435 lines
12 KiB
Text
435 lines
12 KiB
Text
\documentclass[letter]{report}
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\usepackage[margin=0.5in]{geometry}
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\usepackage{Sweave}
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\usepackage{graphicx}
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\usepackage[francais]{babel}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{verbatim}
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\usepackage{float}
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\usepackage{hyperref}
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\usepackage{scrtime}
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\begin{document}
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\title{GAL Buckle 95}
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\author{François Pelletier}
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\maketitle
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\tableofcontents
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\chapter{Initialisation}
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\section{Chargement des paquets}
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<<>>=
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setwd("~/maitrise/GAL-Buckle95/")
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library(actuar)
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library(MASS)
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library(xtable)
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library(parallel)
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library(moments)
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library(FourierStuff)
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library(GeneralizedAsymmetricLaplace)
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library(GMMStuff)
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library(OptionPricingStuff)
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library(QuadraticEstimatingEquations)
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@
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\section{Constantes et données}
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<<>>=
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#Nombre de décimales affichées
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options(digits=6)
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#Marge pour intervalles de confiance
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alpha.confint <- 0.05
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#Marge pour test d'hypothèses
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alpha.test <- 0.05
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#Chargement des données
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RETURNS <- head(read.csv("abbeyn.csv",sep="\t",header=TRUE)[,1],-1)
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#Taille de l'échantillon
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n <- length(RETURNS)
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#Nom de l'échantillon
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strData <- "Buckle95"
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@
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\section{Test de normalité}
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<<>>=
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EppsPulley.test(RETURNS)
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@
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\chapter{Estimation}
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\section{Données mises à l'échelle}
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<<>>=
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sRET <- as.vector(scale(RETURNS))
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@
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\section{Première estimation par QEE}
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<<>>=
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## Point de départ
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pt.depart <- startparamGAL(sRET)
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## Fonctions pour les moments
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meanQEE <- function(param) mGAL(param,1)
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varianceQEE <- function(param) cmGAL(param,2)
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sdQEE <- function(param) sqrt(cmGAL(param,2))
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skewnessQEE <- function(param) cmGAL(param,3)
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kurtosisQEE <- function(param) cmGAL(param,4)
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## Fonctions pour les dérivées
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dmeanQEE <- function(param) dmGAL(param,1)
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dsdQEE <- function(param) dmGAL(param,2)
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## Estimation gaussienne
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optim1 <- optim(pt.depart,obj.gauss,gr=NULL,sRET,
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meanQEE,varianceQEE,dmeanQEE,dsdQEE)
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pt.optim1 <- optim1$par
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## Estimation de crowder
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optim2 <- optim(pt.depart,obj.Crowder,gr=NULL,sRET,
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meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE)
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pt.optim2 <- optim2$par
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## Estimation de crowder modifiée
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optim3 <- optim(pt.depart,obj.Crowder.Mod,gr=NULL,sRET,
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meanQEE,varianceQEE,dmeanQEE,dsdQEE)
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pt.optim3 <- optim3$par
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@
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\section{Résultats de la première estimation par QEE}
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<<>>=
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cov.optim1 <- covariance.QEE(M.gauss(pt.optim1,sRET,
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meanQEE,varianceQEE,dmeanQEE,dsdQEE),
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V.gauss(pt.optim1,sRET,meanQEE,varianceQEE,
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skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n)
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cov.optim2 <- covariance.QEE(M.Crowder(pt.optim2,sRET,
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varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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V.Crowder(pt.optim2,sRET,varianceQEE,
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skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n)
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cov.optim3 <- covariance.QEE(M.Crowder.Mod(pt.optim3,sRET,
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varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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V.Crowder.Mod(pt.optim3,sRET,varianceQEE,dmeanQEE,dsdQEE),n)
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confidence.interval.QEE(pt.optim1,cov.optim1,n)
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confidence.interval.QEE(pt.optim2,cov.optim2,n)
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confidence.interval.QEE(pt.optim3,cov.optim3,n)
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@
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\section{Seconde estimation par QEE}
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<<>>=
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## Estimation gaussienne
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optim4 <- optim(pt.optim1,obj.gauss,gr=NULL,sRET,
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meanQEE,varianceQEE,dmeanQEE,dsdQEE,
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ginv(V.gauss(pt.optim1,sRET,meanQEE,
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varianceQEE,skewnessQEE,kurtosisQEE,
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dmeanQEE,dsdQEE)))
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pt.optim4 <- optim4$par
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## Estimation de crowder
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optim5 <- optim(pt.optim2,obj.Crowder,gr=NULL,sRET,
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meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE,
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ginv(V.Crowder(pt.optim2,sRET,varianceQEE,skewnessQEE,
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kurtosisQEE,dmeanQEE,dsdQEE)))
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pt.optim5 <- optim5$par
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## Estimation de crowder modifiée
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optim6 <- optim(pt.optim3,obj.Crowder.Mod,gr=NULL,sRET,
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meanQEE,varianceQEE,dmeanQEE,dsdQEE,
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ginv(V.Crowder.Mod(pt.optim3,sRET,varianceQEE,
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dmeanQEE,dsdQEE)))
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pt.optim6 <- optim6$par
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@
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\section{Résultats de la seconde estimation par QEE}
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<<>>=
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cov.optim4 <- covariance.QEE(M.gauss(pt.optim4,sRET,
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meanQEE,varianceQEE,dmeanQEE,dsdQEE),
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V.gauss(pt.optim4,sRET,meanQEE,varianceQEE,
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skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n)
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cov.optim5 <- covariance.QEE(M.Crowder(pt.optim5,sRET,
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varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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V.Crowder(pt.optim5,sRET,varianceQEE,skewnessQEE,
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kurtosisQEE,dmeanQEE,dsdQEE),n)
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cov.optim6 <- covariance.QEE(M.Crowder.Mod(pt.optim6,sRET,
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varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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V.Crowder.Mod(pt.optim6,sRET,varianceQEE,dmeanQEE,dsdQEE),n)
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confidence.interval.QEE(pt.optim4,cov.optim4,n)
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confidence.interval.QEE(pt.optim5,cov.optim5,n)
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confidence.interval.QEE(pt.optim6,cov.optim6,n)
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@
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\section{Estimation par GMM}
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<<>>=
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## GMM régulier
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optim7 <- optim.GMM(pt.depart,
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conditions.vector=meanvariance.gmm.vector,
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data=sRET,W=diag(2),
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meanf=meanQEE,variancef=varianceQEE)
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pt.optim7 <- optim7$par
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cov.optim7 <- mean.variance.GMM.gradient.GAL(pt.optim7,sRET) %*%
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covariance.GMM(pt.optim7,meanvariance.gmm.vector,sRET,
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meanf=meanQEE,variancef=varianceQEE) %*%
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t(mean.variance.GMM.gradient.GAL(pt.optim7,sRET)) / n
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## GMM itératif
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optim8 <- iterative.GMM(pt.depart,
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conditions.vector=meanvariance.gmm.vector,
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data=sRET,W=diag(2),
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meanf=meanQEE,variancef=varianceQEE)
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pt.optim8 <- optim8$par
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cov.optim8 <- mean.variance.GMM.gradient.GAL(pt.optim8,sRET) %*%
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optim8$cov %*%
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t(mean.variance.GMM.gradient.GAL(pt.optim8,sRET)) / n
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confidence.interval.QEE(pt.optim7,cov.optim7,n)
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confidence.interval.QEE(pt.optim8,cov.optim8,n)
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@
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\chapter{Comparaison des résultats}
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<<>>=
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# Aggrégation des estimateurs (pour simplifier les calculs)
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pts.estim <- cbind(pt.optim1,pt.optim2,pt.optim3,pt.optim4,
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pt.optim5,pt.optim6,pt.optim7,pt.optim8)
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l.pts.estim <- as.list(data.frame(pts.estim))
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@
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\section{Fonction de répartition}
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<<>>=
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# Points d'évaluation
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xi <- seq(2*min(sRET),2*max(sRET),length.out=2^6)
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# Fonction de répartition par intégration de la fonction caractéristique
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dist1 <- cbind(cftocdf(xi,cfGAL,param=pt.optim1),
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cftocdf(xi,cfGAL,param=pt.optim2),
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cftocdf(xi,cfGAL,param=pt.optim3),
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cftocdf(xi,cfGAL,param=pt.optim4),
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cftocdf(xi,cfGAL,param=pt.optim5),
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cftocdf(xi,cfGAL,param=pt.optim6),
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cftocdf(xi,cfGAL,param=pt.optim7),
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cftocdf(xi,cfGAL,param=pt.optim8))
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# Fonction de répartition par point de selle
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dist2 <- cbind(psaddleapproxGAL(xi,pt.optim1),
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psaddleapproxGAL(xi,pt.optim2),
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psaddleapproxGAL(xi,pt.optim3),
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psaddleapproxGAL(xi,pt.optim4),
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psaddleapproxGAL(xi,pt.optim5),
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psaddleapproxGAL(xi,pt.optim6),
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psaddleapproxGAL(xi,pt.optim7),
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psaddleapproxGAL(xi,pt.optim8))
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# Fonction de répartition par intégration de la fonction de densité
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dist3 <- cbind(pGAL(xi,pt.optim1),
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pGAL(xi,pt.optim2),
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pGAL(xi,pt.optim3),
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pGAL(xi,pt.optim4),
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pGAL(xi,pt.optim5),
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pGAL(xi,pt.optim6),
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pGAL(xi,pt.optim7),
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pGAL(xi,pt.optim8))
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@
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\pagebreak
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\subsection{Graphiques}
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<<results=tex>>=
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for (i in 1:8)
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{
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file<-paste(strData,"-repart-",i,".pdf",sep="")
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pdf(file=file,paper="special",width=6,height=6)
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plot.ecdf(sRET,main=paste("Fonction de répartition ",i))
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lines(xi,dist1[,i],col="green")
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lines(xi,dist2[,1],col="red")
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lines(xi,dist3[,1],col="pink")
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lines(xi,pnorm(xi),type="l",col="blue")
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dev.off()
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cat("\\includegraphics[height=4in,width=4in]{",
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file,"}\n",sep="")
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}
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@
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\subsection{Statistiques}
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Test du $\chi^2$, Méthode avec intégration
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<<results=tex>>=
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chisquare.test1 <- function(param,DATA.hist,FUN,method)
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{
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chisquare.test(DATA.hist,FUN,param,method=method)
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}
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xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),
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cfGAL,"integral")),digits=6)
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@
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Test du $\chi^2$, Méthode avec point de selle
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<<results=tex>>=
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xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),
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pGAL,"saddlepoint")),digits=6)
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@
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Statistique de Kolmogorov-Smirnov
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<<results=tex>>=
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ks.test1 <- function(param,x,y) ks.test(x,y,param)
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xtable(do.call(rbind,mclapply(l.pts.estim,ks.test1,sRET,"pGAL")),digits=6)
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@
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Statistique de distance minimale
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<<results=tex>>=
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tvariate1 <- seq(-.1,.1,by=0.01)
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xtable(do.call(rbind,mclapply(l.pts.estim,
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md.test,sRET,tvariate1,cfGAL,empCF)),digits=6)
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@
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\section{Fonction de densité}
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Intégration de la fonction de densité approximée avec le point de selle, pour la
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normaliser en fonction qui intègre à 1.
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<<>>=
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f_integrale_saddle <- function(param,f,lower,upper)
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integrate(f,lower,upper,param)$value
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norm_int_saddle <- sapply(l.pts.estim,f_integrale_saddle,
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f=dsaddleapproxGAL,lower=-Inf,upper=Inf)
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@
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Séquence de points pour les graphiques
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<<>>=
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x_sRET <- seq(min(sRET)-sd(sRET),max(sRET)+sd(sRET),length.out=50)
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@
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Graphique de la fonction de densité
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<<results=tex>>=
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colors2=c("black","red","green","blue","grey")
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for (i in 1:dim(pts.estim)[2])
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{
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file=paste(strData,"-densite-", i, ".pdf", sep="")
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pdf(file=file, paper="special", width=6, height=6)
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plot(density(sRET),ylim=c(0,.7),type="l",
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main=paste("Densité de",strData, i),xlab=strData,
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ylab="f",lwd=2,lty=1)
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points(x_sRET,
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dGAL(x_sRET,pts.estim[,i]),
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type="b",ylim=c(0,4),col="red",pch=19,lwd=2,lty=2)
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points(x_sRET,
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dsaddleapproxGAL(x_sRET,pts.estim[,i])/norm_int_saddle[i],
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type="b",ylim=c(0,4),col="green",pch=20,lwd=2,lty=3)
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lines(x_sRET,dnorm(x_sRET),type="b",col="blue",
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pch=21,lwd=2,lty=4)
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points(seq(-2,4,length.out=1000)[seq(40,1000,by=40)],
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cftodensity.fft(cfGAL,1000,-2,4,pts.estim[,i])$dens[seq(40,1000,by=40)],
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type="b",col="grey",pch=23,lty=6)
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legend(quantile(sRET,0.9),0.7, c("emp","est.GAL","pt.selle","appx.nrm","fft"),
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cex=0.8, col=colors2, pch=c(NA,19:23), lty=1:6, title="Courbes")
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dev.off()
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cat("\\includegraphics[height=4in, width=4in]{"
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,file, "}\n", sep="")
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}
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@
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\subsection{Tests avec contraintes}
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Test de Wald
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<<>>=
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R <- matrix(c(0,0,1,0,
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0,0,0,1),ncol=4)
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r <- matrix(c(0,0),ncol=1)
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V <- lapply(l.pts.estim,covariance.GMM,meanvariance.gmm.vector,
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sRET,meanQEE,varianceQEE)
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D <- lapply(l.pts.estim,mean.variance.GMM.gradient.GAL,sRET)
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xtable(mapply(Wald.Test,l.pts.estim,n,list(R),list(r),V,D),
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caption="Test de Wald", digits=2)
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@
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\subsection{Vrais paramètres}
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Comme nous avons estimé avec des données centrées et réduites, nous utilisons
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une propriété de la distribution GAL qui nous permet d'obtenir les paramètres
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des rendements non réduits.
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<<>>=
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pts.estim.ns <- apply(pts.estim,2,scaleGAL,type="mu",
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mean(RETURNS),sd(RETURNS))
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@
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<<results=tex>>=
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xtable(pts.estim.ns,
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caption="Paramètres des données non centrées et réduites",
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digits=4)
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@
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\section{Prix d'options}
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\subsection{Données de base}
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<<>>=
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#Taux sans risque
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rfrate <- .05/365
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#Échéance
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T <- 30
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#Pas de discrétisation courbe des prix
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pas <- 0.005
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#Prix initial
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stock0 <- 299
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#Prix d'exercice dans le cours (put)
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strike1 <- stock0*seq(0.98,1,pas)
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#Prix d'exercice hors le cours (put)
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strike2 <- stock0*seq(1+pas,1.02,pas)
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#Prix d'exercice combinés
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strike <- c(strike1,strike2)
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#Damping parameter
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alpha <- 3
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@
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\subsection{Paramètres neutres au risque}
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<<>>=
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pts.estim.ns.rn <- apply(pts.estim.ns,2,riskneutralparGAL,rfrate)
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l.pts.estim.ns.rn <- as.list(data.frame(pts.estim.ns.rn))
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@
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<<results=tex>>=
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xtable(pts.estim.ns.rn,caption="Paramètres neutres au risque",digits=4)
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@
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\subsection{Méthode de Epps}
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<<>>=
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f_putEpps <- function(param,strikeprice,char.fn,eval.time,expiry.time,rate,...)
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putEpps(strikeprice,char.fn,param,eval.time,expiry.time,rate,...)
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prix_put_Epps <- as.data.frame(cbind(strike/stock0,sapply(l.pts.estim.ns.rn,f_putEpps,strike/stock0,cfLM,0,T,rfrate)))
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@
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<<results=tex>>=
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xtable(prix_put_Epps,caption="Prix unitaire de l'option de vente, Méthode de Epps",digits=6)
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@
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\section{Méthode de Carr-Madan}
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<<>>=
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f_callCarrMadan <- function(param,strikeprice,char.fn,eval.time,expiry.time,rate,alpha,...)
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{
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callCarrMadan(strikeprice,char.fn,param,eval.time,expiry.time,rate,alpha,...)
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}
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prix_call_CarrMadan <- as.data.frame(cbind(strike/stock0,sapply(l.pts.estim.ns.rn,f_callCarrMadan,strike/stock0,cfLM,0,T,rfrate,alpha)))
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@
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<<results=tex>>=
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xtable(prix_call_CarrMadan,caption="Prix unitaire de l'option d'achat, Méthode de Carr-Madan",digits=6)
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@
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\end{document}
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