The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

APPENDIX B

Solving for the Green’s Function of the Black-Scholes Equation

We want to solve

We can set the reference coordinates to zero because they can be added later by subtracting from the solution’s arguments. This is because the differential equation does not contain any explicit functions of the coordinates—that is, it is translation invarian. Thus

Take the 2-D Fourier transform of the equation, that is, (x, t) (k, ω),

Then

The residue theorem from complex analysis means that the result is only nonzero for closing the integral over ω in the lower half of the complex ω-plane. This happens only if t < 0. Thus

By completing the square (recalling t < 0), we can factor out a function of x and t times an integral,

Finally substitute t → t − T, and x → x − x_T to get