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memoire/ABBEYN-callGAL-5.pdf → graphiques/ABBEYN-callGAL-5.pdf
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## Exemples de courbes de densité
library(MASS)
source("fonctions.r")
dGAL1 <- function(x) dGALkappa(x,param=c(0,1,1,1))
dGAL2 <- function(x) dGALkappa(x,param=c(0,1,2,1))
dGAL3 <- function(x) dGALkappa(x,param=c(0,1,2,2))
dGAL4 <- function(x) dGALkappa(x,param=c(0,1,.5,1))
pdf("dGAL-exemples.pdf")
curve(dGAL1,from=-5,to=5,n=500,xlab="y",ylab="f(y)")
curve(dGAL2,from=-5,to=5,n=500,add=TRUE)
curve(dGAL3,from=-5,to=5,n=500,add=TRUE)
curve(dGAL4,from=-5,to=5,n=500,add=TRUE)
points(-2,dGAL1(-2),pch=21)
points(1,dGAL1(1),pch=21)
points(-2,dGAL2(-2),pch=22)
points(0.49,dGAL2(0.49),pch=22)
points(-2,dGAL3(-2),pch=15)
points(0.25,dGAL3(0.25),pch=15)
points(-2,dGAL4(-2),pch=16)
points(1.5,dGAL4(1.5),pch=16)
pchgraph <- c(21,22,15,16)
noms <- c("GAL(x; 0, 1, 1, 1)","GAL(x; 0, 1, 2, 1)","GAL(x; 0, 1, 2, 2)","GAL(x; 0, 1, 0.5, 1)")
legend(-5, 0.7, noms, cex=1.0,
pch=pchgraph, lty=1)
dev.off()

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2
memoire/chapitre1.tex
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@ -385,7 +385,7 @@ titres boursiers, entre 1890 et 1915 (figure \ref{fig:mitchell1}).
Une asymétrie négative des rendements sera alors présente.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.75]{../varia/mitchell1.pdf}
\includegraphics[scale=0.75]{../graphiques/mitchell1.pdf}
\caption{Distribution des rendements annuels de 40 titres boursiers,
de 1890 à 1915, Table XVIII de \cite{mitchell1916critique}}
\label{fig:mitchell1}

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2
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@ -548,7 +548,7 @@ avec différents paramètres d'asymétrie et d'aplatissement à la figure
\ref{fig:densiteGAL}.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.8]{../contenus/r/code/dGAL-exemples.pdf}
\includegraphics[scale=0.8]{../graphiques/dGAL-exemples.pdf}
\caption{Fonction de densité de la distribution Laplace asymétrique
généralisée avec différents paramètres:
$GAL(y;\theta,\sigma,\kappa,\tau)$}

182
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@ -18,7 +18,7 @@ la figure \ref{fig:seriechronoR1} sous forme de série chronologique.
\begin{figure}[!ht]
\centering
\includegraphics[height=4in,
width=4in]{../contenus/r/code/ABBEYN-chronologie.pdf}
width=4in]{../graphiques/ABBEYN-chronologie.pdf}
\caption{Représentation en série chronologique de l'échantillon
$R_1$}
\label{fig:seriechronoR1}
@ -69,7 +69,7 @@ d'une courbe de densité à la figure \ref{fig:distributionR1}.
\begin{figure}[!ht]
\centering
\includegraphics[height=4in,width=4in]{../contenus/r/code/ABBEYN-histogramme.pdf}
\includegraphics[height=4in,width=4in]{../graphiques/ABBEYN-histogramme.pdf}
\caption{Distribution de la variable aléatoire $R_1$}
\label{fig:distributionR1}
\end{figure}
@ -109,7 +109,7 @@ quantiles empiriques avec ceux de la loi normale présenté à la figure
\begin{figure}[!ht]
\centering
\includegraphics[height=4in,
width=4in]{../contenus/r/code/ABBEYN-qq.pdf}
width=4in]{../graphiques/ABBEYN-qq.pdf}
\caption{Graphique Quantile-Quantile}
\label{fig:qqplotR1}
\end{figure}
@ -395,7 +395,7 @@ On résout d'abord l'équation du point de selle
$r=0.01$ à la figure \ref{fig:equationptselle0.01R1} :
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.5]{../contenus/maxima/pointdeselleGMM.png}
\includegraphics[scale=0.5]{../graphiques/pointdeselleGMM.png}
\caption{Équation du point de selle pour $r=0.01$}
\label{fig:equationptselle0.01R1}
\end{figure}
@ -513,7 +513,7 @@ la table \ref{tab:intapproxpointselleR1}.
\begin{figure}[!ht]
\centering
\includegraphics[height=6in,
width=6in]{../contenus/r/code/ABBEYN-densiteGALmu-7.pdf}
width=6in]{../graphiques/ABBEYN-densiteGALmu-7.pdf}
\caption{Densité de $R_1^{*}$ selon la méthode des moments
généralisée}
\label{fig:densite1R1}
@ -522,7 +522,7 @@ la table \ref{tab:intapproxpointselleR1}.
\begin{figure}[!ht]
\centering
\includegraphics[height=6in,
width=6in]{../contenus/r/code/ABBEYN-densiteGALmu-5.pdf}
width=6in]{../graphiques/ABBEYN-densiteGALmu-5.pdf}
\caption{Densité de $R_1^{*}$ selon la méthode de l'équation
d'estimation optimale}
\label{fig:densite3R1}
@ -681,7 +681,7 @@ d'ordre 1 est très précise dans ce contexte.
\begin{figure}[!ht]
\centering
\includegraphics[height=6in,
width=6in]{../contenus/r/code/ABBEYN-callGAL-7.pdf}
width=6in]{../graphiques/ABBEYN-callGAL-7.pdf}
\caption{Prix de l'option selon les paramètres estimés avec la
méthode des moments généralisée}
\label{fig:prix1R1-1}
@ -690,7 +690,7 @@ d'ordre 1 est très précise dans ce contexte.
\begin{figure}[!ht]
\centering
\includegraphics[height=6in,
width=6in]{../contenus/r/code/ABBEYN-callGAL-5.pdf}
width=6in]{../graphiques/ABBEYN-callGAL-5.pdf}
\caption{Prix de l'option selon les paramètres estimés avec la
méthode de l'équation d'estimation optimale}
\label{fig:prix1R1-3}

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\bibitem[Wooldridge(2001)]{wooldridge2001econometric}
Jeffrey~M Wooldridge.
\newblock \emph{Econometric Analysis Of Cross Section And Panel Data}.
\newblock MIT press, 2001.
\end{thebibliography}

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\BOOKMARK [0][-]{section*.2}{R\351sum\351}{}% 1
\BOOKMARK [0][-]{section*.4}{Abstract}{}% 2
\BOOKMARK [0][-]{section*.5}{Table des mati\350res}{}% 3
\BOOKMARK [0][-]{section*.6}{Liste des tableaux}{}% 4
\BOOKMARK [0][-]{section*.7}{Liste des figures}{}% 5
\BOOKMARK [0][-]{section*.9}{Remerciements}{}% 6
\BOOKMARK [0][-]{section*.11}{Introduction}{}% 7
\BOOKMARK [0][-]{chapter.1}{Les mod\350les de rendements financiers}{}% 8
\BOOKMARK [1][-]{section.1.1}{L'utilisation de mod\350les en finance}{chapter.1}% 9
\BOOKMARK [2][-]{subsection.1.1.1}{Diff\351rents types de mod\350les}{section.1.1}% 10
\BOOKMARK [2][-]{subsection.1.1.2}{Le risque de mod\351lisation}{section.1.1}% 11
\BOOKMARK [1][-]{section.1.2}{Les rendements financiers}{chapter.1}% 12
\BOOKMARK [2][-]{subsection.1.2.1}{D\351finitions et notations}{section.1.2}% 13
\BOOKMARK [2][-]{subsection.1.2.2}{Rendements cumul\351s}{section.1.2}% 14
\BOOKMARK [2][-]{subsection.1.2.3}{Donn\351es disponibles}{section.1.2}% 15
\BOOKMARK [1][-]{section.1.3}{Les premiers mod\350les}{chapter.1}% 16
\BOOKMARK [2][-]{subsection.1.3.1}{Le mod\350le de Bachelier}{section.1.3}% 17
\BOOKMARK [2][-]{subsection.1.3.2}{Proposition de Mandelbrot}{section.1.3}% 18
\BOOKMARK [2][-]{subsection.1.3.3}{Le mod\350le de Press}{section.1.3}% 19
\BOOKMARK [2][-]{subsection.1.3.4}{Le mod\350le de Praetz}{section.1.3}% 20
\BOOKMARK [1][-]{section.1.4}{Conditions essentielles de Madan et Seneta}{chapter.1}% 21
\BOOKMARK [0][-]{chapter.2}{La distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 22
\BOOKMARK [1][-]{section.2.1}{Le processus de Laplace}{chapter.2}% 23
\BOOKMARK [2][-]{subsection.2.1.1}{Le processus gamma}{section.2.1}% 24
\BOOKMARK [2][-]{subsection.2.1.2}{Le processus de Wiener}{section.2.1}% 25
\BOOKMARK [2][-]{subsection.2.1.3}{Le processus de Laplace est un processus subordonn\351}{section.2.1}% 26
\BOOKMARK [1][-]{section.2.2}{Distribution de Laplace asym\351trique g\351n\351ralis\351e}{chapter.2}% 27
\BOOKMARK [2][-]{subsection.2.2.1}{Fonction caract\351ristique}{section.2.2}% 28
\BOOKMARK [2][-]{subsection.2.2.2}{Invariance d'\351chelle}{section.2.2}% 29
\BOOKMARK [2][-]{subsection.2.2.3}{Fonctions g\351n\351ratrices}{section.2.2}% 30
\BOOKMARK [2][-]{subsection.2.2.4}{Moments et r\364le des param\350tres}{section.2.2}% 31
\BOOKMARK [2][-]{subsection.2.2.5}{Changement d'\351chelle et de localisation}{section.2.2}% 32
\BOOKMARK [2][-]{subsection.2.2.6}{Repr\351sentation alternative et simulation}{section.2.2}% 33
\BOOKMARK [2][-]{subsection.2.2.7}{Fonction de Bessel et densit\351}{section.2.2}% 34
\BOOKMARK [1][-]{section.2.3}{Cas particuliers}{chapter.2}% 35
\BOOKMARK [2][-]{subsection.2.3.1}{Distribution de Laplace asym\351trique}{section.2.3}% 36
\BOOKMARK [1][-]{section.2.4}{Relation avec le mod\350le de madan1990variance}{chapter.2}% 37
\BOOKMARK [0][-]{chapter.3}{Approximation de la densit\351 et de la fonction de r\351partition}{}% 38
\BOOKMARK [1][-]{section.3.1}{L'approximation de Laplace}{chapter.3}% 39
\BOOKMARK [1][-]{section.3.2}{L'approximation de Temme}{chapter.3}% 40
\BOOKMARK [1][-]{section.3.3}{La m\351thode du point de selle}{chapter.3}% 41
\BOOKMARK [2][-]{subsection.3.3.1}{Approximation de la densit\351}{section.3.3}% 42
\BOOKMARK [2][-]{subsection.3.3.2}{Unicit\351 du point de selle}{section.3.3}% 43
\BOOKMARK [2][-]{subsection.3.3.3}{Approximation de la fonction de r\351partition}{section.3.3}% 44
\BOOKMARK [2][-]{subsection.3.3.4}{Quelques propri\351t\351s des approximations}{section.3.3}% 45
\BOOKMARK [1][-]{section.3.4}{Application de la m\351thode du point de selle}{chapter.3}% 46
\BOOKMARK [2][-]{subsection.3.4.1}{Approximation de la densit\351}{section.3.4}% 47
\BOOKMARK [2][-]{subsection.3.4.2}{Approximation de la fonction de r\351partition}{section.3.4}% 48
\BOOKMARK [0][-]{chapter.4}{M\351thode des moments g\351n\351ralis\351e}{}% 49
\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}% 50
\BOOKMARK [2][-]{subsection.4.1.1}{M\351thode classique des moments}{section.4.1}% 51
\BOOKMARK [1][-]{section.4.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.4}% 52
\BOOKMARK [2][-]{subsection.4.2.1}{D\351finition}{section.4.2}% 53
\BOOKMARK [2][-]{subsection.4.2.2}{Convergence}{section.4.2}% 54
\BOOKMARK [2][-]{subsection.4.2.3}{Matrice de pond\351ration optimale}{section.4.2}% 55
\BOOKMARK [2][-]{subsection.4.2.4}{M\351thode des moments g\351n\351ralis\351e it\351rative}{section.4.2}% 56
\BOOKMARK [2][-]{subsection.4.2.5}{Distribution asymptotique des estimateurs}{section.4.2}% 57
\BOOKMARK [1][-]{section.4.3}{Estimation sous contraintes}{chapter.4}% 58
\BOOKMARK [2][-]{subsection.4.3.1}{Distribution asymptotique des estimateurs contraints}{section.4.3}% 59
\BOOKMARK [1][-]{section.4.4}{Tests d'hypoth\350ses param\351triques}{chapter.4}% 60
\BOOKMARK [2][-]{subsection.4.4.1}{Test de Wald}{section.4.4}% 61
\BOOKMARK [2][-]{subsection.4.4.2}{Test du multiplicateur de Lagrange}{section.4.4}% 62
\BOOKMARK [2][-]{subsection.4.4.3}{Test bas\351 sur la statistique de m\351trique de distance}{section.4.4}% 63
\BOOKMARK [2][-]{subsection.4.4.4}{En r\351sum\351}{section.4.4}% 64
\BOOKMARK [0][-]{chapter.5}{M\351thode de l'\351quation d'estimation optimale}{}% 65
\BOOKMARK [1][-]{section.5.1}{\311quation d'estimation optimale}{chapter.5}% 66
\BOOKMARK [1][-]{section.5.2}{\311quation d'estimation optimale modifi\351e}{chapter.5}% 67
\BOOKMARK [0][-]{chapter.6}{Estimation des param\350tres de la distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 68
\BOOKMARK [1][-]{section.6.1}{Vecteur de param\350tres initiaux}{chapter.6}% 69
\BOOKMARK [1][-]{section.6.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.6}% 70
\BOOKMARK [2][-]{subsection.6.2.1}{Matrice de pond\351ration optimale}{section.6.2}% 71
\BOOKMARK [2][-]{subsection.6.2.2}{Variance-covariance des param\350tres}{section.6.2}% 72
\BOOKMARK [2][-]{subsection.6.2.3}{Contraintes lin\351aires}{section.6.2}% 73
\BOOKMARK [1][-]{section.6.3}{M\351thode de l'\351quation d'estimation optimale}{chapter.6}% 74
\BOOKMARK [0][-]{chapter.7}{Tests statistiques}{}% 75
\BOOKMARK [1][-]{section.7.1}{Test de normalit\351}{chapter.7}% 76
\BOOKMARK [2][-]{subsection.7.1.1}{Test de Shapiro-Wilk}{section.7.1}% 77
\BOOKMARK [2][-]{subsection.7.1.2}{Test d\220Epps-Pulley}{section.7.1}% 78
\BOOKMARK [1][-]{section.7.2}{Tests d'ad\351quation}{chapter.7}% 79
\BOOKMARK [2][-]{subsection.7.2.1}{Test 2 de Pearson}{section.7.2}% 80
\BOOKMARK [2][-]{subsection.7.2.2}{Test de Kolmogorov-Smirnov}{section.7.2}% 81
\BOOKMARK [2][-]{subsection.7.2.3}{Test de distance minimale bas\351 sur la fonction g\351n\351ratrice des moments}{section.7.2}% 82
\BOOKMARK [0][-]{chapter.8}{\311valuation d'options}{}% 83
\BOOKMARK [1][-]{section.8.1}{D\351finitions}{chapter.8}% 84
\BOOKMARK [2][-]{subsection.8.1.1}{\311quation martingale}{section.8.1}% 85
\BOOKMARK [2][-]{subsection.8.1.2}{Param\350tres neutres au risque}{section.8.1}% 86
\BOOKMARK [1][-]{section.8.2}{Aper\347u du mod\350le de Black-Scholes}{chapter.8}% 87
\BOOKMARK [1][-]{section.8.3}{M\351thodes d'\351valuation pour options europ\351ennes}{chapter.8}% 88
\BOOKMARK [2][-]{subsection.8.3.1}{M\351thode de Heston}{section.8.3}% 89
\BOOKMARK [2][-]{subsection.8.3.2}{M\351thode de Carr et Madan}{section.8.3}% 90
\BOOKMARK [2][-]{subsection.8.3.3}{Prix d'exercice hors du cours}{section.8.3}% 91
\BOOKMARK [2][-]{subsection.8.3.4}{Critique de la m\351thode de Carr-Madan}{section.8.3}% 92
\BOOKMARK [2][-]{subsection.8.3.5}{M\351thode d\220Epps}{section.8.3}% 93
\BOOKMARK [1][-]{section.8.4}{Particularit\351s}{chapter.8}% 94
\BOOKMARK [2][-]{subsection.8.4.1}{Option sur actions avec dividendes}{section.8.4}% 95
\BOOKMARK [2][-]{subsection.8.4.2}{Options sur contrats \340 terme et taux de change}{section.8.4}% 96
\BOOKMARK [0][-]{chapter.9}{Exemple d'application}{}% 97
\BOOKMARK [1][-]{section.9.1}{Description des donn\351es}{chapter.9}% 98
\BOOKMARK [1][-]{section.9.2}{Estimation}{chapter.9}% 99
\BOOKMARK [1][-]{section.9.3}{Approximation}{chapter.9}% 100
\BOOKMARK [1][-]{section.9.4}{Graphiques}{chapter.9}% 101
\BOOKMARK [1][-]{section.9.5}{Tests statistiques}{chapter.9}% 102
\BOOKMARK [1][-]{section.9.6}{\311valuation d'options}{chapter.9}% 103
\BOOKMARK [0][-]{section*.52}{Conclusion}{}% 104
\BOOKMARK [0][-]{appendix.A}{\311l\351ments de th\351orie des probabilit\351s}{}% 105
\BOOKMARK [1][-]{section.A.1}{D\351finitions de base}{appendix.A}% 106
\BOOKMARK [1][-]{section.A.2}{Transform\351es d'une variable al\351atoire}{appendix.A}% 107
\BOOKMARK [2][-]{subsection.A.2.1}{La fonction caract\351ristique}{section.A.2}% 108
\BOOKMARK [3][-]{section*.53}{Transform\351e de Fourier}{subsection.A.2.1}% 109
\BOOKMARK [3][-]{section*.54}{D\351finition}{subsection.A.2.1}% 110
\BOOKMARK [3][-]{section*.55}{Les moments}{subsection.A.2.1}% 111
\BOOKMARK [2][-]{subsection.A.2.2}{Inversion de la fonction caract\351ristique}{section.A.2}% 112
\BOOKMARK [3][-]{section*.56}{La densit\351}{subsection.A.2.2}% 113
\BOOKMARK [3][-]{section*.57}{La fonction de r\351partition}{subsection.A.2.2}% 114
\BOOKMARK [2][-]{subsection.A.2.3}{La fonction g\351n\351ratrice des moments}{section.A.2}% 115
\BOOKMARK [2][-]{subsection.A.2.4}{La fonction g\351n\351ratrice des cumulants}{section.A.2}% 116
\BOOKMARK [2][-]{subsection.A.2.5}{La transform\351e d'Esscher}{section.A.2}% 117
\BOOKMARK [1][-]{section.A.3}{La transform\351e de Fourier rapide}{appendix.A}% 118
\BOOKMARK [1][-]{section.A.4}{Processus de L\351vy}{appendix.A}% 119
\BOOKMARK [2][-]{subsection.A.4.1}{D\351finition et propri\351t\351s}{section.A.4}% 120
\BOOKMARK [3][-]{section*.58}{Repr\351sentation de L\351vy-Khintchine}{subsection.A.4.1}% 121
\BOOKMARK [3][-]{section*.59}{Repr\351sentation de L\351vy-It\364}{subsection.A.4.1}% 122
\BOOKMARK [2][-]{subsection.A.4.2}{Processus subordonn\351}{section.A.4}% 123
\BOOKMARK [1][-]{section.A.5}{Th\351or\350mes d'int\351gration}{appendix.A}% 124
\BOOKMARK [2][-]{subsection.A.5.1}{Th\351or\350me de convergence domin\351e de Lebesgue}{section.A.5}% 125
\BOOKMARK [2][-]{subsection.A.5.2}{Th\351or\350me de Fubini}{section.A.5}% 126
\BOOKMARK [0][-]{appendix.B}{\311l\351ments de statistique math\351matique}{}% 127
\BOOKMARK [1][-]{section.B.1}{Loi faible des grands nombres}{appendix.B}% 128
\BOOKMARK [1][-]{section.B.2}{Th\351or\350me central limite}{appendix.B}% 129
\BOOKMARK [2][-]{subsection.B.2.1}{Cas univari\351}{section.B.2}% 130
\BOOKMARK [2][-]{subsection.B.2.2}{Cas multivari\351}{section.B.2}% 131
\BOOKMARK [1][-]{section.B.3}{M\351thode delta multivari\351e}{appendix.B}% 132
\BOOKMARK [0][-]{appendix.C}{Donn\351es}{}% 133
\BOOKMARK [0][-]{section*.62}{Bibliographie}{}% 134
\BOOKMARK [0][-]{section*.64}{Contrat de partage}{}% 135

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\select@language {french}
\select@language {french}
\contentsline {chapter}{R\IeC {\'e}sum\IeC {\'e}}{iii}{section*.2}
\contentsline {chapter}{Abstract}{v}{section*.4}
\contentsline {chapter}{Table des mati{\`e}res}{vii}{section*.5}
\contentsline {chapter}{Liste des tableaux}{xi}{section*.6}
\contentsline {chapter}{Liste des figures}{xiii}{section*.7}
\contentsline {chapter}{Remerciements}{xvii}{section*.9}
\contentsline {chapter}{Introduction}{1}{section*.11}
\contentsline {chapter}{\chapternumberline {1}Les mod\IeC {\`e}les de rendements financiers}{3}{chapter.1}
\contentsline {section}{\numberline {1.1}L'utilisation de mod\IeC {\`e}les en finance}{3}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}Diff\IeC {\'e}rents types de mod\IeC {\`e}les}{3}{subsection.1.1.1}
\contentsline {subsection}{\numberline {1.1.2}Le risque de mod\IeC {\'e}lisation}{4}{subsection.1.1.2}
\contentsline {section}{\numberline {1.2}Les rendements financiers}{5}{section.1.2}
\contentsline {subsection}{\numberline {1.2.1}D\IeC {\'e}finitions et notations}{5}{subsection.1.2.1}
\contentsline {subsection}{\numberline {1.2.2}Rendements cumul\IeC {\'e}s}{7}{subsection.1.2.2}
\contentsline {subsection}{\numberline {1.2.3}Donn\IeC {\'e}es disponibles}{8}{subsection.1.2.3}
\contentsline {section}{\numberline {1.3}Les premiers mod\IeC {\`e}les}{8}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}Le mod\IeC {\`e}le de Bachelier}{8}{subsection.1.3.1}
\contentsline {subsection}{\numberline {1.3.2}Proposition de Mandelbrot}{11}{subsection.1.3.2}
\contentsline {subsection}{\numberline {1.3.3}Le mod\IeC {\`e}le de Press}{12}{subsection.1.3.3}
\contentsline {subsection}{\numberline {1.3.4}Le mod\IeC {\`e}le de Praetz}{15}{subsection.1.3.4}
\contentsline {section}{\numberline {1.4}Conditions essentielles de Madan et Seneta}{16}{section.1.4}
\contentsline {chapter}{\chapternumberline {2}La distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{19}{chapter.2}
\contentsline {section}{\numberline {2.1}Le processus de Laplace}{19}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Le processus gamma}{20}{subsection.2.1.1}
\contentsline {subsection}{\numberline {2.1.2}Le processus de Wiener}{21}{subsection.2.1.2}
\contentsline {subsection}{\numberline {2.1.3}Le processus de Laplace est un processus subordonn\IeC {\'e}}{22}{subsection.2.1.3}
\contentsline {section}{\numberline {2.2}Distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{25}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Fonction caract\IeC {\'e}ristique}{25}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Invariance d'\IeC {\'e}chelle}{26}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}Fonctions g\IeC {\'e}n\IeC {\'e}ratrices}{28}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}Moments et r\IeC {\^o}le des param\IeC {\`e}tres}{28}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}Changement d'\IeC {\'e}chelle et de localisation}{30}{subsection.2.2.5}
\contentsline {subsection}{\numberline {2.2.6}Repr\IeC {\'e}sentation alternative et simulation}{32}{subsection.2.2.6}
\contentsline {subsection}{\numberline {2.2.7}Fonction de Bessel et densit\IeC {\'e}}{33}{subsection.2.2.7}
\contentsline {section}{\numberline {2.3}Cas particuliers}{35}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}Distribution de Laplace asym\IeC {\'e}trique}{35}{subsection.2.3.1}
\contentsline {section}{\numberline {2.4}Relation avec le mod\IeC {\`e}le de \cite {madan1990variance}}{37}{section.2.4}
\contentsline {chapter}{\chapternumberline {3}Approximation de la densit\IeC {\'e} et de la fonction de r\IeC {\'e}partition}{39}{chapter.3}
\contentsline {section}{\numberline {3.1}L'approximation de Laplace}{39}{section.3.1}
\contentsline {section}{\numberline {3.2}L'approximation de Temme}{41}{section.3.2}
\contentsline {section}{\numberline {3.3}La m\IeC {\'e}thode du point de selle}{42}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}Approximation de la densit\IeC {\'e}}{42}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}Unicit\IeC {\'e} du point de selle}{44}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}Approximation de la fonction de r\IeC {\'e}partition}{44}{subsection.3.3.3}
\contentsline {subsection}{\numberline {3.3.4}Quelques propri\IeC {\'e}t\IeC {\'e}s des approximations}{45}{subsection.3.3.4}
\contentsline {section}{\numberline {3.4}Application de la m\IeC {\'e}thode du point de selle}{46}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}Approximation de la densit\IeC {\'e}}{46}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}Approximation de la fonction de r\IeC {\'e}partition}{47}{subsection.3.4.2}
\contentsline {chapter}{\chapternumberline {4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{49}{chapter.4}
\contentsline {section}{\numberline {4.1}Introduction}{49}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}M\IeC {\'e}thode classique des moments}{50}{subsection.4.1.1}
\contentsline {section}{\numberline {4.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{50}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}D\IeC {\'e}finition}{51}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Convergence}{52}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}Matrice de pond\IeC {\'e}ration optimale}{53}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e it\IeC {\'e}rative}{55}{subsection.4.2.4}
\contentsline {subsection}{\numberline {4.2.5}Distribution asymptotique des estimateurs}{55}{subsection.4.2.5}
\contentsline {section}{\numberline {4.3}Estimation sous contraintes}{58}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Distribution asymptotique des estimateurs contraints}{59}{subsection.4.3.1}
\contentsline {section}{\numberline {4.4}Tests d'hypoth\IeC {\`e}ses param\IeC {\'e}triques}{61}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}Test de Wald}{61}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}Test du multiplicateur de Lagrange}{62}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}Test bas\IeC {\'e} sur la statistique de m\IeC {\'e}trique de distance}{63}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}En r\IeC {\'e}sum\IeC {\'e}}{63}{subsection.4.4.4}
\contentsline {chapter}{\chapternumberline {5}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{65}{chapter.5}
\contentsline {section}{\numberline {5.1}\IeC {\'E}quation d'estimation optimale}{67}{section.5.1}
\contentsline {section}{\numberline {5.2}\IeC {\'E}quation d'estimation optimale modifi\IeC {\'e}e}{70}{section.5.2}
\contentsline {chapter}{\chapternumberline {6}Estimation des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{73}{chapter.6}
\contentsline {section}{\numberline {6.1}Vecteur de param\IeC {\`e}tres initiaux}{73}{section.6.1}
\contentsline {section}{\numberline {6.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{74}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}Matrice de pond\IeC {\'e}ration optimale}{75}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}Variance-covariance des param\IeC {\`e}tres}{76}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}Contraintes lin\IeC {\'e}aires}{77}{subsection.6.2.3}
\contentsline {section}{\numberline {6.3}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{78}{section.6.3}
\contentsline {chapter}{\chapternumberline {7}Tests statistiques}{81}{chapter.7}
\contentsline {section}{\numberline {7.1}Test de normalit\IeC {\'e}}{81}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}Test de Shapiro-Wilk}{81}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}Test d\IeC {\textquoteright }Epps-Pulley}{82}{subsection.7.1.2}
\contentsline {section}{\numberline {7.2}Tests d'ad\IeC {\'e}quation}{82}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}Test $\chi ^2$ de Pearson}{83}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}Test de Kolmogorov-Smirnov}{84}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{85}{subsection.7.2.3}
\contentsline {chapter}{\chapternumberline {8}\IeC {\'E}valuation d'options}{87}{chapter.8}
\contentsline {section}{\numberline {8.1}D\IeC {\'e}finitions}{87}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\IeC {\'E}quation martingale}{88}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}Param\IeC {\`e}tres neutres au risque}{89}{subsection.8.1.2}
\contentsline {section}{\numberline {8.2}Aper\IeC {\c c}u du mod\IeC {\`e}le de Black-Scholes}{90}{section.8.2}
\contentsline {section}{\numberline {8.3}M\IeC {\'e}thodes d'\IeC {\'e}valuation pour options europ\IeC {\'e}ennes}{90}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}M\IeC {\'e}thode de Heston}{91}{subsection.8.3.1}
\contentsline {subsection}{\numberline {8.3.2}M\IeC {\'e}thode de Carr et Madan}{91}{subsection.8.3.2}
\contentsline {subsection}{\numberline {8.3.3}Prix d'exercice hors du cours}{93}{subsection.8.3.3}
\contentsline {subsection}{\numberline {8.3.4}Critique de la m\IeC {\'e}thode de Carr-Madan}{93}{subsection.8.3.4}
\contentsline {subsection}{\numberline {8.3.5}M\IeC {\'e}thode d\IeC {\textquoteright }Epps}{93}{subsection.8.3.5}
\contentsline {section}{\numberline {8.4}Particularit\IeC {\'e}s}{95}{section.8.4}
\contentsline {subsection}{\numberline {8.4.1}Option sur actions avec dividendes}{95}{subsection.8.4.1}
\contentsline {subsection}{\numberline {8.4.2}Options sur contrats \IeC {\`a} terme et taux de change}{95}{subsection.8.4.2}
\contentsline {chapter}{\chapternumberline {9}Exemple d'application}{97}{chapter.9}
\contentsline {section}{\numberline {9.1}Description des donn\IeC {\'e}es}{97}{section.9.1}
\contentsline {section}{\numberline {9.2}Estimation}{99}{section.9.2}
\contentsline {section}{\numberline {9.3}Approximation}{104}{section.9.3}
\contentsline {section}{\numberline {9.4}Graphiques}{105}{section.9.4}
\contentsline {section}{\numberline {9.5}Tests statistiques}{108}{section.9.5}
\contentsline {section}{\numberline {9.6}\IeC {\'E}valuation d'options}{109}{section.9.6}
\contentsline {chapter}{Conclusion}{113}{section*.52}
\contentsline {appendix}{\chapternumberline {A}\IeC {\'E}l\IeC {\'e}ments de th\IeC {\'e}orie des probabilit\IeC {\'e}s}{115}{appendix.A}
\contentsline {section}{\numberline {A.1}D\IeC {\'e}finitions de base}{115}{section.A.1}
\contentsline {section}{\numberline {A.2}Transform\IeC {\'e}es d'une variable al\IeC {\'e}atoire}{116}{section.A.2}
\contentsline {subsection}{\numberline {A.2.1}La fonction caract\IeC {\'e}ristique}{116}{subsection.A.2.1}
\contentsline {subsubsection}{Transform\IeC {\'e}e de Fourier}{116}{section*.53}
\contentsline {subsubsection}{D\IeC {\'e}finition}{116}{section*.54}
\contentsline {subsubsection}{Les moments}{117}{section*.55}
\contentsline {subsection}{\numberline {A.2.2}Inversion de la fonction caract\IeC {\'e}ristique}{117}{subsection.A.2.2}
\contentsline {subsubsection}{La densit\IeC {\'e}}{117}{section*.56}
\contentsline {subsubsection}{La fonction de r\IeC {\'e}partition}{117}{section*.57}
\contentsline {subsection}{\numberline {A.2.3}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{118}{subsection.A.2.3}
\contentsline {subsection}{\numberline {A.2.4}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des cumulants}{119}{subsection.A.2.4}
\contentsline {subsection}{\numberline {A.2.5}La transform\IeC {\'e}e d'Esscher}{120}{subsection.A.2.5}
\contentsline {section}{\numberline {A.3}La transform\IeC {\'e}e de Fourier rapide}{120}{section.A.3}
\contentsline {section}{\numberline {A.4}Processus de L\IeC {\'e}vy}{121}{section.A.4}
\contentsline {subsection}{\numberline {A.4.1}D\IeC {\'e}finition et propri\IeC {\'e}t\IeC {\'e}s}{121}{subsection.A.4.1}
\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-Khintchine}{122}{section*.58}
\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-It\IeC {\^o}}{122}{section*.59}
\contentsline {subsection}{\numberline {A.4.2}Processus subordonn\IeC {\'e}}{123}{subsection.A.4.2}
\contentsline {section}{\numberline {A.5}Th\IeC {\'e}or\IeC {\`e}mes d'int\IeC {\'e}gration}{123}{section.A.5}
\contentsline {subsection}{\numberline {A.5.1}Th\IeC {\'e}or\IeC {\`e}me de convergence domin\IeC {\'e}e de Lebesgue}{124}{subsection.A.5.1}
\contentsline {subsection}{\numberline {A.5.2}Th\IeC {\'e}or\IeC {\`e}me de Fubini}{124}{subsection.A.5.2}
\contentsline {appendix}{\chapternumberline {B}\IeC {\'E}l\IeC {\'e}ments de statistique math\IeC {\'e}matique}{125}{appendix.B}
\contentsline {section}{\numberline {B.1}Loi faible des grands nombres}{125}{section.B.1}
\contentsline {section}{\numberline {B.2}Th\IeC {\'e}or\IeC {\`e}me central limite}{125}{section.B.2}
\contentsline {subsection}{\numberline {B.2.1}Cas univari\IeC {\'e}}{126}{subsection.B.2.1}
\contentsline {subsection}{\numberline {B.2.2}Cas multivari\IeC {\'e}}{126}{subsection.B.2.2}
\contentsline {section}{\numberline {B.3}M\IeC {\'e}thode delta multivari\IeC {\'e}e}{127}{section.B.3}
\contentsline {appendix}{\chapternumberline {C}Donn\IeC {\'e}es}{129}{appendix.C}
\contentsline {chapter}{Bibliographie}{131}{section*.62}
\contentsline {chapter}{Contrat de partage}{137}{section*.64}

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@article{KOUTROUVELIS01011980,
author = {Koutrouvelis, I. A.},
title = {A goodness-of-fit test of simple hypotheses based on
the empirical characteristic function},
volume = 67,
number = 1,
pages = {238-240},
year = 1980,
doi = {10.1093/biomet/67.1.238},
abstract = {SUMMARY The empirical characteristic function φn(t)
and its asymptotic distribution are utilized to find
a chi-squared goodness-of-fit test for simple null
hypotheses. In addition the optimum number and
location of points t at which φn(t) has to be
evaluated is investigated for specified alternative
hypotheses.},
URL =
{http://biomet.oxfordjournals.org/content/67/1/238.abstract},
eprint =
{http://biomet.oxfordjournals.org/content/67/1/238.full.pdf+html},
journal = {Biometrika}
}
@Manual{RpackageVarianceGamma,
title = {VarianceGamma: The Variance Gamma Distribution},
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year = 2012,
note = {R package version 0.3-1},
url = {http://CRAN.R-project.org/package=VarianceGamma},
}
@Manual{Rsoftware,
title = {R: A Language and Environment for Statistical
Computing},
author = {{R Core Team}},
organization = {R Foundation for Statistical Computing},
address = {Vienna, Austria},
year = 2012,
note = {{ISBN} 3-900051-07-0},
url = {http://www.R-project.org/},
}
@ARTICLE{Singer:2009,
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JOURNAL = {Scholarpedia},
VOLUME = 4,
NUMBER = 7,
PAGES = 2928,
URL = {http://dx.doi.org/10.4249/scholarpedia.2928}
}
@book{abramowitz1965handbook,
title = {Handbook of mathematical functions: with formulas,
graphs, and mathematical tables},
author = {Abramowitz, Milton and Stegun, Irene A},
volume = 55,
year = 1965,
publisher = {Dover publications}
}
@article{applebaum2004levy,
title = {L{\'e}vy processes: From probability to finance and
quantum groups},
author = {Applebaum, David},
journal = {Notices of the AMS},
volume = 51,
number = 11,
pages = {1336--1347},
year = 2004
}
@book{bachelier1900theorie,
title = {Th{\'e}orie de la sp{\'e}culation},
author = {Bachelier, Louis},
year = 1900,
publisher = {Gauthier-Villars}
}
@book{barndorff2001levy,
title = {L{\'e}vy Processes: Theory and Applications},
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