analytic solution of Black-Scholes PDE
Here we present an analytical solution for the Black-Scholes partial differential equation,
We begin by making the substitution:
which is motivated by the fact that it is the portfolio value discounted by the interest rate (see the derivation of the Black-Scholes formula) that is a martingale. Using the product rule on , we derive the PDE that the function must satisfy:
Next, we make the substitutions:
These changes of variables can be motivated by observing that:
The underlying process described by the variable is a geometric Brownian motion (as explained in the derivation of the Black-Scholes formula itself), so that describes a Brownian motion, possibly with a drift. Then should satisfy some sort of diffusion equation (well-known in physics).
The evolution of the system is backwards from the terminal state of the system. Indeed, the boundary condition is given as a terminal state, and the coefficient of is positive in equation (2). (Compare with the standard heat equation, , which describes a temperature evolving forwards in time.) So to get to the heat equation, we have to use a substitution to reverse time.
substituting in equation (2), we find:
The first partial derivative with respect to does not cancel (unless ) because we have not take into account the drift of the Brownian motion. To cancel the drift (which is linear in time), we make the substitutions:
Under the new coordinate system , we have the relations amongst vector fields:
leading to the following of equation (3):
which is one form of the diffusion equation. The domain is on and ; the initial condition is to be:
The original function can be recovered by
The fundamental solution of the PDE (4) is known to be:
In terms of the original function :
() which agrees with the result derived using probabilistic methods (http://planetmath.org/BlackScholesFormula).
|Title||analytic solution of Black-Scholes PDE|
|Date of creation||2013-03-22 16:31:34|
|Last modified on||2013-03-22 16:31:34|
|Last modified by||stevecheng (10074)|