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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 well-known 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 delta-hedging portfolio.
We are able to estimate the corresponding discrete time hedging error that leads to complete justification of our hedging method for the non-increasing 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.

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308 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

v
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2

1

FORMULATION OF THE PROBLEM AND THE STATEMENT OF THE MAIN RESULTS 5
132 KB
3 2 SOME REGULARITY PROPERTIES OF THE AMERICAN OPTION VALUE FUNCTION 14
138 KB
4 3 THE WEIGHTED SQUARE INTEGRAL ESTIMATES FOR THE DIFFERENCE OF DERIVATIVES OF TWO CONVEX FUNCTIONS

3.1 The Weighted Square Integral Estimates for the First Derivative of a Twice Continuously Differentiable Function

3.2 Weighted Energy Estimates for the Difference of Two Convex Functions

26
168 KB
5 4 DERIVATION OF THE BASIC ESTIMATES FOR THE DISCRETE TIME HEDGING ERROR

 

52
144 KB
6 5 BIBLIOGRAPHY

65


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