Abstract:
The classical Garman-Kohlhagen model for the currency exchange assumes that the
domestic and foreign currency risk-free interest rates are constant and the exchange
rate follows a log-normal diffusion process.
In this thesis we consider the general case, when exchange rate evolves according
to arbitrary one-dimensional diffusion process with local volatility that is the function
of time and the current exchange rate and where the domestic and foreign currency
risk-free interest rates may be arbitrary continuous functions of time. First non-trivial
problem we encounter in time-dependent case is the continuity in time argument of
the value function of the American put option and the regularity properties of the
optimal exercise boundary. We establish these properties based on systematic use
of the monotonicity in volatility for the value functions of the American as well
as European options with convex payoffs together with the Dynamic Programming
Principle and we obtain certain type of comparison result for the value functions
and corresponding exercise boundaries for the American puts with different strikes,
maturities and volatilities.
Starting from the latter regularity property that the optimal exercise boundary
curve is left continuous with right-hand limits we give a mathematically rigorous and
transparent derivation of the significant early exercise premium representation for
the value function of the American foreign exchange put option as the sum of the
European put option value function and the early exercise premium.
The proof essentially relies on the particular property of the stochastic integral
with respect to arbitrary continuous semimartingale over the predictable subsets of
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its zeros. We derive from the latter the nonlinear integral equation for the optimal
exercise boundary which can be studied by numerical methods.
We establish several continuity estimates for the American option value process,
the optimal hedging portfolio and the corresponding consumption process with respect
to volatility function. For these estimates the volatility is assumed to be arbitrary
strictly positive bounded function of time.