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Q1

The estimation result is reported below.

Note that the numerical optimization employed is extremely sensitive to the initial value, the initial vector for the parameters is specified referring to “Björk, T. (2004). Arbitrage theory in continuous time. Oxford university press”.

Both of the call prices calculated for Vasicek model and CIR model are close to zero.

R code:

rm(list=ls())

data=read.table("http://stanford.edu/~xing/statfinbook/_BookData/Chap10/bonds_yield_dec2006.txt",header=T)

data=as.matrix(data)

Vasicek=function(para){

theta=para[1]

sigma=para[2]

kp=para[3]

rt=para[4]

T=data[,1]

Rt.obs=data[,2]/100

Bt=(1-exp(-kp*T))/kp

logAt=(theta/kp-sigma^2/2/kp^2)*(Bt-T)-sigma^2/4/kp*Bt^2

Rti=(-logAt+rt*Bt)/T

return(sum((Rti-Rt.obs)^2))

}

CIR=function(para){

theta=para[1]

sigma=para[2]

kp=para[3]

rt=para[4]

T=data[,1]

Rt.obs=data[,2]/100

h=sqrt(kp^2+2*sigma^2)

Bt=2*(exp(T*h)-1)/(2*h+(kp+h)*(exp(T*h)-1))

logAt=2*kp*theta/sigma^2*((kp+h)*T/2+log(2*h)-log(2*h+(kp+h)*(exp(T*h)-1)))

Rti=(-logAt+rt*Bt)/T

return(sum((Rti-Rt.obs)^2))

}

##initial values are set meticulous referring to

##Björk, T. (2004). Arbitrage theory in continuous time. Oxford university press.

obs=data[,2]

N=length(obs)-1

dt=mean(diff(data[,1]))

dataobs=obs[2:(N+1)]

lagdata=obs[1:N]

bhat=(sum(dataobs*lagdata) - sum(dataobs)*sum(lagdata)/N)/(sum(lagdata*lagdata) - sum(lagdata)*sum(lagdata)/N)

kappahat=-log(bhat)/dt

ahat=sum(dataobs)/N-bhat*sum(lagdata)/N

thetahat=ahat/(1-bhat)

s2hat=sum((dataobs-lagdata*bhat-ahat)^2)/N

sigmahat=2*kappahat*s2hat/(1-bhat^2)

VAS=optim(par=c(thetahat,sigmahat,kappahat,0.04),fn=Vasicek,method="L-BFGS-B",lower=c(0,0,0,0))

CIRp=optim(par=VAS$par,fn=CIR,method="L-BFGS-B",lower=c(0,0,0,0))

theta_vas=VAS$par[1]

sigma_vas=VAS$par[2]

kp_vas=VAS$par[3]

r_vas=4.75/100

B25_v=(1-exp(-kp_vas*2.5))/kp_vas

logA25_v=(theta_vas/kp_vas-sigma_vas^2/2/kp_vas^2)*(B25_v-2.5)-sigma_vas^2/4/kp_vas*B25_v^2

p25_v=exp(logA25_v)*exp(-B25_v*r_vas)

B30_v=(1-exp(-kp_vas*3))/kp_vas

logA30_v=(theta_vas/kp_vas-sigma_vas^2/2/kp_vas^2)*(B30_v-3)-sigma_vas^2/4/kp_vas*B30_v^2

p30_v=exp(logA30_v)*exp(-B30_v*r_vas)

sigmap=sigma_vas*sqrt((1-exp(-2*kp_vas*2.5))/2/kp_vas)*(1-exp(-kp_vas*(3-2.5)))/kp_vas

h_v=1/sigmap*log(p30_v/p25_v/0.99)+sigmap/2

Z_v=p30_v*pnorm(h_v)-0.99*p25_v*pnorm(h_v-sigmap)

P_v=Z_v*exp(5/100*2.5*2)*100

theta_cir=CIRp$par[1]

sigma_cir=CIRp$par[2]

kp_cir=CIRp$par[3]

r_cir=4.75/100

h_cir=sqrt(kp_cir^2+2*sigma_cir^2)

B25_c=2*(exp(2.5*h_cir)-1)/(2*h_cir+(kp_cir+h_cir)*(exp(2.5*h_cir)-1))

logA25_c=2*kp_cir*theta_cir/sigma_cir^2*((kp_cir+h_cir)*2.5/2+log(2*h_cir)-log(2*h_cir+(kp_cir+h_cir)*(exp(2.5*h_cir)-1)))

p25_c=exp(logA25_c)*exp(-B25_c*r_cir)

B30_c=2*(exp(3*h_cir)-1)/(2*h_cir+(kp_cir+h_cir)*(exp(3*h_cir)-1))

logA30_c=2*kp_cir*theta_cir/sigma_cir^2*((kp_cir+h_cir)*3/2+log(2*h_cir)-log(2*h_cir+(kp_cir+h_cir)*(exp(3*h_cir)-1)))

p30_c=exp(logA30_c)*exp(-B30_c*r_cir)

logA05_c=2*kp_cir*theta_cir/sigma_cir^2*((kp_cir+h_cir)*0.5/2+log(2*h_cir)-log(2*h_cir+(kp_cir+h_cir)*(exp(0.5*h_cir)-1)))

B05_c=2*(exp(0.5*h_cir)-1)/(2*h_cir+(kp_cir+h_cir)*(exp(0.5*h_cir)-1))

psi=(h_cir+kp_cir)/sigma_cir^2

rho=2*h_cir/(sigma_cir^2*exp(h_cir*2.5)-1)

mu=log(exp(logA05_c)/99)/(B05_c)

Z_c=p30_c*pchisq(2*mu*(rho+psi+B05_c),4*kp_cir*theta_cir/sigma_cir^2,2*rho^2*r_cir*exp(h_cir*2.5)/(rho+psi+B05_c))-99*p25_c*pchisq(2*mu*(rho+psi),4*k

p_cir*theta_cir/sigma_cir^2,2*rho^2*r_cir*exp(h_cir*2.5)/(rho+psi))

P_c=Z_c*exp(5/100*2.5*2)*100

###result

if(1>0){

cat("estimation of Vasicek Model","\n")

cat("theta:",VAS$par[1]*VAS$par[3],"\n")

cat("volatility (sigma):",VAS$par[2],"\n")

cat("reversion speed (kappa):",VAS$par[3],"\n")

cat("short term rate:",VAS$par[4],"\n")

cat("call price:",P_v,"\n")

cat("estimation of Cox, Ingersoll & Ross (CIR) Model","\n")

cat("theta:",CIRp$par[1]*CIRp$par[3],"\n")

cat("volatility (sigma):",CIRp$par[2],"\n")

cat("reversion speed (kappa):",CIRp$par[3],"\n")

cat("short term rate:",CIRp$par[4],"\n")

cat("call price:",P_c,"\n")

}