WHY IS PARTIAL DIFFERENTIAL EQUATIONS IMPORTANT TO FINANCE?

I’m extraordinary because people in financial discuss it we to investigate prejudiced differential equations. I can’t appear to find the answer. Anyone know?

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You can all breathe now November 2, 2010 at 3:12 am

I’m not in finance so I can’t say for sure what applications there are for PDEs in finance.

However, PDEs are normally encountered in the physical sciences when you have a model of system or physical phenomenon where a variable (dependent) is dependent on more than one other variables (independent).

So for instance, you have z = f(x,y), here z is a function dependent on the two independent variables x and y. To find the derivative or rate of change of z with respect to say, time.

dz/dt = ∂z/∂x dx/dt + ∂z/∂y dy/dt

Anyhow, I hope you get the idea. The same applies to PDEs, whenever there is a financial model say the current stock price of MSFT, z which is dependent of x = revenues and y = retail cost, etc. Depending on how the model is setup, you could potentially end up with a PDE.

An example of a financial model stated in terms of PDE is the Black-Scholes equation that describes or models the propagation (like wave equation in mechanics or physics) of option prices, where f(t,x) is the probability density function or probability that price of a stock will have value of x at time t. The PDE describes how the change in probability varies over time.

∂f/ ∂t = a² ∂²f/∂x² < -- second order wave equation.

Here is a good book of reference on financial mathematical modeling.

The Mathematics of Financial Modeling and Investment Management - By Sergio Focardi, Frank J. Fabozzi

http://books.google.com/books?id=tekI8gcmxBwC&source=gbs_navlinks_s
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