Is the BlackScholes equation a differential equation?by TheGhostInside Tags: blackscholes, differential, equation 

#1
Jan2813, 04:30 PM

P: 1

Hi everyone, first post. To anyone who has had experience with the background of the BlackScholes 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? 



#2
Jan2813, 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 FokkerPlanck 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 FokkerPlanck wikipedia page for a brief intro and some further references). 


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