258 lines
13 KiB
TeX
258 lines
13 KiB
TeX
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\documentclass[landscape,final,a0paper,fontscale=0.285]{baposter}
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\usepackage{amsmath}
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%\usepackage{times}
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\newcommand{\captionfont}{\footnotesize}
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\graphicspath{{images/}{../images/}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Begin of Document
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Here starts the poster
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%%%---------------------------------------------------------------------------
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%%% Format it to your taste with the options
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% Define some colors
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%\definecolor{lightblue}{cmyk}{0.83,0.24,0,0.12}
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\definecolor{lightblue}{rgb}{0.145,0.6666,1}
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\hyphenation{resolution occlusions}
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%%
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\begin{poster}%
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% Poster Options
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{
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% Show grid to help with alignment
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grid=false,
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% Column spacing
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colspacing=1em,
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% Color style
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bgColorOne=white,
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bgColorTwo=white,
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borderColor=lightblue,
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headerColorOne=black,
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headerColorTwo=lightblue,
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boxColorOne=white,
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textborder=roundedleft,
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eyecatcher=true,
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headerborder=closed,
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headerheight=0.1\textheight,
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% textfont=\sc, An example of changing the text font
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headershape=roundedright,
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headershade=shadelr,
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headerfont=\Large\bf\textsc, %Sans Serif
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textfont={\setlength{\parindent}{1.5em}},
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boxshade=plain,
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% background=shade-tb,
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linewidth=2pt
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}
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% Eye Catcher
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{\includegraphics[height=5em]{images/ullong.pdf}}
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% Title
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{\bf\textsc{Distribution de Laplace asymétrique généralisée}\vspace{0.5em}}
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% Authors
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{\textsc{François Pelletier, Andrew Luong} \\ \small{École d'actuariat, Université Laval}}
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% University logo
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{% The makebox allows the title to flow into the logo, this is a hack because of the L shaped logo.
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Now define the boxes that make up the poster
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%%%---------------------------------------------------------------------------
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%%% Each box has a name and can be placed absolutely or relatively.
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%%% The only inconvenience is that you can only specify a relative position
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%%% towards an already declared box. So if you have a box attached to the
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%%% bottom, one to the top and a third one which should be in between, you
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%%% have to specify the top and bottom boxes before you specify the middle
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%%% box.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% A coloured circle useful as a bullet with an adjustably strong filling
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\newcommand{\colouredcircle}{%
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\tikz{\useasboundingbox (-0.2em,-0.32em) rectangle(0.2em,0.32em); \draw[draw=black,fill=lightblue,line width=0.03em] (0,0) circle(0.18em);}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Introduction}{name=problem,column=0,row=0}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\small{On définit le prix au temps $t$ d'un titre financier $S(t)$ et le rendement cumulé sur ce titre
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\begin{equation*}L_t = R_1 + \ldots + R_{t} = \log S(t) - \log S(0). \end{equation*}
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On veut modéliser $R_t$ par la distribution de Laplace asymétrique généralisée, aussi appelée variance-gamma. L'estimation paramétrique se fera à partir de la méthode GMM.
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On pourra ainsi obtenir la distribution de $L_t$ par convolution, afin d'évaluer le prix d'options européennes.}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Simulation}{name=contribution,column=1,row=0,bottomaligned=problem}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\small{Le processus de Laplace $Y(t)$ peut être représenté comme la différence de deux processus gamma $G(t)$ \cite{kotz2001laplace}.
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\begin{equation*} Y \stackrel{d}{=} \theta + \frac{\sigma}{\sqrt{2}} \left( \frac{1}{\kappa} G_1 - \kappa G_2 \right)
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\end{equation*} où $ G_1,G_2 \sim \Gamma\left(\alpha=\tau,\beta=1 \right)$.
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Il suffit donc de simuler deux réalisations de cette variable
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aléatoire pour obtenir une réalisation de la distribution Laplace
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asymétrique généralisée.
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}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Estimation du modèle}{name=results,column=2,span=1,row=0}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\small{On utilise les conditions de moment basées sur l'espérance et la variance
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\begin{equation*}
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g(Y_t,\theta) =
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\left[\begin{array}[]{c}
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\bar{Y} - \left(\theta+\mu\tau\right) \\
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\frac{1}{T}\sum_{t=1}^T \left(Y_t - \bar{Y} \right)^2 - \tau\left(\sigma^2+\mu^2\right)
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\end{array}\right]
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\end{equation*}
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La matrice optimale $W$ est estimée en utilisant la méthode du GMM itératif \cite{hall2005generalized}:
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\begin{align*}
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\hat{W}_{(1)} &= \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(0)})g(Y_t,\hat\theta_{(0)})^{\prime}\bigg)^{-1} \\
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\hat{W}_{(i)} &= \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(i-1)})g(Y_t,\hat\theta_{(i-1)})^{\prime}\bigg)^{-1}
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\end{align*}}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Graphique}{name=references,column=3,span=1,bottomaligned=results}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\includegraphics[width=64mm,height=56mm]{GraphiqueAffiche1}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Processus de Laplace}{name=references,column=0,above=bottom}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\small{Le processus de Laplace $Y(t)$ est un processus de Lévy subordonné, c'est-à-dire qu'il est formé de deux processus de Lévy, l'un qui détermine l'amplitude des évènements (sauts) et l'autre qui détermine le temps entre les évènements.
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Le premier est un processus de Wiener $W(t+\tau)-W(t) \sim N(0,\tau\sigma^2)$ et le second un processus gamma $G(t+\tau)-G(t) \sim \Gamma(\tau,\nu\tau)$.
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\begin{equation*}Y(t) = W(G(t))\end{equation*}}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Méthode GMM}{name=questions,column=1,span=1,aligned=references,above=bottom}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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La méthode GMM a pour objectif d'estimer les paramètres $\theta$ d'une distribution en minimisant une norme quadratique
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\begin{equation*}
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\|\hat{m}\left(\theta\right) \|_W^2 = \hat{m}\left(\theta\right)^{\prime} W \hat{m}\left(\theta\right)
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\end{equation*}
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pondérée par une matrice définie positive $W$ et où $\hat{m}\left(\theta\right) = \left(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)\right)$ est la moyenne empirique des conditions de moment $g(Y_t,\theta)$.
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Références}{name=references,column=2,span=2,aligned=references,above=bottom}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliographystyle{plain} % (uses file "plain.bst")
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\bibliography{biblio} % expects file "myrefs.bib"
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Prix d'options}{name=speed,column=2,span=2,row=0,below=results,above=references}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\small{L'évaluation du prix d'un option de vente européenne équivaut à calculer la valeur espérée de la réclamation contingente définie par le contrat
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\begin{equation}
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P(Y(t),T-t) = B(t,T) \int_{0}^{K} (K-Y(T)) d\hat{F}_t(Y(T))
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\end{equation}
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où $B(t,T)$ est la valeur d'une obligation zéro-coupon au taux sans
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risque $r$ d'échéance $T$ et $K$ le prix d'exercice. Cette évaluation doit de faire sous une mesure neutre au risque, selon laquelle les investisseurs n'exigent pas une prime de risque. Cette mesure doit répondre à la propriété martingale. Pour la distribution étudiée, ceci se fait par une modification du paramètre de dérive $\theta$ en $\theta^{\star}$
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\begin{align*}
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e^{rt} = E\left[exp(L_t) \right] = M_{Y_1+\ldots+Y_t}(1) = (\phi_{Y}(-i))^t = \left(\frac{e^{\theta^{\star}}}{(1-\mu-\sigma^2/2)^{\tau}} \right)^t \Rightarrow \theta^{\star} = r + \tau\log(1-\mu-\frac{\sigma^2}{2})
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\end{align*}
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On pourra ensuite évaluer la valeur du prix de l'option de vente avec la formule de Heston (1993) décrite dans \cite{epps2007pricing}
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\begin{align*}
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P(Y(t),T-t) &= B(t,T)K \int_{0}^{K} d{F}_t(Y(T)) - B(t,T)\int_{0}^{K} Y(T)\cdot d{F}_t(Y(T)) = B(t,T)K \hat{F}_t(K) - Y(t)\hat{G}_t(K)
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\end{align*}
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où $\hat{G}_t(K)$ est la fonction de répartition de la transformée d'Esscher (h=1) de la mesure neutre au risque.
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}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Choix du modèle}{name=method,column=0,below=problem,bottomaligned=speed}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\small{Inspirés par les modèles Mandelbrot, Press et Praetz , Madan et Seneta \cite{madan1990variance} présentent un ensemble de caractéristiques essentielles pour un modèle de rendements financiers:}
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\small{\begin{enumerate}
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\item Distribution de $R_t$ ayant une queue longue
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\item Distribution de $R_t$ ayant des moments finis
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\item Processus en temps continu ayant des accroissements stationnaires et indépendants. Distribution des accroissement de même famille peu importe la longueur.
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\item Extension multivariée afin de conserver la validité du modèle CAPM.
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\end{enumerate}}
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\small{Ce tableau décrit le respect des conditions pour les différents modèles étudiés par les auteurs}
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\begin{center}
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\begin{tabular}{|l|cccc|}
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\hline
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& \multicolumn{4}{c|}{\scriptsize{Conditions}} \\
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\hline
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\scriptsize{Modèle} & \scriptsize{1} & \scriptsize{2} & \scriptsize{3} & \scriptsize{4} \\
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\hline
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\scriptsize{Mandelbrot} & \scriptsize{X} & & & \scriptsize{X} \\
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\scriptsize{Press} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} \\
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\scriptsize{Praetz} & \scriptsize{X} & \scriptsize{X} & & \scriptsize{X} \\
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\scriptsize{Madan et Seneta} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} \\
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\hline
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\end{tabular}
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\end{center}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\headerbox{Fonction caractéristique}{name=background model,column=1,below=problem,bottomaligned=speed}{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\small{On définit la variable aléatoire $Y$
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\begin{equation*}
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\label{eq:defvarY-GAL}
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Y = \theta + \mu W + \sigma \sqrt{W} Z
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\end{equation*}
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La fonction caractéristique de la distribution de $Y$ peut être
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obtenue en utilisant la formule de l'espérance conditionnelle.
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\begin{align*}
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\phi_Y&(t;\theta,\sigma,\mu,\tau) = E\left[e^{ity}\right] \\
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&= E\left[E\left[e^{ity} | W \right] \right] \\
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&= \int_0^{\infty} E \left[ e^{it(\theta + \mu w+\sigma\sqrt{w}Z)} \right] g(w) dw \\
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&= e^{i\theta t}\int_0^{\infty} e^{ i\mu w t-\frac{\sigma^2t^2w}{2}} \times \frac{1}{\Gamma (\tau)} w^{\tau-1}e^{-w} \\
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&= \frac{e^{i\theta t}}{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}}
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\end{align*}
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On retrouve la fonction de répartition par le théorème de Gil-Pelaez
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\begin{equation*}
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\label{eq:inversionfncaract}
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F_Y(y) = \frac{1}{2} - \frac{1}{\pi}\int_{0}^{\infty} \frac{Im\left[e^{-ity}\phi_Y(t)\right]}{t} dt
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\end{equation*}}
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}
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\end{poster}
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\end{document}
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