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Sur les lois de diffusions et de modèles financiers avec coefficients non globalement réguliers, On probability distributions of diffusions and financial models with non-globally smooth coefficients

De
171 pages
Sous la direction de Vlad Bally
Thèse soutenue le 23 novembre 2010: Scuola Normale Superiore di Pisa (ENS-PISE), Paris Est
Des travaux récents dans le domaine des mathématiques financières ont fait émerger l'importance de l'étude de la régularité et du comportement fin des queues de distribution pour certaines classes de diffusions à coefficients non globalement réguliers. Dans cette thèse, nous traitons des problèmes issus de ce contexte. Nous étudions d'abord l'existence, la régularité et l'asymptotique en espace de densités pour les solutions d'équations différentielles stochastiques en n'imposant que des conditions locales sur les coefficients de l'équation. Notre analyse se base sur les outils du calcul de Malliavin et sur des estimations pour les processus d'Ito confinés dans un tube autour d'une courbe déterministe. Nous obtenons des estimations significatives de la fonction de répartition et de la densité dans des classes de modèles comprenant des généralisations du CIR et du CEV et des modèles à volatilité locale-stochastique : dans ce deuxième cas, les estimations entraînent l'explosion des moments du sous-jacent et ont ainsi un impact sur le comportement asymptotique en strike de la volatilité implicite. La modélisation paramétrique de la surface de volatilité, à son tour, fait l'objet de la deuxième partie. Nous considérons le modèle SVI de J. Gatheral, en proposant une nouvelle stratégie de calibration quasi-explicite, dont nous illustrons les performances sur des données de marché. Ensuite, nous analysons la capacité du SVI à générer des approximations pour les smiles symétriques, en le généralisant à un modèle dépendant du temps. Nous en testons l'application à un modèle de Heston (sans et avec déplacement), en générant des approximations semi-fermées pour le smile de volatilité
-Equations différentielles Stochastiques
-Mathématiques financières
-Estimation de densités
-Calcul de Malliavin
-Volatilité stochastique
-Volatilité implicite
Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on « tube estimates » for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization. We provide and test the numerical application to the Heston model (without and with displacement), for which we generate semi-closed expressions of the smile
-Stochastic differential equations
-Mathematical Finance
-Density estimation
-Malliavin calculus
-Stochastic Volatility
-Implied volatility
Source: http://www.theses.fr/2010PEST1017/document
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Scuola Normale Superiore
Tesi di Perfezionamento in Matematica per la Finanza
Thèse présentée pour obtenir le grade de
Docteur de l’Université Paris-Est
Spécialité: Mathématiques
parStefano De Marco
of Diffusions and Financial Models
with non-globally smooth coefficients
dirigée par Vlad Bally et Maurizio Pratelli
Rapporteurs: Emmanuel Gobet et Arturo Kohatsu-Higa.
Soutenue le 23 novembre 2010 devant le jury composé de:
Vlad Bally Directeur de thèse
Emmanuel Gobet Rapporteur
Giorgio Letta Examinateur
Stefano Marmi
Claude Martini Examinateur
Maurizio Pratelli Directeur de thèse
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Acknowledgments
The achievement of this thesis would not have been possible without the guidance and con-
stant advice of my supervisors, Prof. Vlad Bally and Prof. Maurizio Pratelli. To them goes
my deep and sincere gratitude, for the time and knowledge (and patience) they have shared
with me.
I am grateful to Prof. Emmanuel Gobet and Prof. Arturo Kohatsu-Higa, who accepted
to report on this thesis. Their capability of understanding and summarizing the sense and
motivation of the problems I have tackled, together with their precious remarks and sugges-
tions, have been an important help. They allowed me to correct and clarify the presentation
of the results, and to look for new directions of research and development.
I am thankful to Prof. Giorgio Letta and Prof. Stefano Marmi, for accepting to take
part in my Ph.D defense: I am honoured by their presence. I would like to express the same
gratitude to Claude Martini. His presence at the concluding moment of my doctoral work is
particularly significant to me.
My work has surely benefited from the friendly and stimulating atmosphere at Scuola
NormaleSuperioreandattheLaboratoired’AnalyseMathématiquesAppliquéesatUniversité
Paris-Est Marne-la-Vallé. I have been kindly welcomed by the latter since the year of my
Master in Mathématiques et Applications (which was a great experience. I have to thank
Prof. Damien Lamberton for his amazing teaching skills in stochastic calculus - to the end
of my Ph.D. The same environment I found wiithin the C.E.R.M.I.C.S. research group, who
hosted me during the months preceding my defense. I will have the opportunity to benefit
from their experience, and to develop a fruitful collaboration with them in the framework of
my ongoing post-doc project.
I shall not forget how much I owe to Zeliade System’s team: from their knowledge of
mathematical finance and information technology, to their kindness and the friendly and
energizing atmosphere they always contribute to create. For their ability and patience in
making the basis of quantitative financial modeling more clear to me, and for helping me
with Python and C# (and I did need help with that), I say a great thank you to all of them.
I shared with them some very pleasant moments. Thank you Steve for the never-lacking
coffee!
I shall always be truly grateful to my parents, for their ever-present care and encourage-
ment. Among others, my interest for sciences would perhaps have never come to light if it
had not been for them. I am thankful to my brother, and to the rest of my family (one of
the best I could wish for!), for being always there.
My days in France have been so much bright since when I met Lola. I would like to say
a big thank you also to her family, who gave me such a warm welcome.
During the preparation of this thesis in Italy and France, I had the great luck to meet
and be surrounded by so many kind and smart people. I say thank you to all the friends in
tel-00588686, version 1 - 26 Apr 20114
Padova, to the perfezionandi of Scuola Normale, to the Ph.D candidates of L.A.M.A., to the
Italian friends in Paris, to the French friends.
Apart from the Scuola Normale Superiore, my work in France has been supported by the
following institutions: the Fondazione “Ing. Aldo Gini” with a scholarship for foreign
studies in 2007, the Università Italo-Francese - Université Franco-Italienne in the
framework of the Vinci project in 2008, and the European Science Foundation in the
framework of the AMaMeF research activity in 2009. Their support is gratefully acknowl-
edged.
tel-00588686, version 1 - 26 Apr 2011Abstract
In this Ph.D. dissertation we deal with some issues of regularity and estimation of probability
laws for diffusions with non-globally smooth coefficients, with particular focus on financial
models.
The analysis of probability laws for the solutions of Stochastic Differential Equations (SDEs)
driven by the Brownian motion is among the main applications of the Malliavin calculus
on the Wiener space: typical issues involve the existence and smoothness of a density, and
the study of the asymptotic behaviour of the distribution’s tails. The classical results in
this area are stated assuming global regularity conditions on the coefficients of the SDE: an
assumption which fails to be fulfilled by several financial models, whose coefficients involve
square-root or other non-Lipschitz continuous functions. Then, in the first part of this thesis
(chapters 2, 3 and 4) we study the existence, smoothness and space asymptotics of densities
when only local conditions on the coefficients of the SDE are considered. Our analysis is
based on Malliavin calculus tools and on tube estimates for Itô processes, namely estimates
on the probability that an Itô process remains around a deterministic curve up to a given
time. We give applications of our results to general classes of option pricing models, including
generalisations of CIR and CEV processes and some Local Stochastic Volatility models. In
the latter case, the estimates we derive on the law of the underlying price have an impact on
moment explosion and, consequently, on the large-strike asymptotic behaviour of the implied
volatility.
Implied volatility modeling, in its turn, makes the object of the second part of this thesis
(chapters 5 and 6). We deal with some issues related to the problem of an efficient and
economical parametric modeling of the volatility surface. We focus on J. Gatheral’s SVI
model, first tackling the problem of its calibration to the market smile. We propose an ef-
fective quasi-explicit calibration procedure and display its performances on financial data.
Then, we analyse the capability of SVI to generate efficient time-dependent approximations
of symmetric smiles in general continuous models, building an explicit time-dependent pa-
rameterization. We provide and test the numerical application to the uncorrelated Heston
model (without and with displacement), generating semi-closed expressions for the smile.
Keywords: SDEs, Smoothness of densities, Local regularity, Tail asymptotics, Malliavin
calculus, Tube estimates for Itô processes, Law of the Stock price, Implied Volatility, SVI,
Heston, Calibration.
tel-00588686, version 1 - 26 Apr 2011Sommario
In questa tesi di perfezionamento, trattiamo dei problemi di regolarità e di stima di leggi
di probabilità per le diffusioni a coefficienti non globalmente regolari, con una particolare
attenzione per le applicazioni ai modelli finanziari.
Lo studio delle distribuzioni delle soluzioni di Equazioni Differenziali Stocastiche dirette dal
moto Browniano è una dei principali settori di applicazione del calcolo di Malliavin sullo
spazio di Wiener: delle problematique tipiche in quest’area sono l’esistenza e la regolarità
della densità e il comportamento asintotico delle code della distribuzione. I risultati classici
in questo settore sono formulati sotto condizioni di regolarità globale sui coefficienti dell’e-
quazione, un’ipotesicherisultaviolatanelcasodinumerosimodellifinanziari, icuicoefficienti
fanno intervenire delle radici quadrate o altre funzioni non globalmente lipschitziane. Di con-
seguenza, nella prima parte di questa tesi (capitoli 2, 3 e 4) studiamo l’esistenza, la regolarità
e il comportamento asintotico spaziale delle densità nel caso in cui si assumano solo delle
condizioni locali sui coefficienti dell’equazione. La nostra analisi è basata sugli strumenti del
calcolo di Malliavin e su delle stime per i processi di Itô confinati a restare attorno ad una
curva deterministica (“tube estimates”). Forniamo applicazione di questi risultati a delle classi
generali di modelli per la valutazione delle opzioni, includendo delle estensioni dei processi
CIR e CEV e dei modelli a volatilità locale-stocastica (LSV). Per questi ultimi, le stime che
otteniamo hanno un impatto sull’esplosione dei momenti dell’attivo sottostante, e quindi sul
comportamento asintotico in strike della volatilità implicita.
La modellizzazione della volatilità implicita, a sua volta, constituisce l’oggetto della seconda
parte della tesi (capitoli 5 e 6), nella quale affrontiamo alcune questioni legate alla costruzione
di una parametrizzazione economica ed efficiente della superficie di volatilità. Consideriamo
in particolare il modello SVI di J. Gatheral, per il quale proponiamo una nuova strategia
di calibrazione semi-esplicita, illustrandone le prestazioni su dei dati di mercato. Quindi,
analizziamo la capacità del modello SVI di generare delle approssimazioni parametriche per
gli smiles simmetrici, estendendolo ad un modello a coefficienti dipendenti dal tempo. In
particolare, ne formuliamo e implementiamo l’applicazione numerica ad un modello di Hes-
ton (senza e con displacement), generando delle approssimazioni semi-esplicite dello smile di
volatilità.
Parole chiave: Equazioni differenziali stocastiche, Regolarità della densità, Code della dis-
tribuzione, Calcolo di Malliavin, Tube estimates per i processi di Itô, Distribuzione del prezzo
del sottostante, Volatilità implicita, SVI, Heston, Calibrazione.
tel-00588686, version 1 - 26 Apr 2011Résumé
Dans cette thèse, nous traitons des problèmes de régularité et d’estimation de lois pour des
diffusions avec coefficients non globalement réguliers, avec une attention particulière pour les
modèles financiers.
L’étude des lois des solutions d’Equations Différentielles Stochastiques (EDS) dirigées par
le mouvement Brownien est un des principaux secteurs d’application du calcul de Malliavin
sur l’espace de Wiener : des problématiques typiques dans ce domaine concernent l’existence
et la régularité d’une densité et l’étude du comportement asymptotique des queues de la
distribution.Lesrésultatsclassiquessurcesujetrequièrentdesconditionsderégularitéglobale
sur les coefficients de l’EDS, une condition qui n’est pas satisfaite par plusieurs modèles
financiers, dont les coefficients font intervenir des racines carrées ou d’autre fonctions non
globalement lipschitziennes. Par conséquent, dans la première partie de cette thèse (chapitres
2, 3 et 4), nous étudions l’existence, la régularité et l’asymptotique en espace de densités
lorsqu’on n’impose que des conditions locales sur les coefficients de l’EDS. Notre analyse
dans cette partie se base sur les outils du calcul de Malliavin et sur des estimations pour les
processus d’Ito confinés dans un tube autour d’une courbe déterministe (“tube estimates”).
Nous appliquons ces résultats à des classes générales de modèles pour l’évaluation d’options,
comprenant des généralisations des processus CIR et CEV et des modèles à volatilité locale-
stochastique (LSV). Dans ce deuxième cas, les estimations que nous obtenons pour la loi du
sous-jacent entraînent l’explosion des moments et ont ainsi un impact sur le comportement
asymptotique en strike de la volatilité implicite.
La modélisation de la volatilité implicite, à son tour, fait l’objet de la deuxième partie de cette
thèse (chapitres 5 et 6), où nous abordons des questions liées au problème d’une modélisation
paramétriqueefficaceetéconomiquedelasurfacedevolatilité.Nousconsidéronsenparticulier
lemodèleSVIdeJ.Gatheral,pourlequelnousproposonsunenouvellestratégiedecalibration
quasi-explicite, dont nous illustrons les performances sur des données de marché. Ensuite,
nous analysons la capacité du SVI à générer de bonnes approximations paramétriques pour
lessmilessymétriques,enlegénéralisantàunmodèledépendantdutemps.Nousenformulons
et testons l’application à un modèle de Heston (sans et avec déplacement), en générant des
approximations semi-fermées du smile de volatilité.
Motsclé:Equationsdifférentiellesstochastiques,Régularitédesdensités,Asymptotiquesdes
queues, Calcul de Malliavin, Tube estimates pour les processus d’Itô, loi du Stock, Volatilité
implicite, SVI, Heston, Calibration.
tel-00588686, version 1 - 26 Apr 20112
tel-00588686, version 1 - 26 Apr 2011Contents 3
Contents
1 Introduction 7
1.1 Analysis of probability laws for SDEs . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 A motivation for extensions : financial models . . . . . . . . . . . . . . . . . . 9
1.2.1 A framework of non globally well-behaved coefficients . . . . . . . . . . 9
1.2.2 Connections between tail asymptotics and Implied Volatility . . . . . . 11
1.3 Outline of results : Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Upper and lower estimates under local assumptions . . . . . . . . . . . 13
1.4 Outline of results : Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 A closer look to implied volatility : Heston and SVI . . . . . . . . . . . 20
I Tail Asymptotics for SDEs with locally smooth coefficients 25
2 Smoothness and upper bounds on densities for SDEs 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Smoothness and upper bounds on densities for SDEs with locally smooth co-
efficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Elements of Malliavin Calculus . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Explicit bounds on integration by parts formula for diffusion processes 34
2.2.4 Proof of Theorems 2.2.1 and 2.2.2 . . . . . . . . . . . . . . . . . . . . . 41
2.3 Application to square root-type diffusions : a CIR/CEV process with local
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Exponential decay at1 . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.2 Asymptotics at 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Technical proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4.1 Proofs of Lemmas 2.2.1 and 2.2.2 . . . . . . . . . . . . . . . . . . . . . 59
2.4.2 Proof of Proposition 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 65
tel-00588686, version 1 - 26 Apr 20114 CONTENTS
3 Lower bounds on distribution functions and densities : the case of Local-
Stochastic Volatility models 67
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.1 Estimates around a Deterministic Curve . . . . . . . . . . . . . . . . . 74
3.2.2 Lower bounds for Cumulative Distribution Function and Moments . . . 75
3.2.3 Lower bounds for the density . . . . . . . . . . . . . . . . . . . . . . . 76
3.3 Proof of results in 3.2.1 and 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.1 A Lagrangian minimization problem . . . . . . . . . . . . . . . . . . . 81
3.4 Proof of results in 3.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6 Technical proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.6.1 Preliminary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.6.2 Proof of Lemma 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.6.3 Tools of Conditional Malliavin calculus . . . . . . . . . . . . . . . . . . 97
3.6.4 Proof of Proposition 3.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Tube estimates and general lower and upper bounds via time-change tech-
niques 103
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 The basic estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Lower and upper estimates of distribution functions : application to diffusions 110
4.3.1 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.2 Lower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.4 A lemma on suprema of Brownian motion . . . . . . . . . . . . . . . . . . . . 118
II Implied volatility and the SVI parametric model : calibration
and time-dependent extensions 121
5 Quasi-Explicit Calibration of Gatheral’s SVI model 123
5.1 A simple model and a delicate calibration . . . . . . . . . . . . . . . . . . . . 123
5.2 Parameter constraints and limiting cases . . . . . . . . . . . . . . . . . . . . . 125
5.2.1 Slopes and minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.2 Arbitrage constraints (b and ) . . . . . . . . . . . . . . . . . . . . . . 125
5.2.3 Limiting cases ! 0 and !1 (almost-affine smiles) . . . . . . . . . 126
5.2.4 Constraints on a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3 A convex optimization problem with Linear Program . . . . . . . . . . . . . . 128
5.3.1 Dimension reduction : drawing out the linear objective . . . . . . . . . 128
5.3.2 Explicit solution of the reduced problem . . . . . . . . . . . . . . . . . 129
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