Why is partial differential equations important to finance?
I'm curious why people in finance tell you to study partial differential equations. I can't seem to find the answer. Anyone know?
Finance best answer:
Answer by You can all breathe now
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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Finance
Dept. of Finance /07
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