corrections rapport de Claire
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François Pelletier
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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.
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Certaines d'entre elles permettent de déterminer entièrement leur
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distribution. Parmi celles-ci, on retrouve la fonction caractéristique
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et les fonctions génératrice des moments et des cumulants, qui sont
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et les fonctions génératrices des moments et des cumulants, qui sont
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les plus couramment utilisées.
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Certaines transformées permettent de modifier la distribution d'une
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@ -226,7 +226,7 @@ distribuées, la fonction caractéristique de $Z$ est la $n^e$ puissance
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de celle de $X$:
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\begin{equation}
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\label{eq:convocaractIID}
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\phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \left[\phi_{X_i}(s)\right]^n.
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\phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \left[\phi_{X}(s)\right]^n.
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\end{equation}
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Cette fonction est donc une solution de rechange intéressante à
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@ -467,7 +467,7 @@ cas, $N(t,dx)$ est une mesure aléatoire de Poisson.
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On considère les processus de Lévy $\lbrace X(t) \rbrace$ et $\lbrace
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Z(t) \rbrace$. Celui qui suit est défini comme étant un processus
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subordonné et aussi un processus de Lévy, comme le démontre
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subordonné et aussi un processus de Lévy, comme le démontrent
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\cite{sato1999levy} et \cite{schoutens2003levy}:
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\begin{equation}
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\label{eq:processussubordonne}
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@ -540,7 +540,8 @@ inférieure ou égale à $g(x)$:
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g(x).
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\end{align}
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Alors, la fonction $f$ est intégrable.
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Alors, la fonction $f$ est intégrable dans l'intervalle
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$\mathit{I}$.
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\subsection{Théorème de Fubini}
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\label{sec:theoreme-de-fubini}
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@ -78,7 +78,7 @@ vers une variable aléatoire de distribution normale multivariée
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centrée de matrice de variance-covariance $\mathbf{\Sigma}$:
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\begin{align}
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\label{eq:TCLmulti2}
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\sqrt{T}\left(\mathbf{Y}_T - \mu\right)\ \stackrel{L}{\rightarrow}\
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\sqrt{T}\left(\mathbf{Y}_T - \boldsymbol{\mu}\right)\ \stackrel{L}{\rightarrow}\
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\mathcal{N}_k(0,\mathbf{\Sigma})
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\end{align}
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@ -119,7 +119,7 @@ retrouver dans le code source. On considère entre autres des erreurs
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d'arrondissement, de logique et de clarté du code, ainsi que des
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particularités du matériel qui n'auraient pas été prises en compte par
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le programmeur. Ces erreurs peuvent être difficiles à détecter, c'est
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pourquoi un grand nombre de tests devrait être effectués avant de
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pourquoi un grand nombre de tests devraient être effectués avant de
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publier un logiciel de modélisation financière.
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\section{Les rendements financiers}
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@ -425,7 +425,7 @@ rendements financiers devrait posséder:
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temps.
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\end{enumerate}
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La famille de distributions L stable semble être celle qui répond le
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La famille de distributions L stables semble être celle qui répond le
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mieux à l'ensemble de ces conditions \citep{walterlevy}. L'équation
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suivante définit la propriété de L-stabilité de la distribution de la
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variable aléatoire des rendements sur une période $R$:
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@ -442,11 +442,11 @@ la forme suivante:
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\left[1+\frac{i\beta \xi}{|\xi|} \tan{\frac{\alpha\pi}{2}} \right].
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\end{align}
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Le domaine et le rôle des paramètres de la distribution L stable est
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décrit à la table \ref{tab:roleparam}. La flexibilité apportée par les
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Le domaine et le rôle des paramètres de la distribution L stable sont
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décrits à la table \ref{tab:roleparam}. La flexibilité apportée par les
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quatre paramètres permet de remplir les quatre conditions établies au
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début de cette section. De plus, l'absence, dans la majorité des cas,
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de moments finis d'ordre supérieur à l'espérance, permet de tenir
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de moments finis d'ordre supérieur à l'espérance permet de tenir
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compte du mouvement erratique des prix et ainsi produire de larges
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discontinuités de son processus. Elle permet aussi d'expliquer
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l'apparence de corrélation sérielle, en considérant une probabilité
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@ -46,7 +46,7 @@ stationnaires. Cependant, la plupart des caractéristiques diffèrent:
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\begin{itemize}
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\item Discontinuité des trajectoires (processus de sauts);
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\item Distribution asymétrique des accroissements;
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\item Paramètre d'échelle et de temps entièrement dissociés.
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\item Paramètres d'échelle et de temps entièrement dissociés.
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\end{itemize}
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Enfin, il possède une représentation alternative qui n'implique aucun
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@ -125,7 +125,7 @@ Cette distribution est définie par la fonction de densité
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$f_{\mu,\sigma}(x)$ et la fonction caractéristique
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$\phi_{\mu,\sigma}(\xi)$:
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\begin{align}
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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} \\
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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} \\
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\phi_{\mu,\sigma}(\xi) &= \exp\left\{
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i\mu\xi-\frac{\sigma^2\xi^2}{2}
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\right\} \label{eq:fncaractnormale}.
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@ -17,11 +17,11 @@ suivant une distribution particulière. Le vecteur de paramètres de
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celle-ci, $\theta$, de longueur $a$, appartenant à l'espace $\Omega
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\subset \mathbb{R}^a$, doit être estimé à partir de l'échantillon. Une
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première approche consiste à maximiser la fonction de vraisemblance
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$LL(\theta;\mathbf{y})$, qui équivaut au produit de la densité
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$L(\theta;\mathbf{y})$, qui équivaut au produit de la densité
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$f(y;\theta)$ évaluée à chacune des réalisations $y(t)$:
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\begin{align}
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\label{eq:vraisemblance}
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LL(\theta;\mathbf{y}) = \prod_{t=1}^T
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L(\theta;\mathbf{y}) = \prod_{t=1}^T
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f(y(t);\theta),\quad\theta\in\Omega.
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\end{align}
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@ -872,12 +872,12 @@ que la somme de deux variables aléatoires normales l'est aussi:
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E\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &=
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E\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] -
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E\left[\sqrt{T}\left(\tilde\theta - \theta_0\right) \right]\nonumber\\
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&= \theta_0 - \theta_0 \nonumber\\
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&= 0 - 0 \nonumber\\
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&= 0 \\
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V\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &=
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V\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] -
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V\left[\sqrt{T}\left(\tilde\theta - \theta_0\right) \right]\nonumber\\
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&= \left(I-(I-P)\right)\mathcal{J}_0^{-1}\nonumber\\
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V\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] +
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V\left[\sqrt{T}\left(\theta_0 - \tilde\theta\right) \right]\nonumber\\
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&= \left(I+(P-I)\right)\mathcal{J}_0^{-1}\nonumber\\
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&= P\mathcal{J}_0^{-1}.
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\end{align}
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@ -180,9 +180,12 @@ où
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\subsection{Variance-covariance des paramètres}
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On obtient la variance-covariance asymptotique des paramètres à partir
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de la variance-covariance associée aux conditions de moment, en
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utilisant la méthode delta multivariée. Pour ce faire, on évalue
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d'abord la valeur théorique du gradient $D(\theta)$.
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de la variance-covariance associée aux conditions de moment en
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utilisant la méthode delta multivariée (Annexe
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\ref{sec:deltamethod}).
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Pour ce faire, on évalue d'abord la valeur
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théorique du gradient $D(\theta)$:
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\begin{align}
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D(\theta) &= E \left[ \begin{array}{cc}
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-1 & -2\,\left( Y-\theta-\mu\,\tau\right) \\
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@ -19,13 +19,13 @@ En se référant à \cite{bingham2004risk}, on définit:
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\item Une \textbf{option d'achat (de vente) européenne} donne le
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\textbf{droit} d'acheter (de vendre) un actif au \textbf{prix
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d'exercice} $K$ au temps $T$.
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\item Lorsque la valeur actuelle du titre est, par rapport au prix
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\item Lorsque sa valeur actuelle est, par rapport au prix
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d'exercice:
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\begin{itemize}
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\item supérieure $(S(t)>K)$, l'option est dite \textbf{dans le
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\item supérieure $(S(t)>K)$, l'option d'achat est dite \textbf{dans le
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cours};
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\item égale $(S(t)=K)$, l'option est dite \textbf{au cours};
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\item inférieure $(S(t)<K)$, l'option est dite \textbf{hors du
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\item égale $(S(t)=K)$, l'option d'achat est dite \textbf{au cours};
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\item inférieure $(S(t)<K)$, l'option d'achat est dite \textbf{hors du
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cours}.
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\end{itemize}
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\end{itemize}
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@ -116,7 +116,7 @@ Afin de pouvoir utiliser les résultats de l'estimation paramétrique
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des chapitres précédents pour évaluer le prix de produits dérivés, on
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doit tout d'abord identifier les paramètres neutres au risque de la
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distribution de Laplace asymétrique généralisée associant le
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rendements cumulé $L_t$ au taux d'intérêt sans risque $r$. Pour ce
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rendement cumulé $L_t$ au taux d'intérêt sans risque $r$. Pour ce
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faire, on utilise l'équation martingale \eqref{eq:equationmartingale}
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ainsi que la fonction génératrice des moments \eqref{eq:fgmGAL}. On
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obtient alors une expression pour le paramètre de dérive neutre au
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@ -341,7 +341,7 @@ par exemple).
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L'intégration numérique de \eqref{eq:prixoptionCarr} pose problème
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quand l'échéance $T-t$ est petite, ou encore le prix d'exercice $K$
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est hors du cours ($k > \ln(S(0))$). Dans ce contexte particulier,
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est hors du cours ($k > \ln(S(t))$). Dans ce contexte particulier,
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\cite{carr1999option} développent une formule alternative à l'équation
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\eqref{eq:prixoptionCarr} pour évaluer le prix de l'option d'achat :
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\begin{align}
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@ -436,7 +436,7 @@ $\left[-K,K\right]$, le théorème de Fubini (section
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En remplaçant le résultat \eqref{eq:fubini-integrale-a-EPPS} dans
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l'équation de départ \eqref{eq:putepps-1}, on obtient ainsi une
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expression particulièrement simple pour le prix de l'option d'achat:
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expression particulièrement simple pour le prix de l'option de vente:
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\begin{align}
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P(S(t),K,T) &= B(t,T)K\left[\frac{1}{2}-\frac{1}{2\pi} \int_{-c}^c
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K^{-i\nu} \frac{\phi(\nu)}{\nu(i+\nu)}
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@ -39,7 +39,7 @@ premiers moments.
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Minimum & -0.027500\\
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1er quartile & -0.009790\\
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Médiane & -0.003260\\
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3e Quartile & 0.006620\\
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3e quartile & 0.006620\\
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Maximum & 0.043400\\
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\hline
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@ -674,8 +674,8 @@ pour les méthodes des moments généralisée et de l'équation d'estimation
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optimale aux figures \ref{fig:prix1R1-1} et \ref{fig:prix1R1-3}. On
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peut facilement remarquer le manque de précision de l'approche de
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Carr-Madan, qui s'approche de la courbe de Black-Scholes lorsque le
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titre est dans la monnaie et qui se met à osciller dès que le titre
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est hors de la monnaie. Les méthodes de Epps et de Heston donnent des
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titre est dans le cours et qui se met à osciller dès que le titre
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est hors le cours. Les méthodes de Epps et de Heston donnent des
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résultats très similaires, et l'approximation du point de selle
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d'ordre 1 est très précise dans ce contexte.
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\begin{figure}[!ht]
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@ -85,26 +85,26 @@
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\bibcite{heston1993closed}{{29}{1993}{{Heston}}{{}}}
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\bibcite{hinkley1977estimation}{{30}{1977}{{Hinkley et Revankar}}{{}}}
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\bibcite{hogg1978introduction}{{31}{1978}{{Hogg et Craig}}{{}}}
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\bibcite{sato1999levy}{{32}{1999}{{iti Sato}}{{}}}
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\bibcite{itkin2005pricing}{{33}{2005}{{Itkin}}{{}}}
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\bibcite{wang2003evaluating}{{34}{2003}{{{Jingbo Wang} et Marsaglia}}{{}}}
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\bibcite{kotz2001laplace}{{35}{2001}{{Kotz et~al.}}{{Kotz, Kozubowski, et Podg{\'o}rski}}}
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\bibcite{KOUTROUVELIS01011980}{{36}{1980}{{Koutrouvelis}}{{}}}
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\bibcite{kozubowski1999class}{{37}{1999}{{Kozubowski et Podg{\'o}rski}}{{}}}
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\bibcite{kozubowski2001asymmetric}{{38}{2001}{{Kozubowski et Podg{\'o}rski}}{{}}}
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\bibcite{kyprianou2007introductory}{{39}{2007}{{Kyprianou}}{{}}}
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\bibcite{lugannani1980saddle}{{40}{1980}{{Lugannani et Rice}}{{}}}
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\bibcite{lukacs1960characteristic}{{41}{1960}{{Lukacs}}{{}}}
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\bibcite{luong1987minimum}{{42}{1987}{{Luong et Thompson}}{{}}}
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\bibcite{madan1990variance}{{43}{1990}{{Madan et Seneta}}{{}}}
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\bibcite{mandelbrot1963variation}{{44}{1963}{{Mandelbrot}}{{}}}
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\bibcite{merton1976option}{{45}{1976}{{Merton}}{{}}}
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\bibcite{barndorff2001levy}{{46}{2001}{{Mikosch et~al.}}{{Mikosch, Resnick, et Barndorff-Nielsen}}}
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\bibcite{mitchell1916critique}{{47}{1916}{{Mitchell}}{{}}}
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\bibcite{newey1994large}{{48}{1994}{{Newey et McFadden}}{{}}}
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\bibcite{newey1987hypothesis}{{49}{1987}{{Newey et West}}{{}}}
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\bibcite{praetz1972distribution}{{50}{1972}{{Praetz}}{{}}}
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\bibcite{press1967compound}{{51}{1967}{{Press}}{{}}}
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\bibcite{itkin2005pricing}{{32}{2005}{{Itkin}}{{}}}
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\bibcite{wang2003evaluating}{{33}{2003}{{{Jingbo Wang} et Marsaglia}}{{}}}
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\bibcite{kotz2001laplace}{{34}{2001}{{Kotz et~al.}}{{Kotz, Kozubowski, et Podg{\'o}rski}}}
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\bibcite{KOUTROUVELIS01011980}{{35}{1980}{{Koutrouvelis}}{{}}}
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\bibcite{kozubowski1999class}{{36}{1999}{{Kozubowski et Podg{\'o}rski}}{{}}}
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\bibcite{kozubowski2001asymmetric}{{37}{2001}{{Kozubowski et Podg{\'o}rski}}{{}}}
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\bibcite{kyprianou2007introductory}{{38}{2007}{{Kyprianou}}{{}}}
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\bibcite{lugannani1980saddle}{{39}{1980}{{Lugannani et Rice}}{{}}}
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\bibcite{lukacs1960characteristic}{{40}{1960}{{Lukacs}}{{}}}
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\bibcite{luong1987minimum}{{41}{1987}{{Luong et Thompson}}{{}}}
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\bibcite{madan1990variance}{{42}{1990}{{Madan et Seneta}}{{}}}
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\bibcite{mandelbrot1963variation}{{43}{1963}{{Mandelbrot}}{{}}}
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\bibcite{merton1976option}{{44}{1976}{{Merton}}{{}}}
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\bibcite{barndorff2001levy}{{45}{2001}{{Mikosch et~al.}}{{Mikosch, Resnick, et Barndorff-Nielsen}}}
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\bibcite{mitchell1916critique}{{46}{1916}{{Mitchell}}{{}}}
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\bibcite{newey1994large}{{47}{1994}{{Newey et McFadden}}{{}}}
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\bibcite{newey1987hypothesis}{{48}{1987}{{Newey et West}}{{}}}
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\bibcite{praetz1972distribution}{{49}{1972}{{Praetz}}{{}}}
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\bibcite{press1967compound}{{50}{1967}{{Press}}{{}}}
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\bibcite{sato1999levy}{{51}{1999}{{Sato}}{{}}}
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\bibcite{schoutens2003levy}{{52}{2003}{{Schoutens}}{{}}}
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\bibcite{RpackageVarianceGamma}{{53}{2012}{{Scott et Dong}}{{}}}
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\bibcite{seneta2004fitting}{{54}{2004}{{Seneta}}{{}}}
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@ -180,13 +180,6 @@ Robert~V. Hogg et Allen Craig.
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\newblock \emph{{I}ntroduction to {M}athematical {S}tatistics}.
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\newblock Macmillan, 1978.
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\bibitem[iti Sato(1999)]{sato1999levy}
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Ken iti Sato.
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\newblock \emph{{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible
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{D}istributions}.
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\newblock Cambridge Studies in Advanced Mathematics. Cambridge University
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Press, 1999.
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\bibitem[Itkin(2005)]{itkin2005pricing}
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Andrey Itkin.
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\newblock {P}ricing {O}ptions with {VG} {M}odel {U}sing {FFT}.
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@ -296,6 +289,13 @@ S.~James Press.
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\newblock {A} {C}ompound {E}vents {M}odel for {S}ecurity {P}rices.
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\newblock \emph{Journal of Business}, pages 317--335, 1967.
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\bibitem[{Sato}(1999)]{sato1999levy}
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{Ken-Iti} {Sato}.
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\newblock \emph{{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible
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{D}istributions}.
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\newblock Cambridge Studies in Advanced Mathematics. Cambridge University
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Press, 1999.
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\bibitem[Schoutens(2003)]{schoutens2003levy}
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Wim Schoutens.
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\newblock \emph{{L}{\'e}vy {P}rocesses in {F}inance}.
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@ -306,7 +306,7 @@ David Scott et Christine~Yang Dong.
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\newblock \emph{{{v}ariancegamma: {T}he {V}ariance {G}amma {D}istribution}},
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2012.
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\newblock URL \url{http://CRAN.R-project.org/package=VarianceGamma}.
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\newblock R package version 0.3-1.
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\newblock Consulté le 24 janvier 2014.
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\bibitem[Seneta(2004)]{seneta2004fitting}
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Eugene Seneta.
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@ -339,7 +339,7 @@ Alan Stuart et J.~Keith Ord.
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\bibitem[Teschl(2004)]{teschl2004topics}
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Gerald Teschl.
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\newblock {T}opics in {R}eal and {F}unctional {A}nalysis.
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\newblock 2004.
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\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}
|
||||
|
|
|
|||
|
|
@ -21,33 +21,32 @@ A level-1 auxiliary file: annexe2.aux
|
|||
A level-1 auxiliary file: annexe3.aux
|
||||
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|
||||
Database file #1: memoire.bib
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|||
|
||||
@Manual{RpackageVarianceGamma,
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||||
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|
||||
Note = "R package version 0.3-1",
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||||
Note = "Consulté le 24 janvier 2014",
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|
@ -517,7 +518,7 @@
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}
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||||
|
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|
||||
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||||
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||||
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||||
|
|
@ -580,6 +581,7 @@
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|||
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|
||||
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|
||||
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||||
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|
||||
Url = "http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html",
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||||
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||||
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||||
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|
||||
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|
||||
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||||
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||||
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|
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|
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||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
Number = 7,
|
||||
Pages = 2928,
|
||||
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|
||||
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|
||||
Year = 2009
|
||||
}
|
||||
|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
|
||||
@Article{applebaum2004levy,
|
||||
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|
||||
Journal = "Notices of the American Mathematical Society",
|
||||
Number = 11,
|
||||
Pages = "1336--1347",
|
||||
Title = "{L}{\'e}vy {P}rocesses: {F}rom {P}robability to {F}inance and {Q}uantum {G}roups",
|
||||
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|
||||
Year = 2004
|
||||
}
|
||||
|
||||
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|
||||
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|
||||
Publisher = "Gauthier-Villars",
|
||||
Title = "{T}h{\'e}orie de la sp{\'e}culation",
|
||||
Year = 1900
|
||||
}
|
||||
|
||||
@Book{barndorff2001levy,
|
||||
Author = "Thomas Mikosch and Sidney I. Resnick and Ole E. Barndorff-Nielsen",
|
||||
Isbn = 9780817641672,
|
||||
Publisher = "Birkh{\"a}user",
|
||||
Title = "{L}{\'e}vy {P}rocesses: {T}heory and {A}pplications",
|
||||
Year = 2001
|
||||
}
|
||||
|
||||
@Article{berkson1980minimum,
|
||||
Author = "Joseph Berkson",
|
||||
Journal = "the Annals of Statistics",
|
||||
Pages = "457--487",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{M}inimum {C}hi-Square, not {M}aximum {L}ikelihood!",
|
||||
Year = 1980
|
||||
}
|
||||
|
||||
@Book{bingham2004risk,
|
||||
Author = "Nicholas H. Bingham and R{\"u}diger Kiesel",
|
||||
Publisher = "Springer",
|
||||
Title = "{R}isk-Neutral {V}aluation: {P}ricing and {H}edging of {F}inancial {D}erivatives",
|
||||
Year = 2004
|
||||
}
|
||||
|
||||
@Article{black1973pricing,
|
||||
Author = "Fischer Black and Myron Scholes",
|
||||
Journal = "the Journal of Political Economy",
|
||||
Pages = "637--654",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{T}he {P}ricing of {O}ptions and {C}orporate {L}iabilities",
|
||||
Year = 1973
|
||||
}
|
||||
|
||||
@Article{black1976pricing,
|
||||
Author = "Fischer Black",
|
||||
Journal = "Journal of Financial Economics",
|
||||
Number = 1,
|
||||
Pages = "167--179",
|
||||
Publisher = "Elsevier",
|
||||
Title = "{T}he {P}ricing of {C}ommodity {C}ontracts",
|
||||
Volume = 3,
|
||||
Year = 1976
|
||||
}
|
||||
|
||||
@Article{buckle1995bayesian,
|
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Author = "D. J. Buckle",
|
||||
Journal = "Journal of the American Statistical Association",
|
||||
Number = 430,
|
||||
Pages = "605--613",
|
||||
Publisher = "Taylor \& Francis Group",
|
||||
Title = "{B}ayesian {I}nference for {S}table {D}istributions",
|
||||
Volume = 90,
|
||||
Year = 1995
|
||||
}
|
||||
|
||||
@Book{butler2007saddlepoint,
|
||||
Author = "Ronald W. Butler",
|
||||
Publisher = "Cambridge University Press",
|
||||
Title = "{S}addlepoint {A}pproximations with {A}pplications",
|
||||
Volume = 22,
|
||||
Year = 2007
|
||||
}
|
||||
|
||||
@Article{carr1999option,
|
||||
Author = "Peter P. Carr and Dilip B. Madan",
|
||||
Journal = "Journal of Computational Finance",
|
||||
Number = 4,
|
||||
Pages = "61--73",
|
||||
Title = "{O}ption {V}aluation {U}sing the {F}ast {F}ourier {T}ransform",
|
||||
Volume = 2,
|
||||
Year = 1999
|
||||
}
|
||||
|
||||
@Article{crowder1986consistency,
|
||||
Author = "Martin Crowder",
|
||||
Journal = "Econometric theory",
|
||||
Pages = "305--330",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{O}n {C}onsistency and {I}nconsistency of {E}stimating {E}quations",
|
||||
Year = 1986
|
||||
}
|
||||
|
||||
@Article{crowder1987linear,
|
||||
Author = "Martin Crowder",
|
||||
Journal = "Biometrika",
|
||||
Number = 3,
|
||||
Pages = "591--597",
|
||||
Publisher = "Biometrika Trust",
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||||
Title = "{O}n {L}inear and {Q}uadratic {E}stimating {F}unctions",
|
||||
Volume = 74,
|
||||
Year = 1987
|
||||
}
|
||||
|
||||
@Article{daniels1954saddlepoint,
|
||||
Author = "Henry E. Daniels",
|
||||
Journal = "the Annals of Mathematical Statistics",
|
||||
Pages = "631--650",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{S}addlepoint {A}pproximations in {S}tatistics",
|
||||
Year = 1954
|
||||
}
|
||||
|
||||
@Article{daniels1987tail,
|
||||
Author = "Henry E. Daniels",
|
||||
Journal = "international Statistical Review/Revue internationale de Statistique",
|
||||
Pages = "37--48",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{T}ail {P}robability {A}pproximations",
|
||||
Year = 1987
|
||||
}
|
||||
|
||||
@Techreport{derman1996modelrisk,
|
||||
Author = "Emanuel Derman",
|
||||
Institution = "Goldman Sachs",
|
||||
Title = "{M}odel {R}isk",
|
||||
Year = 1996
|
||||
}
|
||||
|
||||
@Book{dodge2004statistique,
|
||||
Author = "Yadolah Dodge",
|
||||
Publisher = "Springer Verlag France",
|
||||
Title = "{S}tatistique: {D}ictionnaire Encyclop{\'e}dique",
|
||||
Year = 2004
|
||||
}
|
||||
|
||||
@Article{epps1983test,
|
||||
Author = "Thomas W. Epps and Lawrence B. Pulley",
|
||||
Journal = "Biometrika",
|
||||
Number = 3,
|
||||
Pages = "723--726",
|
||||
Publisher = "Biometrika Trust",
|
||||
Title = "{A} {T}est for {N}ormality {B}ased on the {E}mpirical {C}haracteristic {F}unction",
|
||||
Volume = 70,
|
||||
Year = 1983
|
||||
}
|
||||
|
||||
@Book{epps2007pricing,
|
||||
Author = "Thomas W. Epps",
|
||||
Publisher = "World Scientific Publishing Company incorporated",
|
||||
Title = "{P}ricing {D}erivative {S}ecurities",
|
||||
Year = 2007
|
||||
}
|
||||
|
||||
@Book{everitt2006cambridge,
|
||||
Author = "Brian Everitt and Anders Skrondal",
|
||||
Publisher = "Cambridge University Press Cambridge",
|
||||
Title = "{T}he {C}ambridge {D}ictionary of {S}tatistics",
|
||||
Volume = 4,
|
||||
Year = 2006
|
||||
}
|
||||
|
||||
@Article{fama1993common,
|
||||
Author = "Eugene F. Fama and Kenneth R. French",
|
||||
Journal = "Journal of Financial Economics",
|
||||
Number = 1,
|
||||
Pages = "3--56",
|
||||
Publisher = "Elsevier",
|
||||
Title = "{C}ommon {R}isk {F}actors in the {R}eturns on {S}tocks and {B}onds",
|
||||
Volume = 33,
|
||||
Year = 1993
|
||||
}
|
||||
|
||||
@Article{feuerverger1981efficiency,
|
||||
Author = "Andrey Feuerverger and Philip McDunnough",
|
||||
Journal = "Journal of the Royal Statistical Society. Series B (methodological)",
|
||||
Pages = "20--27",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{O}n the {E}fficiency of {E}mpirical {C}haracteristic {F}unction {P}rocedures",
|
||||
Year = 1981
|
||||
}
|
||||
|
||||
@Article{fox1986large,
|
||||
Author = "Robert Fox and Murad S. Taqqu",
|
||||
Journal = "the Annals of Statistics",
|
||||
Number = 2,
|
||||
Pages = "517--532",
|
||||
Publisher = "institute of Mathematical Statistics",
|
||||
Title = "{L}arge-Sample {P}roperties of {P}arameter {E}stimates for {S}trongly {D}ependent {S}tationary {G}aussian {T}ime {S}eries",
|
||||
Volume = 14,
|
||||
Year = 1986
|
||||
}
|
||||
|
||||
@Article{gil1951note,
|
||||
Author = "J. Gil-Pelaez",
|
||||
Journal = "Biometrika",
|
||||
Number = "3-4",
|
||||
Pages = "481--482",
|
||||
Publisher = "Biometrika Trust",
|
||||
Title = "{N}ote on the {I}nversion {T}heorem",
|
||||
Volume = 38,
|
||||
Year = 1951
|
||||
}
|
||||
|
||||
@Book{gourieroux1989statistique,
|
||||
Author = "Christian Gourieroux and Alain Monfort",
|
||||
Publisher = "Economica",
|
||||
Title = "{S}tatistique et Mod{\`e}les {\'E}conom{\'e}triques: Notions G{\'e}n{\'e}rales, Estimation, Pr{\'e}vision, Algorithmes",
|
||||
Volume = 1,
|
||||
Year = 1989
|
||||
}
|
||||
|
||||
@Book{hall2005generalized,
|
||||
Author = "Alastair R. Hall",
|
||||
Publisher = "Oxford University Press Oxford",
|
||||
Title = "{G}eneralized {M}ethod of {M}oments",
|
||||
Year = 2005
|
||||
}
|
||||
|
||||
@Book{hamilton1994time,
|
||||
Author = "James Douglas Hamilton",
|
||||
Publisher = "Cambridge University Press",
|
||||
Title = "{T}ime {S}eries {A}nalysis",
|
||||
Volume = 2,
|
||||
Year = 1994
|
||||
}
|
||||
|
||||
@Article{hansen1982large,
|
||||
Author = "Lars Peter Hansen",
|
||||
Journal = "Econometrica: Journal of the Econometric Society",
|
||||
Pages = "1029--1054",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{L}arge {S}ample {P}roperties of {G}eneralized {M}ethod of {M}oments {E}stimators",
|
||||
Year = 1982
|
||||
}
|
||||
|
||||
@Article{henze1990approximation,
|
||||
Author = "Norbert Henze",
|
||||
Journal = "Metrika",
|
||||
Number = 1,
|
||||
Pages = "7--18",
|
||||
Publisher = "Springer",
|
||||
Title = "{A}n {A}pproximation to the {L}imit {D}istribution of the {E}pps-{P}ulley {T}est {S}tatistic for {N}ormality",
|
||||
Volume = 37,
|
||||
Year = 1990
|
||||
}
|
||||
|
||||
@Article{heston1993closed,
|
||||
Author = "Steven L. Heston",
|
||||
Journal = "Review of Financial Studies",
|
||||
Number = 2,
|
||||
Pages = "327--343",
|
||||
Publisher = "Soc Financial Studies",
|
||||
Title = "{A} {C}losed-{F}orm {S}olution for {O}ptions with {S}tochastic {V}olatility with {A}pplications to {B}ond and {C}urrency {O}ptions",
|
||||
Volume = 6,
|
||||
Year = 1993
|
||||
}
|
||||
|
||||
@Article{hinkley1977estimation,
|
||||
Author = "David V. Hinkley and Nagesh S. Revankar",
|
||||
Journal = "Journal of Econometrics",
|
||||
Number = 1,
|
||||
Pages = "1--11",
|
||||
Publisher = "Elsevier",
|
||||
Title = "{E}stimation of the {P}areto {L}aw from {U}nderreported {D}ata: {A} {F}urther {A}nalysis",
|
||||
Volume = 5,
|
||||
Year = 1977
|
||||
}
|
||||
|
||||
@Book{hogg1978introduction,
|
||||
Author = "Robert V. Hogg and Allen Craig",
|
||||
Isbn = 9780029789902,
|
||||
Publisher = "Macmillan",
|
||||
Title = "{I}ntroduction to {M}athematical {S}tatistics",
|
||||
Year = 1978
|
||||
}
|
||||
|
||||
@Book{hull1999options,
|
||||
Author = "John C. Hull",
|
||||
Publisher = "Pearson Education india",
|
||||
Title = "{O}ptions, {F}utures, and {O}ther {D}erivatives",
|
||||
Year = 1999
|
||||
}
|
||||
|
||||
@Article{itkin2005pricing,
|
||||
Author = "Andrey Itkin",
|
||||
Journal = "arXiv preprint physics/0503137",
|
||||
Title = "{P}ricing {O}ptions with {VG} {M}odel {U}sing {FFT}",
|
||||
Year = 2005
|
||||
}
|
||||
|
||||
@Book{kotz2001laplace,
|
||||
Author = "Samuel Kotz and Tomasz J. Kozubowski and Krzystof Podg{\'o}rski",
|
||||
Isbn = 9780817641665,
|
||||
Publisher = "Birkh{\"a}user",
|
||||
Series = "Progress in Mathematics Series",
|
||||
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
|
||||
}
|
||||
|
||||
@Article{kozubowski1999class,
|
||||
Author = "Tomasz J. Kozubowski and Krzysztof Podg{\'o}rski",
|
||||
Journal = "Actuarial Research Clearing House",
|
||||
Pages = "113--134",
|
||||
Publisher = "Citeseer",
|
||||
Title = "{A} {C}lass of {A}symmetric {D}istributions",
|
||||
Volume = 1,
|
||||
Year = 1999
|
||||
}
|
||||
|
||||
@Article{kozubowski2001asymmetric,
|
||||
Author = "Tomasz J. Kozubowski and Krzysztof Podg{\'o}rski",
|
||||
Journal = "{M}athematical and {C}omputer {M}odeling",
|
||||
Number = 9,
|
||||
Pages = "1003--1021",
|
||||
Publisher = "Elsevier",
|
||||
Title = "{A}symmetric {L}aplace {L}aws and {M}odeling {F}inancial {D}ata",
|
||||
Volume = 34,
|
||||
Year = 2001
|
||||
}
|
||||
|
||||
@Book{kyprianou2007introductory,
|
||||
Author = "Andreas E. Kyprianou",
|
||||
Publisher = "Springer",
|
||||
Title = "{I}ntroductory {L}ectures on {F}luctuations of {L}{\'e}vy {P}rocesses with {A}pplications",
|
||||
Year = 2007
|
||||
}
|
||||
|
||||
@Article{lugannani1980saddle,
|
||||
Author = "Robert Lugannani and Stephen Rice",
|
||||
Journal = "Advances in Applied Probability",
|
||||
Pages = "475--490",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{S}addle {P}oint {A}pproximation for the {D}istribution of the {S}um of {I}ndependent {R}andom {V}ariables",
|
||||
Year = 1980
|
||||
}
|
||||
|
||||
@Book{lukacs1960characteristic,
|
||||
Author = "Eugene Lukacs",
|
||||
Publisher = "Griffin London",
|
||||
Title = "{C}haracteristic {F}unctions",
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||||
Volume = 4,
|
||||
Year = 1960
|
||||
}
|
||||
|
||||
@Article{luong1987minimum,
|
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Author = "Andrew Luong and Mary E. Thompson",
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||||
Journal = "Canadian Journal of Statistics",
|
||||
Number = 3,
|
||||
Pages = "239--251",
|
||||
Publisher = "Wiley online Library",
|
||||
Title = "{M}inimum-{D}istance {M}ethods {B}ased on {Q}uadratic {D}istances for {T}ransforms",
|
||||
Volume = 15,
|
||||
Year = 1987
|
||||
}
|
||||
|
||||
@Article{madan1990variance,
|
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Author = "Dilip B. Madan and Eugene Seneta",
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||||
Journal = "Journal of Business",
|
||||
Pages = "511--524",
|
||||
Publisher = "JSTOR",
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||||
Title = "{T}he {V}ariance {G}amma {M}odel for {S}hare {M}arket {R}eturns",
|
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Year = 1990
|
||||
}
|
||||
|
||||
@Article{madan1998variance,
|
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Author = "Peter P. Carr and Eric C. Chang and Dilip B. Madan",
|
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Journal = "European Finance Review",
|
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Number = 1,
|
||||
Pages = "79--105",
|
||||
Publisher = "Kluwer Academic Publishers",
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Title = "{T}he {V}ariance {G}amma {P}rocess and {O}ption {P}ricing",
|
||||
Volume = 2,
|
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Year = 1998
|
||||
}
|
||||
|
||||
@Article{mandelbrot1963variation,
|
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Author = "Benoit Mandelbrot",
|
||||
Journal = "Journal of Business",
|
||||
Pages = "394--419",
|
||||
Publisher = "University of Chicago Press",
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||||
Title = "{T}he {V}ariation of {C}ertain {S}peculative {P}rices",
|
||||
Year = 1963
|
||||
}
|
||||
|
||||
@Article{merton1976option,
|
||||
Author = "Robert C. Merton",
|
||||
Journal = "Journal of Financial Economics",
|
||||
Number = 1,
|
||||
Pages = "125--144",
|
||||
Publisher = "Elsevier",
|
||||
Title = "{O}ption {P}ricing when {U}nderlying {S}tock {R}eturns {A}re {D}iscontinuous",
|
||||
Volume = 3,
|
||||
Year = 1976
|
||||
}
|
||||
|
||||
@Article{mitchell1916critique,
|
||||
Author = "Wesley C. Mitchell",
|
||||
Journal = "The Journal of Political Economy",
|
||||
Pages = "625--693",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{A} {C}ritique of {I}ndex {N}umbers of the {P}rices of {S}tocks",
|
||||
Year = 1916
|
||||
}
|
||||
|
||||
@Book{musiela2005martingale,
|
||||
Author = "Marek Musiela and Marek Rutkowski",
|
||||
Publisher = "Springer",
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||||
Title = "{M}artingale {M}ethods in {F}inancial {M}odelling",
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||||
Volume = 36,
|
||||
Year = 2005
|
||||
}
|
||||
|
||||
@Article{newey1987hypothesis,
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Author = "Whitney K. Newey and Kenneth D. West",
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Journal = "International Economic Review",
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Number = 3,
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||||
Pages = "777--787",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{H}ypothesis {T}esting with {E}fficient {M}ethod of {M}oments {E}stimation",
|
||||
Volume = 28,
|
||||
Year = 1987
|
||||
}
|
||||
|
||||
@Article{newey1994large,
|
||||
Author = "Whitney K. Newey and Daniel McFadden",
|
||||
Journal = "Handbook of Econometrics",
|
||||
Pages = "2111--2245",
|
||||
Publisher = "Elsevier",
|
||||
Title = "{L}arge {S}ample {E}stimation and {H}ypothesis {T}esting",
|
||||
Volume = 4,
|
||||
Year = 1994
|
||||
}
|
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|
||||
@Article{praetz1972distribution,
|
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Author = "Peter D. Praetz",
|
||||
Journal = "Journal of Business",
|
||||
Pages = "49--55",
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||||
Publisher = "JSTOR",
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||||
Title = "{T}he {D}istribution of {S}hare {P}rice {C}hanges",
|
||||
Year = 1972
|
||||
}
|
||||
|
||||
@Article{press1967compound,
|
||||
Author = "S. James Press",
|
||||
Journal = "Journal of Business",
|
||||
Pages = "317--335",
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||||
Publisher = "JSTOR",
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||||
Title = "{A} {C}ompound {E}vents {M}odel for {S}ecurity {P}rices",
|
||||
Year = 1967
|
||||
}
|
||||
|
||||
@Article{randal2004non,
|
||||
Author = "John A. Randal and Peter J. Thomson and Martin T. Lally",
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||||
Journal = "Quantitative Finance",
|
||||
Number = 4,
|
||||
Pages = "427--440",
|
||||
Publisher = "Taylor \& Francis",
|
||||
Title = "{N}on-parametric {E}stimation of {H}istorical {V}olatility",
|
||||
Volume = 4,
|
||||
Year = 2004
|
||||
}
|
||||
|
||||
@Book{sato1999levy,
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||||
Author = "Ken-iti Sato",
|
||||
Isbn = 9780521553025,
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||||
Publisher = "Cambridge University Press",
|
||||
Series = "Cambridge Studies in Advanced Mathematics",
|
||||
Title = "{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible {D}istributions",
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||||
Year = 1999
|
||||
}
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|
||||
@Book{schoutens2003levy,
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||||
Author = "Wim Schoutens",
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||||
Publisher = "Wiley",
|
||||
Title = "{L}{\'e}vy {P}rocesses in {F}inance",
|
||||
Year = 2003
|
||||
}
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||||
|
||||
@Article{seneta2004fitting,
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||||
Author = "Eugene Seneta",
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||||
Journal = "Journal of Applied Probability",
|
||||
Pages = "177--187",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{F}itting the {V}ariance {G}amma {M}odel to {F}inancial {D}ata",
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||||
Year = 2004
|
||||
}
|
||||
|
||||
@Article{shapiro1965analysis,
|
||||
Author = "Samuel Sanford Shapiro and Martin B. Wilk",
|
||||
Journal = "Biometrika",
|
||||
Number = "3/4",
|
||||
Pages = "591--611",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{A}n {A}nalysis of {V}ariance {T}est for {N}ormality ({C}omplete {S}amples)",
|
||||
Volume = 52,
|
||||
Year = 1965
|
||||
}
|
||||
|
||||
@Article{shephard1991characteristic,
|
||||
Author = "Neil G. Shephard",
|
||||
Journal = "Econometric Theory",
|
||||
Number = 04,
|
||||
Pages = "519--529",
|
||||
Publisher = "Cambridge Univ Press",
|
||||
Title = "{F}rom {C}haracteristic {F}unction to {D}istribution {F}unction: {A} {S}imple {F}ramework for the {T}heory",
|
||||
Volume = 7,
|
||||
Year = 1991
|
||||
}
|
||||
|
||||
@Book{spiegel1999schaum,
|
||||
Author = "Murray R. Spiegel and John Liu",
|
||||
Publisher = "McGraw-Hill",
|
||||
Title = "{S}chaum's {M}athematical {H}andbook of {F}ormulas and {T}ables",
|
||||
Volume = 1000,
|
||||
Year = 1999
|
||||
}
|
||||
|
||||
@Book{stuart1987kendall,
|
||||
Author = "Alan Stuart and J. Keith Ord",
|
||||
Publisher = "Oxford University Press, New York",
|
||||
Title = "{K}endall{\rq}s {A}dvanced {T}heory of {S}tatistics, {V}ol. 1",
|
||||
Year = 1987
|
||||
}
|
||||
|
||||
@Unpublished{teschl2004topics,
|
||||
Author = "Gerald Teschl",
|
||||
Title = "{T}opics in {R}eal and {F}unctional {A}nalysis",
|
||||
Year = 2004
|
||||
}
|
||||
|
||||
@Phdthesis{torczon1989multi,
|
||||
Author = "Virginia Joanne Torczon",
|
||||
Title = "{M}ulti-directional {S}earch: {A} {D}irect {S}earch {A}lgorithm for {P}arallel {M}achines",
|
||||
Year = 1989
|
||||
}
|
||||
|
||||
@Inproceedings{walterlevy,
|
||||
Author = "Christian Walter",
|
||||
Booktitle = "Proceedings of the 5th AFIR colloquium",
|
||||
Title = "{L}{\'e}vy-{S}tability {U}nder {A}ddition and {F}ractal {S}tructure of {M}arkets: {I}mplications for the {A}ctuaries and {E}mphasized {E}xamination of {MATIF} {N}ational {C}ontract",
|
||||
Year = 1995
|
||||
}
|
||||
|
||||
@Article{wang2003evaluating,
|
||||
Author = "Wai Wan Tsang {Jingbo Wang} and George Marsaglia",
|
||||
Journal = "Journal of Statistical Software",
|
||||
Number = 18,
|
||||
Pages = "1--4",
|
||||
Publisher = "American Statistical Association",
|
||||
Title = "{E}valuating {K}olmogorov's {D}istribution",
|
||||
Volume = 8,
|
||||
Year = 2003
|
||||
}
|
||||
|
||||
@Article{wendel1961non,
|
||||
Author = "J. G. Wendel",
|
||||
Journal = "The Annals of Mathematical Statistics",
|
||||
Number = 1,
|
||||
Pages = "338--339",
|
||||
Publisher = "institute of Mathematical Statistics",
|
||||
Title = "{T}he {N}on-{A}bsolute {C}onvergence of {G}il-{P}elaez' {I}nversion {I}ntegral",
|
||||
Volume = 32,
|
||||
Year = 1961
|
||||
}
|
||||
|
||||
@Article{wolfowitz1957minimum,
|
||||
Author = "Jacob Wolfowitz",
|
||||
Journal = "The Annals of Mathematical Statistics",
|
||||
Pages = "75--88",
|
||||
Publisher = "JSTOR",
|
||||
Title = "{T}he {M}inimum {D}istance {M}ethod",
|
||||
Year = 1957
|
||||
}
|
||||
|
||||
@Book{wooldridge2001econometric,
|
||||
Author = "Jeffrey M. Wooldridge",
|
||||
Publisher = "MIT press",
|
||||
Title = "{E}conometric {A}nalysis of {C}ross {S}ection and {P}anel {D}ata",
|
||||
Year = 2001
|
||||
}
|
||||
|
||||
Loading…
Reference in a new issue