\documentclass[letter]{report} \usepackage[margin=0.5in]{geometry} \usepackage{Sweave} \usepackage{graphicx} \usepackage[francais]{babel} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{verbatim} \usepackage{float} \usepackage{hyperref} \usepackage{scrtime} \begin{document} \title{GAL Buckle 95} \author{François Pelletier} \maketitle \tableofcontents \chapter{Initialisation} \section{Chargement des paquets} \begin{Schunk} \begin{Sinput} > setwd("~/maitrise/GAL-Buckle95/") > library(actuar) > library(MASS) > library(xtable) > library(parallel) > library(moments) > library(FourierStuff) > library(GeneralizedAsymmetricLaplace) > library(GMMStuff) > library(OptionPricingStuff) > library(QuadraticEstimatingEquations) \end{Sinput} \end{Schunk} \section{Constantes et données} \begin{Schunk} \begin{Sinput} > #Nombre de décimales affichées > options(digits=6) > #Marge pour intervalles de confiance > alpha.confint <- 0.05 > #Marge pour test d'hypothèses > alpha.test <- 0.05 > #Chargement des données > RETURNS <- head(read.csv("abbeyn.csv",sep="\t",header=TRUE)[,1],-1) > #Taille de l'échantillon > n <- length(RETURNS) > #Nom de l'échantillon > strData <- "Buckle95" \end{Sinput} \end{Schunk} \section{Test de normalité} \begin{Schunk} \begin{Sinput} > EppsPulley.test(RETURNS) \end{Sinput} \begin{Soutput} Epps-Pulley Normality test T: 0.626033 T*: 0.635568 p-value: 0.007178 $Tstat [1] 0.626033 $Tmod [1] 0.635568 $Zscore [1] 2.44824 $Pvalue [1] 0.00717788 $Reject [1] TRUE \end{Soutput} \end{Schunk} \chapter{Estimation} \section{Données mises à l'échelle} \begin{Schunk} \begin{Sinput} > sRET <- as.vector(scale(RETURNS)) \end{Sinput} \end{Schunk} \section{Première estimation par QEE} \begin{Schunk} \begin{Sinput} > ## Point de départ > pt.depart <- startparamGAL(sRET) > ## Fonctions pour les moments > meanQEE <- function(param) mGAL(param,1) > varianceQEE <- function(param) cmGAL(param,2) > sdQEE <- function(param) sqrt(cmGAL(param,2)) > skewnessQEE <- function(param) cmGAL(param,3) > kurtosisQEE <- function(param) cmGAL(param,4) > ## Fonctions pour les dérivées > dmeanQEE <- function(param) dmGAL(param,1) > dsdQEE <- function(param) dmGAL(param,2) > ## Estimation gaussienne > optim1 <- optim(pt.depart,obj.gauss,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE) > pt.optim1 <- optim1$par > ## Estimation de crowder > optim2 <- optim(pt.depart,obj.Crowder,gr=NULL,sRET, + meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE) > pt.optim2 <- optim2$par > ## Estimation de crowder modifiée > optim3 <- optim(pt.depart,obj.Crowder.Mod,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE) > pt.optim3 <- optim3$par \end{Sinput} \end{Schunk} \section{Résultats de la première estimation par QEE} \begin{Schunk} \begin{Sinput} > cov.optim1 <- covariance.QEE(M.gauss(pt.optim1,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE), + V.gauss(pt.optim1,sRET,meanQEE,varianceQEE, + skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) > cov.optim2 <- covariance.QEE(M.Crowder(pt.optim2,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder(pt.optim2,sRET,varianceQEE, + skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) > cov.optim3 <- covariance.QEE(M.Crowder.Mod(pt.optim3,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder.Mod(pt.optim3,sRET,varianceQEE,dmeanQEE,dsdQEE),n) > confidence.interval.QEE(pt.optim1,cov.optim1,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.780018 -0.726048 -0.672077 [2,] 0.436002 0.596316 0.756630 [3,] 0.262650 0.359186 0.455722 [4,] 1.994757 2.021370 2.047982 \end{Soutput} \begin{Sinput} > confidence.interval.QEE(pt.optim2,cov.optim2,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.694457 -0.627404 -0.560351 [2,] 0.413764 0.640292 0.866820 [3,] 0.232650 0.334028 0.435405 [4,] 1.839966 1.878296 1.916626 \end{Soutput} \begin{Sinput} > confidence.interval.QEE(pt.optim3,cov.optim3,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.765288 -0.711439 -0.657589 [2,] 0.455485 0.606642 0.757798 [3,] 0.264669 0.362932 0.461195 [4,] 1.932691 1.960299 1.987906 \end{Soutput} \end{Schunk} \section{Seconde estimation par QEE} \begin{Schunk} \begin{Sinput} > ## Estimation gaussienne > optim4 <- optim(pt.optim1,obj.gauss,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE, + ginv(V.gauss(pt.optim1,sRET,meanQEE, + varianceQEE,skewnessQEE,kurtosisQEE, + dmeanQEE,dsdQEE))) > pt.optim4 <- optim4$par > ## Estimation de crowder > optim5 <- optim(pt.optim2,obj.Crowder,gr=NULL,sRET, + meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE, + ginv(V.Crowder(pt.optim2,sRET,varianceQEE,skewnessQEE, + kurtosisQEE,dmeanQEE,dsdQEE))) > pt.optim5 <- optim5$par > ## Estimation de crowder modifiée > optim6 <- optim(pt.optim3,obj.Crowder.Mod,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE, + ginv(V.Crowder.Mod(pt.optim3,sRET,varianceQEE, + dmeanQEE,dsdQEE))) > pt.optim6 <- optim6$par \end{Sinput} \end{Schunk} \section{Résultats de la seconde estimation par QEE} \begin{Schunk} \begin{Sinput} > cov.optim4 <- covariance.QEE(M.gauss(pt.optim4,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE), + V.gauss(pt.optim4,sRET,meanQEE,varianceQEE, + skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) > cov.optim5 <- covariance.QEE(M.Crowder(pt.optim5,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder(pt.optim5,sRET,varianceQEE,skewnessQEE, + kurtosisQEE,dmeanQEE,dsdQEE),n) > cov.optim6 <- covariance.QEE(M.Crowder.Mod(pt.optim6,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder.Mod(pt.optim6,sRET,varianceQEE,dmeanQEE,dsdQEE),n) > confidence.interval.QEE(pt.optim4,cov.optim4,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.779792 -0.725853 -0.671914 [2,] 0.436017 0.596319 0.756622 [3,] 0.262456 0.358969 0.455482 [4,] 1.995452 2.022048 2.048644 \end{Soutput} \begin{Sinput} > confidence.interval.QEE(pt.optim5,cov.optim5,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.692712 -0.625874 -0.559036 [2,] 0.414139 0.640445 0.866750 [3,] 0.231568 0.332845 0.434122 [4,] 1.842116 1.880376 1.918636 \end{Soutput} \begin{Sinput} > confidence.interval.QEE(pt.optim6,cov.optim6,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.766288 -0.712450 -0.658612 [2,] 0.455051 0.606193 0.757334 [3,] 0.264972 0.363196 0.461419 [4,] 1.934050 1.961614 1.989178 \end{Soutput} \end{Schunk} \section{Estimation par GMM} \begin{Schunk} \begin{Sinput} > ## GMM régulier > optim7 <- optim.GMM(pt.depart, + conditions.vector=meanvariance.gmm.vector, + data=sRET,W=diag(2), + meanf=meanQEE,variancef=varianceQEE) > pt.optim7 <- optim7$par > cov.optim7 <- mean.variance.GMM.gradient.GAL(pt.optim7,sRET) %*% + covariance.GMM(pt.optim7,meanvariance.gmm.vector,sRET, + meanf=meanQEE,variancef=varianceQEE) %*% + t(mean.variance.GMM.gradient.GAL(pt.optim7,sRET)) / n > ## GMM itératif > optim8 <- iterative.GMM(pt.depart, + conditions.vector=meanvariance.gmm.vector, + data=sRET,W=diag(2), + meanf=meanQEE,variancef=varianceQEE) > pt.optim8 <- optim8$par > cov.optim8 <- mean.variance.GMM.gradient.GAL(pt.optim8,sRET) %*% + optim8$cov %*% + t(mean.variance.GMM.gradient.GAL(pt.optim8,sRET)) / n > confidence.interval.QEE(pt.optim7,cov.optim7,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.878702 -0.641646 -0.404589 [2,] -0.469225 0.625908 1.721040 [3,] -0.192234 0.326366 0.844965 [4,] 1.696121 1.965995 2.235869 \end{Soutput} \begin{Sinput} > confidence.interval.QEE(pt.optim8,cov.optim8,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER [1,] -0.874031 -0.636980 -0.399929 [2,] -0.473292 0.626346 1.725984 [3,] -0.193600 0.322895 0.839390 [4,] 1.704166 1.972716 2.241265 \end{Soutput} \end{Schunk} \chapter{Comparaison des résultats} \begin{Schunk} \begin{Sinput} > # Aggrégation des estimateurs (pour simplifier les calculs) > pts.estim <- cbind(pt.optim1,pt.optim2,pt.optim3,pt.optim4, + pt.optim5,pt.optim6,pt.optim7,pt.optim8) > l.pts.estim <- as.list(data.frame(pts.estim)) \end{Sinput} \end{Schunk} \section{Fonction de répartition} \begin{Schunk} \begin{Sinput} > # Points d'évaluation > xi <- seq(2*min(sRET),2*max(sRET),length.out=2^6) > # Fonction de répartition par intégration de la fonction caractéristique > dist1 <- cbind(cftocdf(xi,cfGAL,param=pt.optim1), + cftocdf(xi,cfGAL,param=pt.optim2), + cftocdf(xi,cfGAL,param=pt.optim3), + cftocdf(xi,cfGAL,param=pt.optim4), + cftocdf(xi,cfGAL,param=pt.optim5), + cftocdf(xi,cfGAL,param=pt.optim6), + cftocdf(xi,cfGAL,param=pt.optim7), + cftocdf(xi,cfGAL,param=pt.optim8)) > # Fonction de répartition par point de selle > dist2 <- cbind(psaddleapproxGAL(xi,pt.optim1), + psaddleapproxGAL(xi,pt.optim2), + psaddleapproxGAL(xi,pt.optim3), + psaddleapproxGAL(xi,pt.optim4), + psaddleapproxGAL(xi,pt.optim5), + psaddleapproxGAL(xi,pt.optim6), + psaddleapproxGAL(xi,pt.optim7), + psaddleapproxGAL(xi,pt.optim8)) > # Fonction de répartition par intégration de la fonction de densité > dist3 <- cbind(pGAL(xi,pt.optim1), + pGAL(xi,pt.optim2), + pGAL(xi,pt.optim3), + pGAL(xi,pt.optim4), + pGAL(xi,pt.optim5), + pGAL(xi,pt.optim6), + pGAL(xi,pt.optim7), + pGAL(xi,pt.optim8)) \end{Sinput} \end{Schunk} \pagebreak \subsection{Graphiques} \begin{Schunk} \begin{Sinput} > for (i in 1:8) + { + file<-paste(strData,"-repart-",i,".pdf",sep="") + pdf(file=file,paper="special",width=6,height=6) + plot.ecdf(sRET,main=paste("Fonction de répartition ",i)) + lines(xi,dist1[,i],col="green") + lines(xi,dist2[,1],col="red") + lines(xi,dist3[,1],col="pink") + lines(xi,pnorm(xi),type="l",col="blue") + dev.off() + cat("\\includegraphics[height=4in,width=4in]{", + file,"}\n",sep="") + } \end{Sinput} \includegraphics[height=4in,width=4in]{Buckle95-repart-1.pdf} \includegraphics[height=4in,width=4in]{Buckle95-repart-2.pdf} \includegraphics[height=4in,width=4in]{Buckle95-repart-3.pdf} \includegraphics[height=4in,width=4in]{Buckle95-repart-4.pdf} \includegraphics[height=4in,width=4in]{Buckle95-repart-5.pdf} \includegraphics[height=4in,width=4in]{Buckle95-repart-6.pdf} \includegraphics[height=4in,width=4in]{Buckle95-repart-7.pdf} \includegraphics[height=4in,width=4in]{Buckle95-repart-8.pdf}\end{Schunk} \subsection{Statistiques} Test du $\chi^2$, Méthode avec intégration \begin{Schunk} \begin{Sinput} > chisquare.test1 <- function(param,DATA.hist,FUN,method) + { + chisquare.test(DATA.hist,FUN,param,method=method) + } > xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET), + cfGAL,"integral")),digits=6) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrr} \hline & chisquare.stat & df & p.value \\ \hline pt.optim1 & 5.473824 & 6.000000 & 0.484626 \\ pt.optim2 & 5.329673 & 6.000000 & 0.502277 \\ pt.optim3 & 5.388158 & 6.000000 & 0.495076 \\ pt.optim4 & 5.474310 & 6.000000 & 0.484567 \\ pt.optim5 & 5.337004 & 6.000000 & 0.501372 \\ pt.optim6 & 5.390662 & 6.000000 & 0.494769 \\ pt.optim7 & 5.454256 & 6.000000 & 0.487003 \\ pt.optim8 & 5.476963 & 6.000000 & 0.484245 \\ \hline \end{tabular} \end{table}\end{Schunk} Test du $\chi^2$, Méthode avec point de selle \begin{Schunk} \begin{Sinput} > xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET), + pGAL,"saddlepoint")),digits=6) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrr} \hline & chisquare.stat & df & p.value \\ \hline pt.optim1 & 9.293574 & 6.000000 & 0.157728 \\ pt.optim2 & 8.345592 & 6.000000 & 0.213862 \\ pt.optim3 & 9.050625 & 6.000000 & 0.170751 \\ pt.optim4 & 9.292836 & 6.000000 & 0.157767 \\ pt.optim5 & 8.344140 & 6.000000 & 0.213959 \\ pt.optim6 & 9.062381 & 6.000000 & 0.170100 \\ pt.optim7 & 8.616379 & 6.000000 & 0.196330 \\ pt.optim8 & 8.610490 & 6.000000 & 0.196698 \\ \hline \end{tabular} \end{table}\end{Schunk} Statistique de Kolmogorov-Smirnov \begin{Schunk} \begin{Sinput} > ks.test1 <- function(param,x,y) ks.test(x,y,param) > xtable(do.call(rbind,mclapply(l.pts.estim,ks.test1,sRET,"pGAL")),digits=6) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrrrr} \hline & statistic & p.value & alternative & method & data.name \\ \hline pt.optim1 & 0.158220 & 0.171912 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ pt.optim2 & 0.140346 & 0.289345 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ pt.optim3 & 0.156772 & 0.179751 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ pt.optim4 & 0.158159 & 0.172235 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ pt.optim5 & 0.139916 & 0.292753 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ pt.optim6 & 0.156960 & 0.178718 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ pt.optim7 & 0.141230 & 0.282437 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ pt.optim8 & 0.140016 & 0.291954 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ \hline \end{tabular} \end{table}\end{Schunk} Statistique de distance minimale \begin{Schunk} \begin{Sinput} > tvariate1 <- seq(-.1,.1,by=0.01) > xtable(do.call(rbind,mclapply(l.pts.estim, + md.test,sRET,tvariate1,cfGAL,empCF)),digits=6) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrr} \hline & md.stat & df & p.value \\ \hline pt.optim1 & 0.000422 & 21.000000 & 0.000000 \\ pt.optim2 & 0.120174 & 21.000000 & 0.000000 \\ pt.optim3 & 0.001384 & 21.000000 & 0.000000 \\ pt.optim4 & 0.000388 & 21.000000 & 0.000000 \\ pt.optim5 & 0.123295 & 21.000000 & 0.000000 \\ pt.optim6 & 0.001451 & 21.000000 & 0.000000 \\ pt.optim7 & 0.007980 & 21.000000 & 0.000000 \\ pt.optim8 & 0.010416 & 21.000000 & 0.000000 \\ \hline \end{tabular} \end{table}\end{Schunk} \section{Fonction de densité} Intégration de la fonction de densité approximée avec le point de selle, pour la normaliser en fonction qui intègre à 1. \begin{Schunk} \begin{Sinput} > f_integrale_saddle <- function(param,f,lower,upper) + integrate(f,lower,upper,param)$value > norm_int_saddle <- sapply(l.pts.estim,f_integrale_saddle, + f=dsaddleapproxGAL,lower=-Inf,upper=Inf) \end{Sinput} \end{Schunk} Séquence de points pour les graphiques \begin{Schunk} \begin{Sinput} > x_sRET <- seq(min(sRET)-sd(sRET),max(sRET)+sd(sRET),length.out=50) \end{Sinput} \end{Schunk} Graphique de la fonction de densité \begin{Schunk} \begin{Sinput} > colors2=c("black","red","green","blue","grey") > for (i in 1:dim(pts.estim)[2]) + { + file=paste(strData,"-densite-", i, ".pdf", sep="") + pdf(file=file, paper="special", width=6, height=6) + plot(density(sRET),ylim=c(0,.7),type="l", + main=paste("Densité de",strData, i),xlab=strData, + ylab="f",lwd=2,lty=1) + points(x_sRET, + dGAL(x_sRET,pts.estim[,i]), + type="b",ylim=c(0,4),col="red",pch=19,lwd=2,lty=2) + points(x_sRET, + dsaddleapproxGAL(x_sRET,pts.estim[,i])/norm_int_saddle[i], + type="b",ylim=c(0,4),col="green",pch=20,lwd=2,lty=3) + + lines(x_sRET,dnorm(x_sRET),type="b",col="blue", + pch=21,lwd=2,lty=4) + points(seq(-2,4,length.out=1000)[seq(40,1000,by=40)], + cftodensity.fft(cfGAL,1000,-2,4,pts.estim[,i])$dens[seq(40,1000,by=40)], + type="b",col="grey",pch=23,lty=6) + legend(quantile(sRET,0.9),0.7, c("emp","est.GAL","pt.selle","appx.nrm","fft"), + cex=0.8, col=colors2, pch=c(NA,19:23), lty=1:6, title="Courbes") + dev.off() + cat("\\includegraphics[height=4in, width=4in]{" + ,file, "}\n", sep="") + } \end{Sinput} \includegraphics[height=4in, width=4in]{Buckle95-densite-1.pdf} \includegraphics[height=4in, width=4in]{Buckle95-densite-2.pdf} \includegraphics[height=4in, width=4in]{Buckle95-densite-3.pdf} \includegraphics[height=4in, width=4in]{Buckle95-densite-4.pdf} \includegraphics[height=4in, width=4in]{Buckle95-densite-5.pdf} \includegraphics[height=4in, width=4in]{Buckle95-densite-6.pdf} \includegraphics[height=4in, width=4in]{Buckle95-densite-7.pdf} \includegraphics[height=4in, width=4in]{Buckle95-densite-8.pdf}\end{Schunk} \subsection{Tests avec contraintes} Test de Wald \begin{Schunk} \begin{Sinput} > R <- matrix(c(0,0,1,0, + 0,0,0,1),ncol=4) > r <- matrix(c(0,0),ncol=1) > V <- lapply(l.pts.estim,covariance.GMM,meanvariance.gmm.vector, + sRET,meanQEE,varianceQEE) > D <- lapply(l.pts.estim,mean.variance.GMM.gradient.GAL,sRET) > xtable(mapply(Wald.Test,l.pts.estim,n,list(R),list(r),V,D), + caption="Test de Wald", digits=2) \end{Sinput} \begin{Soutput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrrrrrrr} \hline & pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\ \hline wald.stat & 1861.21 & 1796.75 & 1690.26 & 1865.01 & 1814.81 & 1690.62 & 2111.08 & 2175.45 \\ p.value & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ reject & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline \end{tabular} \caption{Test de Wald} \end{table} \end{Soutput} \end{Schunk} \subsection{Vrais paramètres} Comme nous avons estimé avec des données centrées et réduites, nous utilisons une propriété de la distribution GAL qui nous permet d'obtenir les paramètres des rendements non réduits. \begin{Schunk} \begin{Sinput} > pts.estim.ns <- apply(pts.estim,2,scaleGAL,type="mu", + mean(RETURNS),sd(RETURNS)) \end{Sinput} \end{Schunk} \begin{Schunk} \begin{Sinput} > xtable(pts.estim.ns, + caption="Paramètres des données non centrées et réduites", + digits=4) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrrrrrrr} \hline & pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\ \hline 1 & -0.0092 & -0.0080 & -0.0090 & -0.0092 & -0.0079 & -0.0091 & -0.0081 & -0.0081 \\ 2 & 0.0078 & 0.0083 & 0.0079 & 0.0078 & 0.0083 & 0.0079 & 0.0081 & 0.0081 \\ 3 & 0.0033 & 0.0031 & 0.0033 & 0.0033 & 0.0031 & 0.0033 & 0.0030 & 0.0030 \\ 4 & 2.0214 & 1.8783 & 1.9603 & 2.0220 & 1.8804 & 1.9616 & 1.9660 & 1.9727 \\ \hline \end{tabular} \caption{Paramètres des données non centrées et réduites} \end{table}\end{Schunk} \section{Prix d'options} \subsection{Données de base} \begin{Schunk} \begin{Sinput} > #Taux sans risque > rfrate <- .05/365 > #Échéance > T <- 30 > #Pas de discrétisation courbe des prix > pas <- 0.005 > #Prix initial > stock0 <- 299 > #Prix d'exercice dans le cours (put) > strike1 <- stock0*seq(0.98,1,pas) > #Prix d'exercice hors le cours (put) > strike2 <- stock0*seq(1+pas,1.02,pas) > #Prix d'exercice combinés > strike <- c(strike1,strike2) > #Damping parameter > alpha <- 3 \end{Sinput} \end{Schunk} \subsection{Paramètres neutres au risque} \begin{Schunk} \begin{Sinput} > pts.estim.ns.rn <- apply(pts.estim.ns,2,riskneutralparGAL,rfrate) > l.pts.estim.ns.rn <- as.list(data.frame(pts.estim.ns.rn)) \end{Sinput} \end{Schunk} \begin{Schunk} \begin{Sinput} > xtable(pts.estim.ns.rn,caption="Paramètres neutres au risque",digits=4) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrrrrrrr} \hline & pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\ \hline 1 & -0.0066 & -0.0057 & -0.0065 & -0.0066 & -0.0057 & -0.0065 & -0.0058 & -0.0058 \\ 2 & 0.0078 & 0.0083 & 0.0079 & 0.0078 & 0.0083 & 0.0079 & 0.0081 & 0.0081 \\ 3 & 0.0033 & 0.0031 & 0.0033 & 0.0033 & 0.0031 & 0.0033 & 0.0030 & 0.0030 \\ 4 & 2.0214 & 1.8783 & 1.9603 & 2.0220 & 1.8804 & 1.9616 & 1.9660 & 1.9727 \\ \hline \end{tabular} \caption{Paramètres neutres au risque} \end{table}\end{Schunk} \subsection{Méthode de Epps} \begin{Schunk} \begin{Sinput} > f_putEpps <- function(param,strikeprice,char.fn,eval.time,expiry.time,rate,...) + putEpps(strikeprice,char.fn,param,eval.time,expiry.time,rate,...) > prix_put_Epps <- as.data.frame(cbind(strike/stock0,sapply(l.pts.estim.ns.rn,f_putEpps,strike/stock0,cfLM,0,T,rfrate))) \end{Sinput} \end{Schunk} \begin{Schunk} \begin{Sinput} > xtable(prix_put_Epps,caption="Prix unitaire de l'option de vente, Méthode de Epps",digits=6) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrrrrrrrr} \hline & V1 & pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\ \hline 1 & 0.980000 & 0.015415 & 0.015785 & 0.015431 & 0.015417 & 0.015794 & 0.015427 & 0.015792 & 0.015819 \\ 2 & 0.985000 & 0.017360 & 0.017737 & 0.017377 & 0.017362 & 0.017746 & 0.017372 & 0.017742 & 0.017770 \\ 3 & 0.990000 & 0.019456 & 0.019838 & 0.019474 & 0.019458 & 0.019847 & 0.019469 & 0.019843 & 0.019871 \\ 4 & 0.995000 & 0.021705 & 0.022090 & 0.021724 & 0.021707 & 0.022099 & 0.021719 & 0.022093 & 0.022122 \\ 5 & 1.000000 & 0.024107 & 0.024492 & 0.024126 & 0.024109 & 0.024501 & 0.024121 & 0.024495 & 0.024523 \\ 6 & 1.005000 & 0.026661 & 0.027044 & 0.026681 & 0.026663 & 0.027053 & 0.026676 & 0.027046 & 0.027074 \\ 7 & 1.010000 & 0.029365 & 0.029745 & 0.029385 & 0.029367 & 0.029753 & 0.029381 & 0.029745 & 0.029773 \\ 8 & 1.015000 & 0.032217 & 0.032591 & 0.032238 & 0.032219 & 0.032600 & 0.032233 & 0.032591 & 0.032617 \\ 9 & 1.020000 & 0.035214 & 0.035580 & 0.035235 & 0.035216 & 0.035589 & 0.035231 & 0.035579 & 0.035605 \\ \hline \end{tabular} \caption{Prix unitaire de l'option de vente, Méthode de Epps} \end{table}\end{Schunk} \section{Méthode de Carr-Madan} \begin{Schunk} \begin{Sinput} > f_callCarrMadan <- function(param,strikeprice,char.fn,eval.time,expiry.time,rate,alpha,...) + { + callCarrMadan(strikeprice,char.fn,param,eval.time,expiry.time,rate,alpha,...) + } > prix_call_CarrMadan <- as.data.frame(cbind(strike/stock0,sapply(l.pts.estim.ns.rn,f_callCarrMadan,strike/stock0,cfLM,0,T,rfrate,alpha))) \end{Sinput} \end{Schunk} \begin{Schunk} \begin{Sinput} > xtable(prix_call_CarrMadan,caption="Prix unitaire de l'option d'achat, Méthode de Carr-Madan",digits=6) \end{Sinput} % latex table generated in R 3.1.0 by xtable 1.7-3 package % Tue May 27 23:05:26 2014 \begin{table}[ht] \centering \begin{tabular}{rrrrrrrrrr} \hline & V1 & pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\ \hline 1 & 0.980000 & 0.044899 & 0.045144 & 0.044912 & 0.044900 & 0.045150 & 0.044909 & 0.045145 & 0.045163 \\ 2 & 0.985000 & 0.041847 & 0.042096 & 0.041860 & 0.041848 & 0.042102 & 0.041857 & 0.042096 & 0.042114 \\ 3 & 0.990000 & 0.038895 & 0.039148 & 0.038909 & 0.038896 & 0.039153 & 0.038906 & 0.039147 & 0.039165 \\ 4 & 0.995000 & 0.036045 & 0.036300 & 0.036059 & 0.036046 & 0.036306 & 0.036056 & 0.036299 & 0.036317 \\ 5 & 1.000000 & 0.033297 & 0.033555 & 0.033312 & 0.033298 & 0.033560 & 0.033309 & 0.033553 & 0.033571 \\ 6 & 1.005000 & 0.030653 & 0.030912 & 0.030668 & 0.030654 & 0.030918 & 0.030665 & 0.030910 & 0.030928 \\ 7 & 1.010000 & 0.028114 & 0.028375 & 0.028130 & 0.028115 & 0.028381 & 0.028126 & 0.028372 & 0.028390 \\ 8 & 1.015000 & 0.025681 & 0.025943 & 0.025697 & 0.025682 & 0.025949 & 0.025694 & 0.025940 & 0.025958 \\ 9 & 1.020000 & 0.023356 & 0.023618 & 0.023373 & 0.023357 & 0.023624 & 0.023369 & 0.023615 & 0.023633 \\ \hline \end{tabular} \caption{Prix unitaire de l'option d'achat, Méthode de Carr-Madan} \end{table}\end{Schunk} \end{document}