/Fitting a Jump Diffusion.r
Last active Sep 30, 2017
Calibration of financial models
| ########## This code calculates non parametric estimates of the jump-difusion parameters ####### | |
| simulateJump=function(mu_,ss_,lambda_,mu2_,sigma_,TotalTime,delta) | |
| { | |
| Sn=0 | |
| times <- c(0) | |
| while(Sn <= TotalTime) | |
| { | |
| n <- length(times) | |
| u <- runif(1) | |
| expon <- -log(u)/lambda_ | |
| Sn <- times[n]+expon | |
| times <- c(times, Sn) | |
| } | |
| times=times[-length(times)] | |
| t=seq(from=0,to=TotalTime,by=delta) | |
| #the last time is beyond TotalTime, so i delete from the vector times | |
| indicator=seq(from=0,to=0,length=length(t)) # if there are jumps or not between two times | |
| jumpSize=seq(from=0,to=0,length=length(t)) # stores the size of the jump | |
| for(m in 2:length(times)) | |
| { | |
| for(k in 2: length(t)) | |
| { | |
| if( t[k-1]<=times[m] && times[m]<=t[k]) | |
| { | |
| indicator[k]=1 | |
| jumpSize[k]=rnorm(1,mean=mu2_,sd=sigma_) | |
| } | |
| } | |
| } | |
| stock=seq(from=0,to=0,length=length(t)) | |
| stock[1]=10 | |
| for(i in 2:length(t)) | |
| { | |
| stock[i]=stock[i-1]+(mu_)*delta+ss_*sqrt(delta)*rnorm(1)+jumpSize[i]*indicator[i] | |
| } | |
| final=cbind(t,stock,jumpSize,indicator) | |
| return(final) | |
| } | |
| #### define the moment functions | |
| M1 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <-xdifference/(timediff/delta) | |
| sum(kernel.weights*xdiff)/(delta*sum(kernel.weights)) | |
| } | |
| M2 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <- ( xdifference/(timediff/delta) )^2 | |
| sum(kernel.weights*xdiff)/( delta*sum(kernel.weights) ) | |
| } | |
| M4<- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <-( xdifference/(timediff/delta) )^4 | |
| sum(kernel.weights*xdiff)/(delta*sum(kernel.weights)) | |
| } | |
| M6 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <-( xdifference/(timediff/delta) )^6 | |
| sum(kernel.weights*xdiff)/(delta*sum(kernel.weights)) | |
| } | |
| delta=1/(8*252) # stock exchages are open on average for about 8 hours every businessday, and there are 252 business days. | |
| mu_=0.0 # drift | |
| ss_=0.15 # jump diffusion diffusive vol | |
| lambda_=25# jump intensity | |
| mu2_=0 # mean jump size | |
| sigma_=0.03 # jump size volatility | |
| TotalTime=20# years. In this case 20 years | |
| dataMax=simulateJump(mu_=mu_,ss_=ss_,lambda_=lambda_,mu2_=mu2_,sigma_=sigma_,TotalTime=TotalTime,delta=delta) | |
| data=dataMax[,1:2] | |
| Xdiff = diff(data[,2]) | |
| Xcurrent = data[1:length(Xdiff),2] | |
| TimeDiff = diff(data[,1]) | |
| V=cbind(Xdiff,TimeDiff,Xcurrent) | |
| #### define the kernel function, in this case being the Gausian kernel | |
| K <- function(u) | |
| { | |
| exp(-0.5*u^2)/sqrt(2*pi) | |
| } | |
| ############ Calculating the estimates for the parameters ########## | |
| n=200 | |
| X=Xcurrent | |
| H=sort(X) #values at which we are evaluating the institaneous | |
| # The bandwiths for the moments | |
| h1= 3 | |
| h2= 2 | |
| h4= 1 | |
| h6= 0.9 | |
| #first.est <-sapply(H, M1, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h1) | |
| second.est <-sapply(H, M2, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h2) | |
| fourth.est <-sapply(H, M4, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h4) | |
| sixth.est <-sapply(H, M6, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h6) | |
| ############ Calculating the bootstrap estimates ############ | |
| first.boot.est=c() | |
| second.boot.est=c() | |
| forth.boot.est=c() | |
| sixth.boot.est=c() | |
| for(i in 1:n) | |
| { | |
| sampleRows<- sample(1:length(V[,1]), size=length(V[,1]), replace=TRUE) | |
| NewV=V[sampleRows,] | |
| #first.boot.est<- cbind(first.boot.est, sapply(H, M1,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h1)) | |
| second.boot.est<- cbind(second.boot.est, sapply(H, M2,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h2)) | |
| forth.boot.est<- cbind(forth.boot.est, sapply(H, M4,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h4)) | |
| sixth.boot.est<- cbind(sixth.boot.est, sapply(H, M6,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h6)) | |
| NewV=c() | |
| print(paste("Iteration Number :",i)) | |
| } | |
| ####### plots of the estimated parameters ######## | |
| ### plot for drift | |
| plot(H, first.est, type="l", ylim=c(-0.5,0.5), cex.lab=1.5, lwd=2, ylab=expression(mu), xlab="Log share price") | |
| for(i in 1:n) { lines(H,first.boot.est[,i],col="grey") } | |
| ### confidence interval for drift | |
| lower.first=rep(0,length(H)) | |
| upper.first=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.first[k]=quantile(first.boot.est[k,],prob=c(0.1)) | |
| upper.first[k]=quantile(first.boot.est[k,],prob=c(0.9)) | |
| } | |
| lines(H,upper.first,lty="dashed",lwd=2,col="blue") | |
| lines(H,first.est,lty="dashed",lwd=2,col="green") | |
| lines(H,lower.first,lty="dashed",lwd=2,col="red") | |
| legend("topright", legend=c("90% boostrap bound", "Estimate","10% boostrap bound"), | |
| col=c("red", "green","blue"), lty=1:2, cex=0.8, text.font=4) | |
| ###### Estimatong the jump variance, intensity function and sigma ######## | |
| ## estimates of jump variance, intensity function and sigma using the original data ## | |
| var_y=mean(sixth.est/(5*fourth.est)) | |
| sqrt(var_y) # This is the estimate of the jump size vol | |
| lam=fourth.est/(3*var_y^2) # Estimate of jump intensity | |
| sig=(second.est-lam*var_y ) | |
| ## bootstrap estimates of jump variance, intensity function and sigma ## | |
| lamvec=c() | |
| variancevec=c() | |
| sigmavec=c() | |
| for( i in 1:n) | |
| { | |
| var_y.boot=mean(sixth.boot.est[,i]/(5*forth.boot.est[,i])) | |
| lam.boot=forth.boot.est[,i]/(3*(var_y.boot)^2) | |
| sig.boot= (second.boot.est[,i]-lam.boot*var_y.boot ) | |
| variancevec=rbind(variancevec,var_y.boot) | |
| lamvec=cbind(lamvec,lam.boot) | |
| sigmavec=cbind(sigmavec,sig.boot) | |
| } | |
| ##### confidence interval for the jump size standard deviation#### | |
| quantile(sqrt(variancevec),probs=c(0.1,0.9)) | |
| ########### plots of lamda and sigma ########### | |
| ### plot for lambda### | |
| plot(H, lam, type="l",ylim=c(15,35),cex.lab=1.5,lwd=2, ylab=expression(lambda), xlab="Log share price") | |
| for(i in 1:n) { lines(H,lamvec[,i],col="grey") } | |
| ### confidence interval for lambda ### | |
| lower.lambda=rep(0,length(H)) | |
| upper.lambda=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.lambda[k]=quantile(lamvec[k,],prob=c(0.1)) | |
| upper.lambda[k]=quantile(lamvec[k,],prob=c(0.9)) | |
| } | |
| lines(H,upper.lambda,lty="dashed",lwd=2,col="blue") | |
| lines(H,lam,lty="dashed",lwd=2,col="green") | |
| lines(H,lower.lambda,lty="dashed",lwd=2,col="red") | |
| legend("topright", legend=c("90% boostrap bound", "Estimate","10% boostrap bound"), | |
| col=c("blue", "green","red"), lty=1:2, cex=0.8, text.font=4) | |
| ### plot for sigma ### | |
| plot(H, sqrt(sig), type="l",ylim=c(0.1,0.2),lwd=2, cex.lab=1.5, ylab=expression(sigma[jump-diffusion]), xlab="Log share price") | |
| for(i in 1:n) { lines(H,sqrt(sigmavec[,i]),col="grey") } | |
| ### confidence interval for sigma ### | |
| lower.sig=rep(0,length(H)) | |
| upper.sig=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.sig[k]=quantile(sigmavec[k,],prob=c(0.1)) | |
| upper.sig[k]=quantile(sigmavec[k,],prob=c(0.9)) | |
| } | |
| lines(H,sqrt(upper.sig),lty="dashed",lwd=2,col="blue") | |
| lines(H,sqrt(sig),lty="dashed",lwd=2,col="green") | |
| lines(H,sqrt(lower.sig),lty="dashed",lwd=2,col="red") | |
| legend("topright", legend=c("90% boostrap bound", "Estimate","10% boostrap bound"), | |
| col=c("blue", "green","red"), lty=1:2, cex=0.8, text.font=4) |
| ########## This code calculates non parametric estimates of the standard diffusion parameters ####### | |
| simulateGBM=function(mu_,ss_,TotalTime,delta) | |
| { | |
| time=seq(from=0,to=TotalTime,by=delta) | |
| stock=seq(from=0,to=0,length=length(time)) | |
| stock[1]=10 | |
| for(i in 2:length(time)) | |
| { | |
| stock[i]=stock[i-1]+(mu_)*delta+ss_*sqrt(delta)*rnorm(1) | |
| } | |
| final=cbind(time,stock) | |
| return(final) | |
| } | |
| #### define the sample moment functions | |
| M1 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <-xdifference/(timediff/delta) | |
| sum(kernel.weights*xdiff)/(delta*sum(kernel.weights)) | |
| } | |
| M2 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <- ( xdifference/(timediff/delta) )^2 | |
| sum(kernel.weights*xdiff)/( delta*sum(kernel.weights) ) | |
| } | |
| ### generate the data ##### | |
| delta=1/(8*252) # This is hourly data per day for 252 business days | |
| mu_= 0.055 # drift | |
| ss_= 0.305 # diffusive vol | |
| TotalTime= 20 # years. In this case 20 years | |
| data=simulateGBM(mu_=mu_,ss_=ss_,TotalTime=TotalTime,delta=delta) | |
| Xdiff = diff(data[,2]) | |
| Xcurrent = data[1:length(Xdiff),2] | |
| TimeDiff = diff(data[,1]) | |
| V=cbind(Xdiff,TimeDiff,Xcurrent) | |
| #### define the kernel fucntion, in this case being the Gausian kernel | |
| K <- function(u) | |
| { | |
| exp(-0.5*u^2)/sqrt(2*pi) | |
| } | |
| ############ Calculating the estimates for the parameters ########## | |
| n=200 # number of boostrap samples | |
| X=Xcurrent | |
| H=sort(X) #values at which we are evaluating the institaneous | |
| # The bandwiths for the moments | |
| h1 = 3 | |
| h2 = 2 | |
| first.est <-sapply(H, M1, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h1) | |
| second.est <-sapply(H, M2, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h2) | |
| ############ Calculating the bootstrap estimates ############ | |
| first.boot.est=c() | |
| second.boot.est=c() | |
| forth.boot.est=c() | |
| sixth.boot.est=c() | |
| for(i in 1:n) | |
| { | |
| sampleRows<- sample(1:length(V[,1]), size=length(V[,1]), replace=TRUE) | |
| NewV=V[sampleRows,] | |
| first.boot.est<- cbind(first.boot.est, sapply(H, M1,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h1)) | |
| second.boot.est<- cbind(second.boot.est, sapply(H, M2,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h2)) | |
| NewV=c() | |
| print(paste("Iteration Number :",i)) | |
| } | |
| ####### plots of the estimated parameters ######## | |
| ### plot for drift | |
| plot(H, first.est, type="l", ylim=c(-1,1), cex.lab=1.5, lwd=2, ylab=expression(mu), xlab="Log share price") | |
| for(i in 1:n) { lines(H,first.boot.est[,i],col="grey") } | |
| ### confidence interval for drift | |
| lower.first=rep(0,length(H)) | |
| upper.first=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.first[k]=quantile(first.boot.est[k,],prob=c(0.1)) | |
| upper.first[k]=quantile(first.boot.est[k,],prob=c(0.9)) | |
| } | |
| lines(H,upper.first,lty="dashed",lwd=2,col="blue") | |
| lines(H,first.est,lty="dashed",lwd=2,col="green") | |
| lines(H,lower.first,lty="dashed",lwd=2,col="red") | |
| legend("topright", legend=c("90% boostrap bound", "Estimate","10% boostrap bound"), | |
| col=c("red", "green","blue"), lty=1:2, cex=0.8, text.font=4) | |
| ### plot for the diffsuive volatility | |
| plot(H, sqrt(second.est), cex.lab=1.5, type="l",ylim=c(0.28,0.32),lwd=2, ylab=expression(hat(sigma)[standard-diffusion]), xlab="Log share price") | |
| for(i in 1:n) { lines(H,sqrt(second.boot.est[,i]),col="grey") } | |
| ### confidence interval for var | |
| lower.second=rep(0,length(H)) | |
| upper.second=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.second[k]=quantile(sqrt(second.boot.est[k,]),prob=c(0.1)) | |
| upper.second[k]=quantile(sqrt(second.boot.est[k,]),prob=c(0.9)) | |
| } | |
| lines(H,upper.second,lty="dashed",lwd=2,col="blue") | |
| lines(H,sqrt(second.est),lty="dashed",lwd=2,col="green") | |
| lines(H,lower.second,lty="dashed",lwd=2,col="red") | |
| legend("topright", legend=c("90% boostrap bound", "Estimate","10% boostrap bound"), | |
| col=c("red", "green","blue"), lty=1:2, cex=0.8, text.font=4) |
| ########## This code calculates non parametric estimates of the parameters ####### | |
| ### read in data from excell file Data for model fitting | |
| data=read.table("clipboard",header=TRUE) | |
| attach(data) | |
| Xdiff | |
| Xcurrent | |
| TimeDiff | |
| V=cbind(Xdiff,TimeDiff,Xcurrent) | |
| #### define the kernel fucntion, in this case being the Gausian kernel | |
| K <- function(u) | |
| { | |
| exp(-0.5*u^2)/sqrt(2*pi) | |
| } | |
| #### define the moment functions | |
| M1 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <-xdifference/(timediff/delta) | |
| sum(kernel.weights*xdiff)/(delta*sum(kernel.weights)) | |
| } | |
| M2 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <- ( xdifference/(timediff/delta) )^2 | |
| sum(kernel.weights*xdiff)/( delta*sum(kernel.weights) ) | |
| } | |
| M4<- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <-( xdifference/(timediff/delta) )^4 | |
| sum(kernel.weights*xdiff)/(delta*sum(kernel.weights)) | |
| } | |
| M6 <- function(a, xdifference,timediff,xcurrentValue, h) | |
| { | |
| kernel.weights <- K( (xcurrentValue-a)/h ) | |
| xdiff <-( xdifference/(timediff/delta) )^6 | |
| sum(kernel.weights*xdiff)/(delta*sum(kernel.weights)) | |
| } | |
| ############ Calculating the estimates for the parameters ########## | |
| n=1000 | |
| delta=15/(24*60) # delta is 15 minute | |
| X=Xcurrent | |
| H=sort(X) #values at which we are evaluating the institaneous | |
| # The bandwiths for the moments | |
| h1=0.031 | |
| h2=0.022 | |
| h4=0.027 | |
| h6=0.027 | |
| first.est <-sapply(H, M1, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h1) | |
| second.est <-sapply(H, M2, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h2) | |
| fourth.est <-sapply(H, M4, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h4) | |
| sixth.est <-sapply(H, M6, xdifference=V[,1],timediff=V[,2], xcurrentValue=V[,3], h=h6) | |
| ############ Calculating the bootstrap estimates ############ | |
| first.boot.est=c() | |
| second.boot.est=c() | |
| forth.boot.est=c() | |
| sixth.boot.est=c() | |
| for(i in 1:n) | |
| { | |
| sampleRows<- sample(1:length(V[,1]), size=length(V[,1]), replace=TRUE) | |
| NewV=V[sampleRows,] | |
| first.boot.est<- cbind(first.boot.est, sapply(H, M1,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h1)) | |
| second.boot.est<- cbind(second.boot.est, sapply(H, M2,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h2)) | |
| forth.boot.est<- cbind(forth.boot.est, sapply(H, M4,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h4)) | |
| sixth.boot.est<- cbind(sixth.boot.est, sapply(H, M6,xdifference=NewV[,1],timediff=NewV[,2],xcurrentValue=NewV[,3],h=h6)) | |
| NewV=c() | |
| } | |
| ####### plots of the estimated parameters ######## | |
| ### plot for drift | |
| plot(H, first.est, type="l", ylim=c(-0.05,0.05), cex.lab=1.5, lwd=2, ylab=expression(mu), xlab="Log of the JSE Top 40 index") | |
| for(i in 1:n) { lines(H,first.boot.est[,i],col="grey") } | |
| ### confidence interval for drift | |
| lower.first=rep(0,length(H)) | |
| upper.first=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.first[k]=quantile(first.boot.est[k,],prob=c(0.1)) | |
| upper.first[k]=quantile(first.boot.est[k,],prob=c(0.9)) | |
| } | |
| lines(H,lower.first,lty="dashed",lwd=2) | |
| lines(H,upper.first,lty="dashed",lwd=2) | |
| ### plot for second moment | |
| plot(H, sqrt(second.est), cex.lab=1.5, type="l",ylim=c(0.007,0.018),lwd=2, ylab=expression(hat(sigma)[standard-diffusion]), xlab="Log of the JSE Top 40 index") | |
| for(i in 1:n) { lines(H,sqrt(second.boot.est[,i]),col="grey") } | |
| ### confidence interval for drift | |
| lower.second=rep(0,length(H)) | |
| upper.second=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.second[k]=quantile(sqrt(second.boot.est[k,]),prob=c(0.1)) | |
| upper.second[k]=quantile(sqrt(second.boot.est[k,]),prob=c(0.9)) | |
| } | |
| lines(H,lower.second,lty="dashed",lwd=2) | |
| lines(H,upper.second,lty="dashed",lwd=2) | |
| ###### Estimatong the jump variance, intensity function and sigma ######## | |
| ## estimates of jump variance, intensity function and sigma using the original data ## | |
| var_y=mean(sixth.est/(5*fourth.est)) | |
| lam=fourth.est/(3*var_y^2) | |
| sig=(second.est-lam*var_y ) | |
| ## bootstrap estimates of jump variance, intensity function and sigma ## | |
| lamvec=c() | |
| variancevec=c() | |
| sigmavec=c() | |
| for( i in 1:n) | |
| { | |
| var_y.boot=mean(sixth.boot.est[,i]/(5*forth.boot.est[,i])) | |
| lam.boot=forth.boot.est[,i]/(3*(var_y.boot)^2) | |
| sig.boot= (second.boot.est[,i]-lam.boot*var_y.boot ) | |
| variancevec=rbind(variancevec,var_y.boot) | |
| lamvec=cbind(lamvec,lam.boot) | |
| sigmavec=cbind(sigmavec,sig.boot) | |
| } | |
| ##### confidence interval for the jump size standard deviation#### | |
| quantile(sqrt(variancevec),probs=c(0.1,0.9)) | |
| ########### plots of lamda and sigma ########### | |
| ### plot for lambda### | |
| plot(H, lam, type="l",ylim=c(0.00,90),cex.lab=1.5,lwd=2, ylab=expression(lambda), xlab="Log of the JSE Top 40 index") | |
| for(i in 1:20) { lines(H,lamvec[,i],col="grey") } | |
| ### confidence interval for lambda ### | |
| lower.lambda=rep(0,length(H)) | |
| upper.lambda=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.lambda[k]=quantile(lamvec[k,],prob=c(0.1)) | |
| upper.lambda[k]=quantile(lamvec[k,],prob=c(0.9)) | |
| } | |
| lines(H,lower.lambda,lwd=2, lty="dashed") | |
| lines(H,upper.lambda,lwd=2, lty="dashed") | |
| ### plot for sigma ### | |
| plot(H, sig, type="l",ylim=c(0,0.00025),lwd=2, cex.lab=1.5, ylab=expression(sigma[jump-diffusion]), xlab="Log of the JSE Top 40 index") | |
| for(i in 1:n) { lines(H,sigmavec[,i],col="grey") } | |
| ### confidence interval for sigma ### | |
| lower.sig=rep(0,length(H)) | |
| upper.sig=rep(0,length(H)) | |
| for(k in 1:length(H)) | |
| { lower.sig[k]=quantile(sigmavec[k,],prob=c(0.1)) | |
| upper.sig[k]=quantile(sigmavec[k,],prob=c(0.9)) | |
| } | |
| plot(H, sqrt(sig), type="l",ylim=c(0,0.015),lwd=2,cex.lab=1.5, ylab=expression(sigma[jump-diffusion]), xlab="Log of the JSE Top 40 index") | |
| lines(H,sqrt(lower.sig),lwd=2, lty="dashed") | |
| lines(H,sqrt(upper.sig),lwd=2, lty="dashed") | |
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