2}\frac{\partial^2}{\partial x^2}\left[ b(x,t)^2 f(x,t)\right],$$

where f(x,t) is the probability density of finding the system to have a value of x between x and x+dx at a time t.

This generalizes to more variables (see the Fokker-Planck wikipedia page for a brief intro and some further references).

Register to reply

Related Discussions
Black-Scholes equation (a type of diffusion equation) Calculus & Beyond Homework 1
Black-Scholes PDE and finding the general solution Differential Equations 1
Black-Scholes formula Social Sciences 2
Black-Scholes Research Problem Differential Equations 1
Black Scholes Equation General Discussion 1