Applied Maths Seminar: Buchen -- Pricing Financial Derivatives with the Method of Images

Vanilla options belong to the class of path-independent derivatives and are most
conveniently priced using discounted risk-neutral expectation of the maturity payoff.
Many exotic options including barrier and lookback options however, belong to the class
of path-dependent derivatives.  These latter options are more easily priced using PDE
methods.  This seminar summarises the research of the author and his students on the
problem of pricing such path-dependent options in the classical Black-Scholes
framework.  In particular, we show how the path dependency can be eliminated using the
Method of Images, reducing the problem to pricing an equivalent vanilla option.
Illustrative examples include pricing barrier options, lookback options and
double-barrier options all of which use the Method of Images for absorbing boundaries.
We conclude the seminar by pricing a real estate lease, the only known financial example
to us of a reflecting boundary problem.