Exponentially fitted schemes and Box Methods for Exotic Options
Master's Thesis in Financial Mathematics, Halmstad University, Sweden
Supervisor:
Description
Exotic options derivatives that are especially designed for customers needs.
For these nonstandard options the wellknown transformations of the BlackScholes
equation to the heat equation do not work any more.
Hence, one has to discretize the BlackScholes equation directly.
If one simply applies standard numerical methods for parabolic equations without
taking special care of the financial meaning of the variables then
one faces severe problems.
Here, in this thesis, we will show how to modify standard numerical schemes,
e.g. by nonlocal discretization approaches,
to remove unrealistic restrictions on the time step,
preserve the positivity of the solution obeying a discrete maximum principle,
adequate treatment of nonsmooth initial data
(payoff function), etc. while keeping all the stability properties
of the original scheme.
As an illustrative example we consider a discretely monitored barrier option,
a knockandout European Put option.
Keywords
positivitypreserving finite difference schemes, Mmatrices, nonlocal
discretization, nonstandard finite difference methods (NSFD), discrete maximum
principle, spurious oscillations, discretely monitored Barrier options,
Box scheme, exponentiallyfitted method, error damping, nonsmooth initial conditions
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