Register to reply

Is the Black-Scholes equation a differential equation?

by TheGhostInside
Tags: blackscholes, differential, equation
Share this thread:
TheGhostInside
#1
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 Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
Mute
#2
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,

e.g.,

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

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