

Title of Thesis
Discrete Time Hedging of the American Option 
Author(s)
Sultan Hussain 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2009 
Subject Mathematics 
Number of Pages 76 
Keywords (Extracted from title, table of contents and
abstract of thesis) Discrete, Time, Hedging, American, Option,
convex, payoff, square, integral 
Abstract In a complete
financial market we consider the discrete time hedging of the
American option with a convex payoff. It is wellknown that for the
perfect hedging the writer of the option must trade continuously in
time which is impossible in practice. In reality, the writer hedges
only at some discrete time instants.
The perfect hedging requires the knowledge of the partial derivative
of the value function of the American option in the underlying
asset, explicit form of which is unknown in most cases of practical
importance. At the same time several approximation methods are
developed for the calculation of the value function of the American
option.
We establish in this thesis that, having at hand any uniform
approximation of the American option value function at the
equidistant discrete rebalancing times it is possible to construct a
discrete time hedging portfolio the value process of which uniformly
approximates the value process of the continuous time perfect
deltahedging portfolio.
We are able to estimate the corresponding discrete time hedging
error that leads to complete justification of our hedging method for
the nonincreasing convex payoff functions including the important
case of the American put. It is essentially based on a recently
found new type square integral estimate for the derivative of an
arbitrary convex function by Shashiashvili [23]. We generalize the
latter square integral estimate to the case of the family of the
weight functions, satisfying certain conditions.

