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

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
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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, est