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:
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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 [9]
|
(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