corrections rapport de Claire

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François Pelletier 2014-01-24 20:18:35 -05:00
parent d4101c37e9
commit 9420600d37
15 changed files with 119 additions and 752 deletions

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@ -66,7 +66,7 @@ d'un opérateur intégral, ou à noyau, sur la fonction de densité.
Certaines d'entre elles permettent de déterminer entièrement leur
distribution. Parmi celles-ci, on retrouve la fonction caractéristique
et les fonctions génératrice des moments et des cumulants, qui sont
et les fonctions génératrices des moments et des cumulants, qui sont
les plus couramment utilisées.
Certaines transformées permettent de modifier la distribution d'une
@ -226,7 +226,7 @@ distribuées, la fonction caractéristique de $Z$ est la $n^e$ puissance
de celle de $X$:
\begin{equation}
\label{eq:convocaractIID}
\phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \left[\phi_{X_i}(s)\right]^n.
\phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \left[\phi_{X}(s)\right]^n.
\end{equation}
Cette fonction est donc une solution de rechange intéressante à
@ -467,7 +467,7 @@ cas, $N(t,dx)$ est une mesure aléatoire de Poisson.
On considère les processus de Lévy $\lbrace X(t) \rbrace$ et $\lbrace
Z(t) \rbrace$. Celui qui suit est défini comme étant un processus
subordonné et aussi un processus de Lévy, comme le démontre
subordonné et aussi un processus de Lévy, comme le démontrent
\cite{sato1999levy} et \cite{schoutens2003levy}:
\begin{equation}
\label{eq:processussubordonne}
@ -540,7 +540,8 @@ inférieure ou égale à $g(x)$:
g(x).
\end{align}
Alors, la fonction $f$ est intégrable.
Alors, la fonction $f$ est intégrable dans l'intervalle
$\mathit{I}$.
\subsection{Théorème de Fubini}
\label{sec:theoreme-de-fubini}

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@ -78,7 +78,7 @@ vers une variable aléatoire de distribution normale multivariée
centrée de matrice de variance-covariance $\mathbf{\Sigma}$:
\begin{align}
\label{eq:TCLmulti2}
\sqrt{T}\left(\mathbf{Y}_T - \mu\right)\ \stackrel{L}{\rightarrow}\
\sqrt{T}\left(\mathbf{Y}_T - \boldsymbol{\mu}\right)\ \stackrel{L}{\rightarrow}\
\mathcal{N}_k(0,\mathbf{\Sigma})
\end{align}

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@ -119,7 +119,7 @@ retrouver dans le code source. On considère entre autres des erreurs
d'arrondissement, de logique et de clarté du code, ainsi que des
particularités du matériel qui n'auraient pas été prises en compte par
le programmeur. Ces erreurs peuvent être difficiles à détecter, c'est
pourquoi un grand nombre de tests devrait être effectués avant de
pourquoi un grand nombre de tests devraient être effectués avant de
publier un logiciel de modélisation financière.
\section{Les rendements financiers}
@ -425,7 +425,7 @@ rendements financiers devrait posséder:
temps.
\end{enumerate}
La famille de distributions L stable semble être celle qui répond le
La famille de distributions L stables semble être celle qui répond le
mieux à l'ensemble de ces conditions \citep{walterlevy}. L'équation
suivante définit la propriété de L-stabilité de la distribution de la
variable aléatoire des rendements sur une période $R$:
@ -442,11 +442,11 @@ la forme suivante:
\left[1+\frac{i\beta \xi}{|\xi|} \tan{\frac{\alpha\pi}{2}} \right].
\end{align}
Le domaine et le rôle des paramètres de la distribution L stable est
décrit à la table \ref{tab:roleparam}. La flexibilité apportée par les
Le domaine et le rôle des paramètres de la distribution L stable sont
décrits à la table \ref{tab:roleparam}. La flexibilité apportée par les
quatre paramètres permet de remplir les quatre conditions établies au
début de cette section. De plus, l'absence, dans la majorité des cas,
de moments finis d'ordre supérieur à l'espérance, permet de tenir
de moments finis d'ordre supérieur à l'espérance permet de tenir
compte du mouvement erratique des prix et ainsi produire de larges
discontinuités de son processus. Elle permet aussi d'expliquer
l'apparence de corrélation sérielle, en considérant une probabilité

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@ -46,7 +46,7 @@ stationnaires. Cependant, la plupart des caractéristiques diffèrent:
\begin{itemize}
\item Discontinuité des trajectoires (processus de sauts);
\item Distribution asymétrique des accroissements;
\item Paramètre d'échelle et de temps entièrement dissociés.
\item Paramètres d'échelle et de temps entièrement dissociés.
\end{itemize}
Enfin, il possède une représentation alternative qui n'implique aucun
@ -125,7 +125,7 @@ Cette distribution est définie par la fonction de densité
$f_{\mu,\sigma}(x)$ et la fonction caractéristique
$\phi_{\mu,\sigma}(\xi)$:
\begin{align}
f_{\mu,\sigma}(x) &= \frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left\{\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2\right\}} \label{eq:fndensitenormale} \\
f_{\mu,\sigma}(x) &= \frac{1}{\sqrt{2\pi\sigma^2}}\exp{-\left\{\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2\right\}} \label{eq:fndensitenormale} \\
\phi_{\mu,\sigma}(\xi) &= \exp\left\{
i\mu\xi-\frac{\sigma^2\xi^2}{2}
\right\} \label{eq:fncaractnormale}.

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@ -17,11 +17,11 @@ suivant une distribution particulière. Le vecteur de paramètres de
celle-ci, $\theta$, de longueur $a$, appartenant à l'espace $\Omega
\subset \mathbb{R}^a$, doit être estimé à partir de l'échantillon. Une
première approche consiste à maximiser la fonction de vraisemblance
$LL(\theta;\mathbf{y})$, qui équivaut au produit de la densité
$L(\theta;\mathbf{y})$, qui équivaut au produit de la densité
$f(y;\theta)$ évaluée à chacune des réalisations $y(t)$:
\begin{align}
\label{eq:vraisemblance}
LL(\theta;\mathbf{y}) = \prod_{t=1}^T
L(\theta;\mathbf{y}) = \prod_{t=1}^T
f(y(t);\theta),\quad\theta\in\Omega.
\end{align}
@ -872,12 +872,12 @@ que la somme de deux variables aléatoires normales l'est aussi:
E\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &=
E\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] -
E\left[\sqrt{T}\left(\tilde\theta - \theta_0\right) \right]\nonumber\\
&= \theta_0 - \theta_0 \nonumber\\
&= 0 - 0 \nonumber\\
&= 0 \\
V\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &=
V\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] -
V\left[\sqrt{T}\left(\tilde\theta - \theta_0\right) \right]\nonumber\\
&= \left(I-(I-P)\right)\mathcal{J}_0^{-1}\nonumber\\
V\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] +
V\left[\sqrt{T}\left(\theta_0 - \tilde\theta\right) \right]\nonumber\\
&= \left(I+(P-I)\right)\mathcal{J}_0^{-1}\nonumber\\
&= P\mathcal{J}_0^{-1}.
\end{align}

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@ -180,9 +180,12 @@ où
\subsection{Variance-covariance des paramètres}
On obtient la variance-covariance asymptotique des paramètres à partir
de la variance-covariance associée aux conditions de moment, en
utilisant la méthode delta multivariée. Pour ce faire, on évalue
d'abord la valeur théorique du gradient $D(\theta)$.
de la variance-covariance associée aux conditions de moment en
utilisant la méthode delta multivariée (Annexe
\ref{sec:deltamethod}).
Pour ce faire, on évalue d'abord la valeur
théorique du gradient $D(\theta)$:
\begin{align}
D(\theta) &= E \left[ \begin{array}{cc}
-1 & -2\,\left( Y-\theta-\mu\,\tau\right) \\

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@ -19,13 +19,13 @@ En se référant à \cite{bingham2004risk}, on définit:
\item Une \textbf{option d'achat (de vente) européenne} donne le
\textbf{droit} d'acheter (de vendre) un actif au \textbf{prix
d'exercice} $K$ au temps $T$.
\item Lorsque la valeur actuelle du titre est, par rapport au prix
\item Lorsque sa valeur actuelle est, par rapport au prix
d'exercice:
\begin{itemize}
\item supérieure $(S(t)>K)$, l'option est dite \textbf{dans le
\item supérieure $(S(t)>K)$, l'option d'achat est dite \textbf{dans le
cours};
\item égale $(S(t)=K)$, l'option est dite \textbf{au cours};
\item inférieure $(S(t)<K)$, l'option est dite \textbf{hors du
\item égale $(S(t)=K)$, l'option d'achat est dite \textbf{au cours};
\item inférieure $(S(t)<K)$, l'option d'achat est dite \textbf{hors du
cours}.
\end{itemize}
\end{itemize}
@ -116,7 +116,7 @@ Afin de pouvoir utiliser les résultats de l'estimation paramétrique
des chapitres précédents pour évaluer le prix de produits dérivés, on
doit tout d'abord identifier les paramètres neutres au risque de la
distribution de Laplace asymétrique généralisée associant le
rendements cumulé $L_t$ au taux d'intérêt sans risque $r$. Pour ce
rendement cumulé $L_t$ au taux d'intérêt sans risque $r$. Pour ce
faire, on utilise l'équation martingale \eqref{eq:equationmartingale}
ainsi que la fonction génératrice des moments \eqref{eq:fgmGAL}. On
obtient alors une expression pour le paramètre de dérive neutre au
@ -341,7 +341,7 @@ par exemple).
L'intégration numérique de \eqref{eq:prixoptionCarr} pose problème
quand l'échéance $T-t$ est petite, ou encore le prix d'exercice $K$
est hors du cours ($k > \ln(S(0))$). Dans ce contexte particulier,
est hors du cours ($k > \ln(S(t))$). Dans ce contexte particulier,
\cite{carr1999option} développent une formule alternative à l'équation
\eqref{eq:prixoptionCarr} pour évaluer le prix de l'option d'achat :
\begin{align}
@ -436,7 +436,7 @@ $\left[-K,K\right]$, le théorème de Fubini (section
En remplaçant le résultat \eqref{eq:fubini-integrale-a-EPPS} dans
l'équation de départ \eqref{eq:putepps-1}, on obtient ainsi une
expression particulièrement simple pour le prix de l'option d'achat:
expression particulièrement simple pour le prix de l'option de vente:
\begin{align}
P(S(t),K,T) &= B(t,T)K\left[\frac{1}{2}-\frac{1}{2\pi} \int_{-c}^c
K^{-i\nu} \frac{\phi(\nu)}{\nu(i+\nu)}

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@ -39,7 +39,7 @@ premiers moments.
Minimum & -0.027500\\
1er quartile & -0.009790\\
Médiane & -0.003260\\
3e Quartile & 0.006620\\
3e quartile & 0.006620\\
Maximum & 0.043400\\
\hline
@ -674,8 +674,8 @@ pour les méthodes des moments généralisée et de l'équation d'estimation
optimale aux figures \ref{fig:prix1R1-1} et \ref{fig:prix1R1-3}. On
peut facilement remarquer le manque de précision de l'approche de
Carr-Madan, qui s'approche de la courbe de Black-Scholes lorsque le
titre est dans la monnaie et qui se met à osciller dès que le titre
est hors de la monnaie. Les méthodes de Epps et de Heston donnent des
titre est dans le cours et qui se met à osciller dès que le titre
est hors le cours. Les méthodes de Epps et de Heston donnent des
résultats très similaires, et l'approximation du point de selle
d'ordre 1 est très précise dans ce contexte.
\begin{figure}[!ht]

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@ -85,26 +85,26 @@
\bibcite{heston1993closed}{{29}{1993}{{Heston}}{{}}}
\bibcite{hinkley1977estimation}{{30}{1977}{{Hinkley et Revankar}}{{}}}
\bibcite{hogg1978introduction}{{31}{1978}{{Hogg et Craig}}{{}}}
\bibcite{sato1999levy}{{32}{1999}{{iti Sato}}{{}}}
\bibcite{itkin2005pricing}{{33}{2005}{{Itkin}}{{}}}
\bibcite{wang2003evaluating}{{34}{2003}{{{Jingbo Wang} et Marsaglia}}{{}}}
\bibcite{kotz2001laplace}{{35}{2001}{{Kotz et~al.}}{{Kotz, Kozubowski, et Podg{\'o}rski}}}
\bibcite{KOUTROUVELIS01011980}{{36}{1980}{{Koutrouvelis}}{{}}}
\bibcite{kozubowski1999class}{{37}{1999}{{Kozubowski et Podg{\'o}rski}}{{}}}
\bibcite{kozubowski2001asymmetric}{{38}{2001}{{Kozubowski et Podg{\'o}rski}}{{}}}
\bibcite{kyprianou2007introductory}{{39}{2007}{{Kyprianou}}{{}}}
\bibcite{lugannani1980saddle}{{40}{1980}{{Lugannani et Rice}}{{}}}
\bibcite{lukacs1960characteristic}{{41}{1960}{{Lukacs}}{{}}}
\bibcite{luong1987minimum}{{42}{1987}{{Luong et Thompson}}{{}}}
\bibcite{madan1990variance}{{43}{1990}{{Madan et Seneta}}{{}}}
\bibcite{mandelbrot1963variation}{{44}{1963}{{Mandelbrot}}{{}}}
\bibcite{merton1976option}{{45}{1976}{{Merton}}{{}}}
\bibcite{barndorff2001levy}{{46}{2001}{{Mikosch et~al.}}{{Mikosch, Resnick, et Barndorff-Nielsen}}}
\bibcite{mitchell1916critique}{{47}{1916}{{Mitchell}}{{}}}
\bibcite{newey1994large}{{48}{1994}{{Newey et McFadden}}{{}}}
\bibcite{newey1987hypothesis}{{49}{1987}{{Newey et West}}{{}}}
\bibcite{praetz1972distribution}{{50}{1972}{{Praetz}}{{}}}
\bibcite{press1967compound}{{51}{1967}{{Press}}{{}}}
\bibcite{itkin2005pricing}{{32}{2005}{{Itkin}}{{}}}
\bibcite{wang2003evaluating}{{33}{2003}{{{Jingbo Wang} et Marsaglia}}{{}}}
\bibcite{kotz2001laplace}{{34}{2001}{{Kotz et~al.}}{{Kotz, Kozubowski, et Podg{\'o}rski}}}
\bibcite{KOUTROUVELIS01011980}{{35}{1980}{{Koutrouvelis}}{{}}}
\bibcite{kozubowski1999class}{{36}{1999}{{Kozubowski et Podg{\'o}rski}}{{}}}
\bibcite{kozubowski2001asymmetric}{{37}{2001}{{Kozubowski et Podg{\'o}rski}}{{}}}
\bibcite{kyprianou2007introductory}{{38}{2007}{{Kyprianou}}{{}}}
\bibcite{lugannani1980saddle}{{39}{1980}{{Lugannani et Rice}}{{}}}
\bibcite{lukacs1960characteristic}{{40}{1960}{{Lukacs}}{{}}}
\bibcite{luong1987minimum}{{41}{1987}{{Luong et Thompson}}{{}}}
\bibcite{madan1990variance}{{42}{1990}{{Madan et Seneta}}{{}}}
\bibcite{mandelbrot1963variation}{{43}{1963}{{Mandelbrot}}{{}}}
\bibcite{merton1976option}{{44}{1976}{{Merton}}{{}}}
\bibcite{barndorff2001levy}{{45}{2001}{{Mikosch et~al.}}{{Mikosch, Resnick, et Barndorff-Nielsen}}}
\bibcite{mitchell1916critique}{{46}{1916}{{Mitchell}}{{}}}
\bibcite{newey1994large}{{47}{1994}{{Newey et McFadden}}{{}}}
\bibcite{newey1987hypothesis}{{48}{1987}{{Newey et West}}{{}}}
\bibcite{praetz1972distribution}{{49}{1972}{{Praetz}}{{}}}
\bibcite{press1967compound}{{50}{1967}{{Press}}{{}}}
\bibcite{sato1999levy}{{51}{1999}{{Sato}}{{}}}
\bibcite{schoutens2003levy}{{52}{2003}{{Schoutens}}{{}}}
\bibcite{RpackageVarianceGamma}{{53}{2012}{{Scott et Dong}}{{}}}
\bibcite{seneta2004fitting}{{54}{2004}{{Seneta}}{{}}}

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@ -180,13 +180,6 @@ Robert~V. Hogg et Allen Craig.
\newblock \emph{{I}ntroduction to {M}athematical {S}tatistics}.
\newblock Macmillan, 1978.
\bibitem[iti Sato(1999)]{sato1999levy}
Ken iti Sato.
\newblock \emph{{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible
{D}istributions}.
\newblock Cambridge Studies in Advanced Mathematics. Cambridge University
Press, 1999.
\bibitem[Itkin(2005)]{itkin2005pricing}
Andrey Itkin.
\newblock {P}ricing {O}ptions with {VG} {M}odel {U}sing {FFT}.
@ -296,6 +289,13 @@ S.~James Press.
\newblock {A} {C}ompound {E}vents {M}odel for {S}ecurity {P}rices.
\newblock \emph{Journal of Business}, pages 317--335, 1967.
\bibitem[{Sato}(1999)]{sato1999levy}
{Ken-Iti} {Sato}.
\newblock \emph{{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible
{D}istributions}.
\newblock Cambridge Studies in Advanced Mathematics. Cambridge University
Press, 1999.
\bibitem[Schoutens(2003)]{schoutens2003levy}
Wim Schoutens.
\newblock \emph{{L}{\'e}vy {P}rocesses in {F}inance}.
@ -306,7 +306,7 @@ David Scott et Christine~Yang Dong.
\newblock \emph{{{v}ariancegamma: {T}he {V}ariance {G}amma {D}istribution}},
2012.
\newblock URL \url{http://CRAN.R-project.org/package=VarianceGamma}.
\newblock R package version 0.3-1.
\newblock Consulté le 24 janvier 2014.
\bibitem[Seneta(2004)]{seneta2004fitting}
Eugene Seneta.
@ -339,7 +339,7 @@ Alan Stuart et J.~Keith Ord.
\bibitem[Teschl(2004)]{teschl2004topics}
Gerald Teschl.
\newblock {T}opics in {R}eal and {F}unctional {A}nalysis.
\newblock 2004.
\newblock Consulté le 24 janvier 2014, 2004.
\newblock URL \url{http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html}.
\bibitem[Walter(1995)]{walterlevy}

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@ -21,33 +21,32 @@ A level-1 auxiliary file: annexe2.aux
A level-1 auxiliary file: annexe3.aux
A level-1 auxiliary file: deed.aux
Database file #1: memoire.bib
Warning--empty note in teschl2004topics
You've used 63 entries,
2773 wiz_defined-function locations,
966 strings with 14129 characters,
and the built_in function-call counts, 25310 in all, are:
= -- 2354
966 strings with 14138 characters,
and the built_in function-call counts, 25309 in all, are:
= -- 2355
> -- 954
< -- 81
+ -- 342
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* -- 1694
:= -- 3811
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:= -- 3812
add.period$ -- 194
call.type$ -- 63
change.case$ -- 262
chr.to.int$ -- 63
cite$ -- 127
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duplicate$ -- 1372
empty$ -- 2353
format.name$ -- 390
if$ -- 5428
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int.to.chr$ -- 1
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missing$ -- 80
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num.names$ -- 252
pop$ -- 593
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preamble$ -- 1
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quote$ -- 0
@ -59,8 +58,7 @@ text.length$ -- 21
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warning$ -- 1
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while$ -- 279
width$ -- 0
write$ -- 815
(There was 1 warning)
write$ -- 816

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@Manual{RpackageVarianceGamma,
Author = "David Scott and Christine Yang Dong",
Note = "R package version 0.3-1",
Note = "Consulté le 24 janvier 2014",
Title = "{{v}ariancegamma: {T}he {V}ariance {G}amma {D}istribution}",
Url = "http://CRAN.R-project.org/package=VarianceGamma",
Year = 2012
Year = 2012
}
@Manual{Rsoftware,
Address = "Vienna, Austria",
Author = "{R Core Team}",
Organization = "R Foundation for Statistical Computing",
Note = "Consulté le 24 janvier 2014",
Title = "{{R}: A {L}anguage and {E}nvironment for {S}tatistical {C}omputing}",
Url = "http://www.R-project.org",
Year = 2012
Year = 2014
}
@Article{Singer:2009,
@ -517,7 +518,7 @@
}
@Book{sato1999levy,
Author = "Ken-iti Sato",
Author = "{Ken-Iti} {Sato}",
Publisher = "Cambridge University Press",
Series = "Cambridge Studies in Advanced Mathematics",
Title = "{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible {D}istributions",
@ -580,6 +581,7 @@
@Unpublished{teschl2004topics,
Author = "Gerald Teschl",
Title = "{T}opics in {R}eal and {F}unctional {A}nalysis",
Note = "Consulté le 24 janvier 2014",
Url = "http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html",
Year = 2004
}

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@Comment{x-kbibtex-personnameformatting=<%f ><%l><, %s>}
@Article{KOUTROUVELIS01011980,
Author = "Ioannis A. Koutrouvelis",
Doi = "10.1093/biomet/67.1.238",
Journal = "{B}iometrika",
Number = 1,
Pages = "238--240",
Title = "{{A} {G}oodness-{O}f-{F}it {T}est of {S}imple {H}ypotheses {B}ased on the {E}mpirical {C}haracteristic {F}unction}",
Volume = 67,
Year = 1980
}
@Manual{RpackageVarianceGamma,
Author = "David Scott and Christine Yang Dong",
Note = "R package version 0.3-1",
Title = "{{v}ariancegamma: {T}he {V}ariance {G}amma {D}istribution}",
Url = "http://CRAN.R-project.org/package=VarianceGamma",
Year = 2012
}
@Manual{Rsoftware,
Address = "Vienna, Austria",
Author = "{R Core Team}",
Isbn = "3-900051-07-0",
Organization = "R Foundation for Statistical Computing",
Title = "{{R}: A {L}anguage and {E}nvironment for {S}tatistical {C}omputing}",
Url = "http://www.R-project.org",
Year = 2012
}
@Article{Singer:2009,
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Author = "Milton Abramowitz and Irene A. Stegun",
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@Book{barndorff2001levy,
Author = "Thomas Mikosch and Sidney I. Resnick and Ole E. Barndorff-Nielsen",
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@Book{bingham2004risk,
Author = "Nicholas H. Bingham and R{\"u}diger Kiesel",
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Year = 2004
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@Article{black1973pricing,
Author = "Fischer Black and Myron Scholes",
Journal = "the Journal of Political Economy",
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@Article{black1976pricing,
Author = "Fischer Black",
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Title = "{T}he {P}ricing of {C}ommodity {C}ontracts",
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Author = "Peter P. Carr and Dilip B. Madan",
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Author = "Martin Crowder",
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Author = "Andrey Itkin",
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Author = "Samuel Kotz and Tomasz J. Kozubowski and Krzystof Podg{\'o}rski",
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Title = "{T}he {L}aplace {D}istribution and {G}eneralizations: {A} {R}evisit with {A}pplications to {C}ommunications, {E}conomics, {E}ngineering, and {F}inance",
Year = 2001
}
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Author = "Tomasz J. Kozubowski and Krzysztof Podg{\'o}rski",
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Author = "Tomasz J. Kozubowski and Krzysztof Podg{\'o}rski",
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Author = "Ken-iti Sato",
Isbn = 9780521553025,
Publisher = "Cambridge University Press",
Series = "Cambridge Studies in Advanced Mathematics",
Title = "{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible {D}istributions",
Year = 1999
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@Book{schoutens2003levy,
Author = "Wim Schoutens",
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