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# Green's Function Solution

We now use the Green's function for the Diffusion or Heat equation[4], which is the solution to that equation for a point (or delta function) source at point z' at time t'=0

 (19)

The verification of this Green's function solution is shown in Appendix A. The Green's function shows the Gaussian diffusion of the pointlike input with distance from the input (z-z') increasing as the square root of the time t', as in a random walk.

We can use the Green's function to write the solution for in terms of summing over its input values at points z'on the boundary at the initial time t'=0

 (20)

Putting in the initial condtions at t'=0, where vanishes for negative z', gives

 (21)

To do the integral we change the variable to

 q = (22) dz' = (23)

The lower limit on the q integral is now

 (24)

where substitution gives the dimensionless

 (25)

In the first term we now complete the square to get a new variable

 (26)

The new lower limit on q' in the first term is now -d1 where

 (27)

After completing the square on the first term, the exponent simplifies to

 (28)

Both integrals are now related to the Cumulative Distribution Function of the Normal Distribution

 (29)

If we change t to -t in the above integral and invert the limits we get the form of our integrals

 (30)

We now have our solution for the canonical

 (31)

Finally, we use the facts that , and that , and the conversion

 (32)

to get the Black-Scholes solution

 w(x,t) = x N(d1)-c e-r(t*-t)N(d2). (33)

Next: Post-Analysis Up: Solution of the Black Previous: Conversion of the Black-Scholes
Dennis Silverman
1999-05-20