Model-Independent Arbitrage Bounds on American Put Options

185 Pages·2015·1.33 MB·English

Model-Independent Arbitrage
Bounds on American Put
Options
submitted by
Christoph Hoeggerl
for the degree of Doctor of Philosophy
of the
University of Bath
Department of Mathematical Sciences
December 2014
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Signature of Author.................................................................
Christoph Hoeggerl

Fu¨r meine Eltern

SUMMARY
The standard approach to pricing financial derivatives is to determine the discounted,
risk-neutralexpectedpayoff underamodel. Thismodel-basedapproachleaves usprone
to model risk, as no model can fully capture the complex behaviour of asset prices in
the real world.
Alternatively, we could use the prices of some liquidly traded options to deduce
no-arbitrage conditions on the contingent claim in question. Since the reference prices
are taken from the market, we are not required to postulate a model and thus the
conditions found have to hold under any model.
In this thesis we are interested in the pricing of American put options using the
latter approach. To this end, we will assume that European options on the same
underlyingand with the same maturity are liquidly traded in the market. We can then
usethemarketinformationincorporatedintothesepricestoderiveasetofno-arbitrage
conditions that are valid under any model. Furthermore, we will show that in a market
trading only finitely many American and co-terminal European options it is always
possible to decide whether or not the prices are consistent with a model. If they are
not there has to exist arbitrage in the market.

ACKNOWLEDGEMENTS
Firstandforemost,IwanttothankmysupervisorAlexanderM.G.Coxforhisguidance
and support throughout the years. I am very grateful for his interest in my research
and for always finding time in his increasingly busy schedule.
I would also like to express my gratitude to the members of the ProbL@b for pro-
vidingsuch a friendly and stimulating working environment. Both theweekly seminars
and the reading groups have allowed me to gain a more in-depth understanding of
various topics in probability.
Further I would like to thank all the people that have made these last few years so
enjoyable. In particular, I want to thank Marion, Deepak, Vaios, Curdin and Melina
for the odd beer and for having always been there when I needed them. My office
mates Yi and Matteo for the great chats as well as Kuntalee, Maren and Horatio for
the exciting games of squash and badminton.
A very special thank you to Fani for her love and devotion and for cheering me up
when I needed it the most.
Finally, my parents and my sister who have always supported and encouraged me
— thank you

Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction 2
1.1 Robust pricing of derivatives . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 The potential picture . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 The Chacon-Walsh solution to the Skorokhod embedding problem 7
1.3.3 Connecting the potential of a measure to option prices . . . . . . 9
1.3.4 Arbitrage in the model-free setting . . . . . . . . . . . . . . . . . 11
2 Model-independent no-arbitrage conditions on American put options 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Necessary conditions for the American put price function A . . . . . . . 15
2.3 Sufficiency of the conditions on the American put price function A . . . 21
2.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Existence and calculation of the critical time . . . . . . . . . . . 24
2.3.3 The splitting procedure . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.4 Convergence of the Algorithm . . . . . . . . . . . . . . . . . . . . 37
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Arbitrage situations in markets trading American and co-terminal
European options 46
3.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 No-arbitrage bounds on option prices . . . . . . . . . . . . . . . . . . . . 51
3.4 Construction of the price functions . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 Computation of the prices for strikes in K(P∗) . . . . . . . . . . 54
0
3.4.2 Price-misspecifications and their corrections . . . . . . . . . . . . 55
3.4.3 Expansion of the initial set of prices . . . . . . . . . . . . . . . . 58
i
Contents
3.4.4 Computation of the prices for strikes in K(P∗) . . . . . . . . . . 58
i
3.4.5 Modification to the price corrections . . . . . . . . . . . . . . . . 59
3.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Arbitrage situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6.1 Sub- and super-replicating strategies . . . . . . . . . . . . . . . . 64
3.6.2 Situation I:Violation of theno-arbitrage conditions by theprices
of traded options . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6.3 Situation II: Violation of e ≥ Erhs(K ,P∗) . . . . . . . . . . . 74
i,p lb i,p 0
3.6.4 Situation III: Violation of A (K ,P )≥ a . . . . . . . . . 81
ub i,p i,p−1 i,p
3.7 Admissibility of the price functions . . . . . . . . . . . . . . . . . . . . . 86
3.7.1 Violation of the upper bound under P∗ . . . . . . . . . . . . . . 88
1
3.7.2 Violation of convexity under P∗ . . . . . . . . . . . . . . . . . . . 95
1
3.7.3 Violations under P∗, i ≥ 2 . . . . . . . . . . . . . . . . . . . . . . 114
i
3.8 Convergence of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . 117
3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.10.1 Properties of the Legendre-Fenchel condition . . . . . . . . . . . 121
3.10.2 General properties of the price functions . . . . . . . . . . . . . . 123
3.10.3 General properties of the revised price functions . . . . . . . . . 131
3.10.4 Properties of the price functions when a violation of A≥ A occurs133
3.10.5 Properties of the price functions under P∗ when a violation of
1
A ≥ A occurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.10.6 Properties of the price functions under P∗ when a violation of
1
convexity occurs . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.10.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.11 Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Bibliography 172
ii