用R拟合Vasicek模型

Sun, 10 Apr 2011 00:00:00 +0000
Vasicek模型几乎可以说是最常见的利率模型。在金融领域有着很大的用处，可以用R来做Vasicek模型的人显然不够多。
这里给出一个拟合Vasicek模型的代码：
#Vasicek negloglike函数:
VASICEKnegloglike<-function(param,data,times){ n.obs<-length(data)
 dum<-sort.list(times)
 data<-data[dum]
 times<-times[dum]
 delta<-diff(times,1)
 mv<-data[1:(n.obs-1)]*exp(-param[2]*delta)+param[1]*(1-exp(-param[2]*delta))
 variance<-param[3]^2*(1-exp(-2*param[2]*delta))/(2*param[2])
 VASICEKnegloglike<--sum(log(dnorm(data[2:n.obs],mv,sqrt(variance))))/n.obs
}

# VASICEKyield函数
VASICEKyield<-function(r,tau,Pparam,riskpremium=0)
{ b<-Pparam[1]+riskpremium
 a<-Pparam[2]
 sig<-Pparam[3]
 Btau<-(1-exp(-a*tau))/a
 Atau<-((Btau-tau)*(a^2*b-0.5*sig^2)/a^2 - sig^2*Btau^2/(4*a))
 VASICEKyield<-r*Btau/tau-Atau/tau
}

# 数据实例
maturities<-c(0.25,0.5,1,2,5,10)
spliceddata<-matrix(scan("http://www.math.ku.dk/~rolf/teaching/mfe04/USrates.52to04.dat"),byrow=T,ncol=7)
obs<-spliceddata[,2]
N<-length(obs)-1
dt<-1/12
data<-obs[2:(N+1)]
lagdata<-obs[1:N]
dates<-spliceddata[,1]

#Closed form estimators
bhat<-(sum(data*lagdata) - sum(data)*sum(lagdata)/N)/(sum(lagdata*lagdata) - sum(lagdata)*sum(lagdata)/N)
kappahat<--log(bhat)/dt
ahat<-sum(data)/N-bhat*sum(lagdata)/N
thetahat<-ahat/(1-bhat)
s2hat<-sum((data-lagdata*bhat-ahat)^2)/N
sigmahat<-sqrt(2*kappahat*s2hat/(1-bhat^2))
#plot the data (or more accurately, the short rate)
plot(dates[2:(N+1)],data,type='l', xlab="date", ylab="US 3M rate", main="US short rate 1952-2004")
abline(h=thetahat, lty=3)
text(dates[10], thetahat, "thetahat")

tstVAS<-optim(par=c(thetahat,kappahat,sigmahat),
 fn=VASICEKnegloglike,method = "BFGS", data=obs,times=dates)
#analyze average yield curves
meanyield<-2:7
for(i in 1:6) meanyield[i]<-mean(spliceddata[,i+1])
plot(maturities,meanyield,type='l',ylim=c(0.05,0.068),xlim=c(0,14),
 ylab="mean/average/typical yield",
	 main="Typical yield curves w/ US 1952-2004 data")
text(maturities[6],meanyield[6], "Average observed yield curve")
points(maturities,VASICEKyield(meanyield[1],maturities,tstVAS$par,riskpremium=0),type='l',lty=2)
text(maturities[6],VASICEKyield(meanyield[1],maturities[6],tstVAS$par,riskpremium=0)+0.001,
 "Typical Vasicek model curve, I (r0 = avr. short rate, risk premium=0) ")
points(maturities,VASICEKyield(thetahat,maturities,tstVAS$par,riskpremium=0),type='l',lty=2)
text(maturities[6],VASICEKyield(thetahat,maturities[6],tstVAS$par,riskpremium=0)-0.0005,"Typical Vasicek model curve, II (r0 = thetahat, risk premium=0) ")

rpfit<-function(rp){
rpfit<-sum(abs(meanyield-VASICEKyield(meanyield[1],maturities,tstVAS$par,riskpremium=rp)))
}

rp<-optim(par=c(0),fn=rpfit,method = "BFGS")$par

theory<-VASICEKyield(meanyield[1],maturities,tstVAS$par,riskpremium=rp)
points(maturities,theory,type='l',lty=2)

text(maturities[6],theory[6],"Typical Vasicek model curve, III (r0=avr. short rate, risk premium='what fits best') ")

plot(maturities,spliceddata[(N+1),2:7],type='l',ylab="yield",ylim=c(0.01,0.06),xlim=c(0,13), main="On the last day in the sample")

text(maturities[6],spliceddata[(N+1),7], "Observed yield curve")

points(maturities,VASICEKyield(spliceddata[(N+1),3],maturities,tstVAS$par,riskpremium=rp),type='l',lty=2)

text(maturities[6],VASICEKyield(spliceddata[(N+1),3],maturities[6],tstVAS$par,riskpremium=rp),"Vasicek model 'prediction' (w/ all the estimates) ")
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用R拟合Vasicek模型
