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The Mathematics of Derivatives: Tools for Designing Numerical Algorithms
book

The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

by Robert L. Navin
December 2006
Intermediate to advanced content levelIntermediate to advanced
205 pages
4h 26m
English
Wiley
Content preview from 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

images

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

images

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

images

Then

images

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

images

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

images

Finally substitute ttT, and xxxT to get

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Publisher Resources

ISBN: 9780470047255Purchase book