File:Zins-Vasicek.png
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DescriptionZins-Vasicek.png | a path of a gaussion ornstein uhlenbeck short rate process and the corresponding yield curve at T=0 (purple) and some points in time later. |
Date | |
Source | created by myself with GNU R, see source below |
Author | Thomas Steiner |
Permission (Reusing this file) |
Thomas Steiner put it under the CC-by-SA 2.5. |
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R-source:
## einfachstes bsp, yields in Vasicek, Notation wie Brigo/Mercurio #übernimmt log(z) als Parameter (nichtnegativ) yvas<-function(ttm,para,rt) { lambda=exp(para$lambda) theta=exp(para$theta) sigma2=(exp(para$sigma))^2 B=1/lambda*(1-exp(-lambda*ttm)) lnA=(theta-sigma2/(2*lambda^2))*(B-ttm)-sigma2/(4*lambda)*B^2 return( -(lnA-B*rt)/ttm ) } ### OU-levy processes ################################ ## starting at time t=0 at Z_0=para$r until time t=T in "steps" equidistant steps; mean reversion "mr" ## parameters of the distribution of the bdlp are stored in param process.ou<-function(para,type,T=1,steps=500) { dt=T/(steps-1) ou=array(para$r,dim=steps) if (type=="cauchy") { dLt=rcauchy(steps-1, location=para$location, scale=para$scale*dt) } else if (type=="gamma") { dLt=rgamma(n=steps-1, shape=para$shape, scale=para$scale*dt) } else if (type=="IG") { dLt=rinvGauss(n=steps-1, nu=(para$delta*dt)/para$gamma, lambda=(para$delta*dt)^2) } else if (type=="norm") { dLt=rnorm(n=steps-1, mean=para$lambda*para$theta*dt, sd=para$sigma*dt) } else { print("process.ou:: this type of OU-process is not yet implemented") } for (tm in 1:(steps-1)) { ou[tm+1]=ou[tm]-para$lambda*ou[tm]*dt+dLt[tm] } return (ou) } set.seed(28019180) time=seq(0,14,length=700) pa=list(lambda=2,theta=0.04,sigma=0.2,r=0.08) lpa=list(lambda=log(pa$lambda),theta=log(pa$theta),sigma=log(pa$sigma)) #log-parameter gou=process.ou(pa,"norm",T=14,steps=length(time)) plot(time,gou,type="l",lwd=2,axes=F,xlab="Zeit",ylab="Zins",main="Zinsen im Vasicek-Modell",cex.main=2) abline(h=pa$theta-pa$sigma^2/(2*pa$lambda^2),lty="dotted",col="grey") lines(time,yvas(time,lpa,pa$r),col="purple",lwd=2) yticks=axTicks(2) axis(1) axis(side=2,at=yticks,labels=paste(format(100*yticks,digits=3),"%",sep="")) text(x=10.4,y=0.0325,expression(paste("d",r[t]==2(0.04-r[t]),"dt+",0.2,r[t],"d", W[t])),cex=1.4)
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current | 06:15, 16 October 2011 | 1,003 × 644 (12 KB) | Thire (talk | contribs) | 2 neue kurven | |
13:58, 3 July 2007 | 1,200 × 900 (13 KB) | Thire (talk | contribs) | {{Information |Description = a path of a gaussion ornstein uhlenbeck short rate process and the corresponding yield curve. |Source = created by myself with GNU R, see source below |Date = 3 july 2007 |Author = Thomas Steiner |Permis |
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