   SYLLABUS  Previous: 4.3.1 Transformation to log-normal  Up: 4.3 Methods for European  Next: 4.3.3 Black-Scholes formula

4.3.2 Solution of the normalized diffusion equation [ SLIDE Fourier-Laplace fwd - bck || Solution binary options || same VIDEO as previous section: modem - LAN - DSL]

Readers interested in solving the diffusion equation (4.3.1#eq.5) analytically are likely to be familiar with the Fourier-Laplace transform that is commonly used to solve initial and boundary value problems. Others may skip the whole derivation and verify that the final result (4.3.2#eq.11) indeed satisfies the Black-Scholes equation (3.4#eq.4).
Start by transforming (4.3.1#eq.5) with a Laplace transform in time (4.3.2#eq.1)

Note that the condition is important to ensure causality. Integrate the first term by parts and substitute a Dirac function for the initial condition  (4.3.2#eq.2)

The notation refers to the Laplace transform of . Spatial derivatives are dealt with a Fourier transform (4.3.2#eq.3)

and yields an explicit solution in Fourier-Laplace space (4.3.2#eq.4)

The pole in the complex plane for needs to be taken into account when inverting the Laplace transform in a causal manner (4.3.2#eq.5)

where the residue theorem has been used to evaluate the complex integral, closing the contour in the lower half plane where the phase factor decays exponentially. Invert the Fourier transformation (4.3.2#eq.6)

and use the formula (3.323.2) from Gradshteyn & Ryzhik (4.3.2#eq.7)

here with and to write down a solution of the diffusion equation (4.3.2#eq.8)

This shows that a Dirac function assumed as initial condition in (4.3.2#eq.2) spreads out into a Gaussian as time evolves forward. A superposition of a whole series of Dirac functions can now be used to decompose any arbitrary initial condition  (4.3.2#eq.9)

and after evolving each Dirac functions separately using (4.3.2#eq.8), can again be superposed at a later time when : (4.3.2#eq.10)

Transforming back into financial variables (4.3.1#eq.1), some algebra finally yields a general formula for the price of a binary option with a terminal payoff  (4.3.2#eq.11)

SYLLABUS  Previous: 4.3.1 Transformation to log-normal  Up: 4.3 Methods for European  Next: 4.3.3 Black-Scholes formula