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Applications of least-squares regressions to pricing and hedging of financial derivatives [Elektronische Ressource] / Andreas J. Grau

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¨ ¨Technische Universitat MunchenZentrum MathematikApplications of Least Squares Regressions to Pricingand Hedging of Financial DerivativesAndreas J. GrauVollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Mathematik der Technischen Universitat¨ Munchen¨zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender: Univ. Prof. Dr. Bernd SimeonPrufer¨ der Dissertation: 1. Univ. Prof. Dr. Rudi Zagst2. Prof. Phelim P. Boyle, Ph.D. (em.),Wilfrid Laurier University, Waterloo, Kanada,(nur schriftliche Beurteilung)3. Univ. Prof. Dr. Hans Joachim BungartzDie Dissertation wurde am 12.12.2007 bei der Technischen Universitat¨ eingereicht und durch dieFakultat¨ fur¨ Mathematik am 05.02.2008 angenommen.2Acknowledgements iAcknowledgementsThis thesis would not have been possible without the support of numerous people. First,I would like to thank my supervisor Prof. Dr. Rudi Zagst for assigning challenging tasks to me,discussing my ideas patiently and providing assistance in order to making this thesis readable. Heeven left me enough room for my unconventional ideas. I thank Prof. Dr. Peter Forsyth, Prof. Dr.Ken Vetzal and Prof. Dr. Jan Kallsen for discussions with valuable input and thought provokingimpulses. This is especially true for my co supervisor Prof. Dr. Phelim Boyle and the third refereeof this thesis Prof. Dr. Hans Bungartz.As well, I would like to thank my collegues Dr.

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¨ ¨Technische Universitat Munchen
Zentrum Mathematik
Applications of Least Squares Regressions to Pricing
and Hedging of Financial Derivatives
Andreas J. Grau
Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Mathematik der Technischen Universitat¨ Munchen¨
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender: Univ. Prof. Dr. Bernd Simeon
Prufer¨ der Dissertation: 1. Univ. Prof. Dr. Rudi Zagst
2. Prof. Phelim P. Boyle, Ph.D. (em.),
Wilfrid Laurier University, Waterloo, Kanada,
(nur schriftliche Beurteilung)
3. Univ. Prof. Dr. Hans Joachim Bungartz
Die Dissertation wurde am 12.12.2007 bei der Technischen Universitat¨ eingereicht und durch die
Fakultat¨ fur¨ Mathematik am 05.02.2008 angenommen.2Acknowledgements i
Acknowledgements
This thesis would not have been possible without the support of numerous people. First,
I would like to thank my supervisor Prof. Dr. Rudi Zagst for assigning challenging tasks to me,
discussing my ideas patiently and providing assistance in order to making this thesis readable. He
even left me enough room for my unconventional ideas. I thank Prof. Dr. Peter Forsyth, Prof. Dr.
Ken Vetzal and Prof. Dr. Jan Kallsen for discussions with valuable input and thought provoking
impulses. This is especially true for my co supervisor Prof. Dr. Phelim Boyle and the third referee
of this thesis Prof. Dr. Hans Bungartz.
As well, I would like to thank my collegues Dr. Stefan Dirnstorfer, Christina Niethammer and
Christoph Hanle¨ for making the time working on my dissertation enjoyable.
Last, but not least, a special thanks goes to my parents and my brother for their constant full
support from back home.ii AcknowledgementsContents
Introduction 1
1 Mathematical Foundations 5
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Regression Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Approximation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Pricing and Hedging in Complete Markets . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.3 Exercisable Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Numerical Methods for Option Valuation . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Direct PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 The Challenge of Path Dependency 33
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Pricing Using Feature Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 A Discretely Sampled Asian Option . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.4 Summary of the Feature Extraction . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Pricing Delayed Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.1 Numerical Example: A Parisian Option . . . . . . . . . . . . . . . . . . . . . 47
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Moving Window Asian Options 51
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Moving Window Asian Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Continuous Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4.1 Asian American Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4.2 Exponential Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.3 Moving Window Asian Option . . . . . . . . . . . . . . . . . . . . . . . . . . 57
iii3.5 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.2 Choice of Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5.3 Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6.2 Heuristic Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Callable Convertible Bonds 73
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Models for Convertible Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 No Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.3 Cash Flows, Call and Put Provisions . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4.1 PDE Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4.2 Monte Carlo Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5.1 Convergence Analysis - PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5.2gence - Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 91
4.5.3 Properties of Different Call Strategies . . . . . . . . . . . . . . . . . . . . . . 93
4.5.4 Moving Window and Call Notice Protection . . . . . . . . . . . . . . . . . . 100
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Simulation Based Hedging and Incomplete Markets 105
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Basic Requirements for a Pricing Method . . . . . . . . . . . . . . . . . . . . 107
5.3.2 Hedging and Pricing of a Liquidly Traded Security . . . . . . . . . . . . . . 108
5.3.3 Setting for an Illiquid Market . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.4 Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.5 American Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 Monte Carlo Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.1 Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.2 Simulation Based Hedging in a Black Scholes Market (European Options) . 126
5.4.3 Hedged Monte Carlo (Potters et. al. [96]) in a Black Scholes Market (Euro
pean Options) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4.4 Hedging in a Black Scholes Market (American Put Option) 128
5.4.5 Remarks on the Computational Efficiency . . . . . . . . . . . . . . . . . . . . 129
5.4.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 Conclusions 143
iv7 Appendix 145
7.1 Important Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Notes for the Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Proof of Equation (1.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4 Proof of Equation Set (4.4) (4.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.5 Feature Extraction in Octave/MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6 Simulation Based Hedging in Octave/MATLAB . . . . . . . . . . . . . . . . . . . . 155
Bibliography 156
vviIntroduction 1
Introduction
The introduction of financial options delivered a valuable contribution to the efficiency of the
markets in the world. Investors seeking risks - speculators - can use financial options to obtain
large effects with little money. Investors avoiding risks - hedgers - can now buy insurances for
their portfolios at reasonable prices. In book 1, Chapter 11 of Politics, Aristotle already tells the
story of Thales of Miletus (624 547 BC) basically buying an option on olive crop. But, it took until
the 1970s where large volumes of financial options were traded at derivatives exchanges. Today,
the underlying problem of pricing and hedging options is well known and several approaches of
its solution have been proposed. Assuming a simple complete market without any transaction
cost, the Black Scholes model has been most successful since its introduction 1973 [17].
Despite the beauty and simplicity of the Black Scholes model, the efficient evaluation of many
exotic options remains challenging. It turned out quickly that analytic solutions e.g. from Mer-
ton [87] are by far not sufficient for the evaluation of traded securities. Consequently, a large
variety of procedures has been developed for the solution of the governing partial differential
Equation (PDE). Direct solvers are e.g. a finite differences method by Schwartz [103], the finite el
ement method and the finite volume methods by Forsyth and Vetzal [50] resp. Zvan et al [124] as
well as a mesh less method by Li et al [80]. A popular solver is the Cox Ross Rubinstein method
(CRR) [36], which discretizes the asset price process by a binomial tree and solves for the option
price by a simple recursion, which is easy to implement. But, the CRR method does not have as
good convergence properties as the other PDE methods.
A different approach solving for option prices in the Black Scholes model focuses on the un
derlying stochastic differential equation (SDE). This is done by simulating Monte Carlo paths
of the underlying asset and computing option prices as some expected value, presented first by
Boyle [21, 23, 53]. While option features such as an early exercise can easily be evaluated in a PDE
solver (cp. Forsyth and Vetzal [50]), this is hard for Monte Carlo methods. Carrier` e [32] presented
the first practical Monte Carlo method for the valuation of options with early exercise features in
1996, which became popular after being extended by Longstaff and Schwartz [81] in 2001. This
method is called Least Squares Monte Carlo.
That means the main methods for option valuation are PDE solvers and Monte Carlo simula
tion. On the one hand, for many pricing problems PDE solvers deliver highly accurate solutions2 Introduction
in little time. This is especially true for low dimensional pricing problems. But, not all of them
are low dimensional. Especially path dependent options often require the introduction of addi
tional state variables. Reaching four or five dimensions, the pricing becomes usually infeasible
for current PDE and computer technology. On the other hand, Monte Carlo methods can price
options independent of the dimension of the pricing problem. But, these methods are converging
slowly such that highly accurate solutions often cannot be obtained. Additionally, the valuation
of high dimensional financial derivatives with embedded options like an early exercise is still
challenging, even if the method of Carrier` [32, 81] is used.
This work will focus on pricing and hedging of derivatives with Monte Carlo simulation. In
some cases, direct numerical PDE solutions will be used as a reference. We will provide insight
into the versatile applications of regression methods for the Monte Carlo valuation. As a result,
very fast valuation procedures are developed: In some cases the methods developed in this dis
sertation are the first of its kind which handle specific exotic options. Especially the pricing of a
high dimensional Moving Window Asian option with early exercise and the implementation of a
moving window soft call constraint of convertible bonds are solved for the first time in this thesis.
Prior technology could not cope with the high dimensional pricing problem together with
an early exercise feature. The PDE method can deal with an early exercise feature easily, but
high dimensional problems are unfeasible. Monte Carlo methods can deal with high dimensional
problems, but an early exercise of a high dimensional option pricing problem is hard to treat
correctly in the previous setting.
Another contribution of this thesis is the Simulation Based Hedging method which connects
realistic models for the underlying with suitable pricing and hedging without a detour to a so
called risk neutral measure. The Simulation Based Hedging has extraordinary properties: E.g.
using the Black Scholes assumptions its convergence to the Black Scholes prices is much faster
than the comparable Longstaff Schwartz Least Squares Monte Carlo [81]. Furthermore, the un
derlying can follow any real world process: The algorithm always computes the optimal hedging
strategy and thus attains realistic risk adjusted prices and hedges. This can also be done using
multiple hedge instruments.
Consequently, the new Simulation Based Hedging is a new pricing framework together with
a numerical method for the solution to option pricing problems in so called incomplete markets.
The whole setting of the framework is new, but related to risk minimization techniques for op
timal hedging of financial options presented by several authors [46, 95, 47, 33]). Especially, the
setting of Simulation Based Hedging can be seen as an extension to the variance minimization
presented by Schweizer [104] and the presented numerical solution is related to a method pre
sented by Potters et. al. [96] resp. Pochart and Bouchaud [95].
The main results of this dissertation are:
† The usual Monte Carlo method is altered for a quicker evaluation: Extracting the main
features of the option’s payoff, a simple regression can accelerate the evaluation of path