Copulas and Financial Time Series
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I was recently asked to write a survey on copulas for financial time series. The paper is, so far, unfortunately, in French, and is available on https://hal.archives-ouvertes.fr/. There is a description of various models, including some graphs and statistical outputs, obtained from read data.
To illustrate, I’ve been using weekly log-returns of (crude) oil prices, Brent, Dubaï and Maya.
The dataset is available from an excel file, oil.xls (I thought it was possible to load it direclty from the internet, but it did not work… so I suggest to download the file first, and then load it)
> library(xlsx)
> temp <- tempfile()
> download.file(
+ "http://freakonometrics.free.fr/oil.xls",temp)
trying URL 'http://freakonometrics.free.fr/oil.xls'
Content type 'application/vnd.ms-excel' length 99328 bytes (97 KB)
downloaded 97 KB
> oil=read.xlsx(temp,sheetName="DATA",dec=",")
Error in .jcall("RJavaTools", "Ljava/lang/Object;", "invokeMethod", cl, :
java.io.IOException: block[ 0 ] already removed - does your POIFS have circular or duplicate block references?
> oil=read.xlsx("D:\home\acharpen\mes documents\oil.xls",sheetName="DATA")
Then we can plot those three time series
> head(oil)
Date WTI brent Dubai Maya
1 1997-01-10 2.73672 2.25465 3.3673 1.5400
2 1997-01-17 -3.40326 -6.01433 -3.8249 -4.1076
3 1997-01-24 -4.09531 -1.43076 -6.6375 -4.6166
4 1997-01-31 -0.65789 0.34873 0.7326 -1.5122
5 1997-02-07 -3.14293 -1.97765 -0.7326 -1.8798
6 1997-02-14 -5.60321 -7.84534 -7.6372 -11.0549
> Time=as.Date(oil$Date,"%Y-%m-%d")
> plot(Time,oil[,3],type="l",ylab="Brent, weekly log returns",ylim=range(oil[,3:5]))
The idea is to use some multivariate ARMA-GARCH processes here. The heuristics here is that the first part is used to model the dynamics of the average value of the time series, and the second part is used to model the dynamics of the variance of the time series. Two kinds of models are considered in the paper
- a mutivariate GARCH process (or a model on the dynamics of the variance matrix) on the residuals from the ARMA models
- a multivariate model (based on copulas) on the residuals of the ARMA-GARCH process
So different series will be considered here, obtained as residuals of different models. And we can also normalize those residuals. Either from the ARMA models
> fit1=arima(x=dat[,1],order=c(2,0,1)) > fit2=arima(x=dat[,2],order=c(1,0,1)) > fit3=arima(x=dat[,3],order=c(1,0,1)) > dat_arma <- cbind(residuals(fit1),residuals(fit2),residuals(fit3)) > m <- apply(dat_arma, 2, mean) > v <- apply(dat_arma, 2, var) > dat_arma_std <- t((t(dat_arma)-m)/sqrt(v))
or from the ARMA-GARCH models
> fit1=garchFit(formula = ~ arma(2,1)+garch(1, 1),data=dat[,1],cond.dist ="std") > fit2=garchFit(formula = ~ arma(1,1)+garch(1, 1),data=dat[,2],cond.dist ="std") > fit3=garchFit(formula = ~ arma(1,1)+garch(1, 1),data=dat[,3],cond.dist ="std") > m_res <- apply(dat_res, 2, mean) > v_res <- apply(dat_res, 2, var) > dat_res_std =cbind((dat_res[,1]-m_res[1])/sqrt(v_res[1]),(dat_res[,2]-m_res[2])/sqrt(v_res[2]),(dat_res[,3]-m_res[3])/sqrt(v_res[3]))
- Multivariate GARCH Models
A first model that can be consider is a multivariate EWMA for the covariance matrix,
> ewma=EWMAvol(dat_res_std, lambda = 0.96)
To visualize the volatility, use
> emwa_series_vol=function(i=1){
+ plot(Time,ewma$Sigma.t[,1],type="l",
+ col="white",ylim=c(0,80))
+ lines(Time,dat_arma[,i]+40,col="grey")
+ j=1
+ if(i==2) j=5
+ if(i==3) j=9
+ lines(Time,ewma$Sigma.t[,j],lwd=1.5)
+ }
The implied correlation is here
> emwa_series_cor=function(i=1,j=2){
+ if((min(i,j)==1)&(max(i,j)==2)){
+ a=1; b=5; ab=2}
+ if((min(i,j)==1)&(max(i,j)==3)){
+ a=1; b=9; ab=3}
+ if((min(i,j)==2)&(max(i,j)==3)){
+ a=5; b=9; ab=6}
+ r=ewma$Sigma.t[,ab]/sqrt(ewma$Sigma.t[,a]*
+ ewma$Sigma.t[,b])
+ plot(Time,r,type="l",ylim=c(0,1))
+ }
It is also possible possible some multivariate GARCH, namely a BEKK(1,1) model, e.g. using
> library(MTS)
> bekk=BEKK11(dat_arma)
> bekk_series_vol=function(i=1){
+ plot(Time,bk$Sigma.t[,1],type="l",
+ ylab=names(dat)[i],col="white",ylim=c(0,80))
+ lines(Time,dat_arma[,i]+40,col="grey")
+ j=1
+ if(i==2) j=5
+ if(i==3) j=9
+ lines(Time,bk$Sigma.t[,j],lwd=1.5)
+ }
> bekk_series_cor=function(i=1,j=2){
+ if((min(i,j)==1)&(max(i,j)==2)){
+ a=1; b=5; ab=2}
+ if((min(i,j)==1)&(max(i,j)==3)){
+ a=1; b=9; ab=3}
+ if((min(i,j)==2)&(max(i,j)==3)){
+ a=5; b=9; ab=6}
+ r=bk$Sigma.t[,ab]/sqrt(bk$Sigma.t[,a]*
+ bk$Sigma.t[,b])
+ plot(Time,r,type="l",
+ ylim=c(0,1))
+ }
The implied corelation is here
- Modeling Residuals from Univariate GARCH Models
The first step might be to consider some static (joint) distribution for the residuals. The univariate marginal distributions are
while the contour of the joint density (obtained using a bivariate kernel estimator) is
It is also possible to visualise the copula density (with some nonparametric estimator on top, and parametric copula below)
> copula_NP=function(i=1,j=2){
+ uv=dat_garch_std[,c(i,j)]
+ n=nrow(uv)
+ uv=cbind(rank(uv[,1]),rank(uv[,2]))/(n+1)
+ xy=qnorm(uv)
+ s=0.3
+ dc=Vectorize(function(x,y) mean(dnorm(rep(qnorm(x),n),xy[,1],s)*dnorm(rep(qnorm(y),n),xy[,2],s))/
+ dnorm(qnorm(x))/dnorm(qnorm(y)))
+ vx=seq(1/30,29/30,by=1/30)
+ vz=outer(vx,vx,dc)
+ zl=c(0,8)
+ persp(vx,vx,vz,theta=20,phi=10,col="yellow",shade=TRUE,xlab=names(dat)[j],
+ ylab=names(dat)[j],zlab="",ticktype="detailed",zlim=zl)
+
+ library(copula)
+ norm.cop <- normalCopula(0.5)
+ norm.cop <- normalCopula(fitCopula(norm.cop,uv)@estimate)
+ dc=function(x,y) dCopula(cbind(x,y), norm.cop)
+ vz=outer(vx,vx,dc)
+ persp(vx,vx,vz,theta=20,phi=10,col="yellow",shade=TRUE,xlab=names(dat)[j],
+ ylab=names(dat)[j],zlab="copule Gaussienne",ticktype="detailed",zlim=zl)
+
+ t.cop <- tCopula(0.5,df=3)
+ t.fit <- fitCopula(t.cop,uv)@estimate
+ t.cop <- tCopula(t.fit[1],df=t.fit[2])
+ dc=function(x,y) dCopula(cbind(x,y), t.cop)
+ vz=outer(vx,vx,dc)
+ persp(vx,vx,vz,theta=20,phi=10,col="yellow",shade=TRUE,xlab=names(dat)[j],
+ ylab=names(dat)[j],zlab="copule de Student",ticktype="detailed",zlim=zl)
+ }
One can consider the function,
compute its empirical version on the three pairs of series, and compare it with some parametric version,
>
> lambda=function(C){
+ l=function(u) pcopula(C,cbind(u,u))/u
+ u=seq(.001,.5,by=.001)
+ v=Vectorize(l)(u)
+ return(c(v,rev(v)))
+ }
>
> graph_lambda=function(i,j){
+ X=dat_res
+ U=rank(X[,i])/(nrow(X)+1)
+ V=rank(X[,j])/(nrow(X)+1)
+ library(copula)
+ normal.cop <- normalCopula(.5,dim=2)
+ t.cop <- tCopula(.5,dim=2,df=3)
+ gumbel.cop <- gumbelCopula(1.5,dim=2)
+ fit1=fitCopula(normal.cop, cbind(U,V), method="ml")
+ fit2=fitCopula(t.cop, cbind(U,V), method="ml")
+ C1=normalCopula(fit1@copula@parameters,dim=2)
+ C2=tCopula(fit2@copula@parameters[1],dim=2,df=trunc(fit2@copula@parameters[2]))
+
+ Lemp=function(z) sum((U<=z)&(V<=z))/sum(U<=z)
+ Remp=function(z) sum((U>=1-z)&(V>=1-z))/sum(U>=1-z)
+ u=seq(.001,.5,by=.001)
+ L=Vectorize(Lemp)(u)
+ R=Vectorize(Remp)(rev(u))
+ nom=paste("Indice de dépendance de queue ",names(dat)[i]," vs. ",names(dat)[j],sep="")
+ plot(c(u,u+.5-u[1]),c(L,R),type="l",
+ ylim=0:1,lwd=2,
+ xlab="",ylab=nom)
+ lines(c(u,u+.5-u[1]),lambda(C1),lty=2)
+ lines(c(u,u+.5-u[1]),lambda(C2),col=gray.colors(1,.4))
+ }
But one could wonder if the correlation is stable with time.
> time_varying_correl_2=function(i=1,j=2,
+ nom_arg="Pearson"){
+ uv=dat_arma[,c(i,j)]
+ n=nrow(uv)
+ correlation=function(t){
+ sub_uv=uv[t+(-26):26,]
+ cor(sub_uv,method = tolower(nom_arg))[1,2]
+ }
+ ic_correlation=function(t){
+ b=1000
+ sub_uv=uv[t+(-26):26,]
+ cr=rep(NA,b)
+ for(sim in 1:b){
+ sub_uv_b=sub_uv[sample(1:53,size=53,replace=TRUE),]+matrix(rnorm(106),53,2)/10000
+ cr[sim]=cor(sub_uv_b,method = tolower(nom_arg))[1,2]}
+ quantile(cr,c(.05,.95))
+ }
+ nom=paste("Corr?lation de ",nom_arg, " ",names(dat)[i]," vs. ",names(dat)[j],sep="")
+ C=sapply(26:(n-26),correlation)
+ IC_C=sapply(27:(n-27),ic_correlation)
+ plot(Time[26:(n-26)],C,type="l",xlab="Time",ylab=nom,ylim=c(.3,1),col="white")
+ polygon(c(Time[27:(n-27)],rev(Time[27:(n-27)])),c(IC_C[1,],rev(IC_C[2,])),col="light blue",border=NA)
+ lines(Time[26:(n-26)],C)
+ abline(h=cor(uv,method = tolower(nom_arg))[1,2],lty=2,col="red")
+ }
>
> time_varying_correl_2(1,2)
> time_varying_correl_2(1,2, "spearman")
> time_varying_correl_2(1,2,"kendall")
Spearman rank correlation with time
or Kendall’s tau, with time
To model that correlation, consider the DCC model(s)
> library(MTS) > m2=dccFit(dat_res_std) > m3=dccFit(dat_res_std,type="Engle") > R2=m2$rho.t > R3=m3$rho.t
To get some forecasts, use for instance
> library(rmgarch) > garch11.spec = ugarchspec(mean.model = list(armaOrder = c(2,1)),variance.model = list(garchOrder = c(1,1), model = "GARCH"), distribution.model = "norm") > dcc.garch11.spec = dccspec(uspec = multispec( replicate(3, garch11.spec) ), dccOrder = c(1,1), distribution = "mvnorm") > dcc.fit= dccfit(dcc.garch11.spec,data = dat) > fcst=dccforecast(dcc.fit,n.ahead=200) > plot(fcst)
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