

Title of Thesis
American Foreign Exchange Put Option Problem 
Author(s)
NASIR REHMAN 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2009 
Subject Mathematics 
Number of Pages 79 
Keywords (Extracted from title, table of contents and
abstract of thesis) American, Foreign, Exchange, Option, Problem,
currency, monotonicity, volatility, strikes, maturities 
Abstract The classical
GarmanKohlhagen model for the currency exchange assumes that the
domestic and foreign currency riskfree interest rates are constant
and the exchange rate follows a lognormal diffusion process.
In this thesis we consider the general case, when exchange rate
evolves according to arbitrary onedimensional di usion process with
local volatility that is the function of time and the current
exchange rate and where the domestic and foreign currency riskfree
interest rates may be arbitrary continuous functions of time. First
nontrivial problem we encounter in timedependent 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 righthand 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 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.

