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Is the Black-Scholes equation a differential equation?

by TheGhostInside
Tags: blackscholes, differential, equation
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Jan28-13, 04:30 PM
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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?
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Jan28-13, 05:39 PM
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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).

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