# Is the Black-Scholes equation a differential equation?

by TheGhostInside
Tags: blackscholes, differential, equation
 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?
 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).

 Related Discussions Calculus & Beyond Homework 1 Differential Equations 1 Social Sciences 2 Differential Equations 1 General Discussion 1