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Chapter4.3Code.R
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Chapter4.3Code.R
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# Chapter 4.3 code --------------------------------------------------------
# Last updated: Aug 24, 2017
set.seed(19930225)
numTime = 50 # number of time points
t0 = 0
t = 1
kappa2 = 0.6 # parameter values from Duffie and Garleanu (2001)
sigma2 = 0.14
theta2 = 0.02
lambda0 = 0.01
npath = 1000 # number of sample paths
cir_paths = replicate(npath,
EM_cir(
n = numTime,
kappa = kappa2,
sigma = sigma2,
theta = theta2,
start = lambda0,
t0 = t0,
t = t))
t2 = 2
theta3 = 0.8 # Increase theta to 0.8
cir_paths_paramchange = t(matrix(unlist(lapply(
cir_paths[dim(cir_paths)[1],], EM_cir,
n = numTime+1,
kappa = kappa2,
sigma = sigma2,
theta = theta3,
t0 = t,
t = t2)),
ncol=numTime+2,
byrow=T))
cir_paths2 = rbind(cir_paths, cir_paths_paramchange[2:(numTime+1),])
t3 = 3
cir_paths_paramchange2 = t(matrix(unlist(lapply(
cir_paths2[dim(cir_paths2)[1],], EM_cir,
n = numTime+1,
kappa = kappa2,
sigma = sigma2,
theta = theta2, # Change theta back to 0.02
t0 = t2,
t = t3)),
ncol=numTime+2,
byrow=T))
cir_paths3 = rbind(cir_paths2, cir_paths_paramchange2[2:(numTime+1),])
# plot figure 4.3
yran <- range(cir_paths3,na.rm=T) # plot range
plot(0:(numTime*3),
cir_paths3[,1],
type = "l",
ylim = yran, # plot the first path
xlab = "Time",
ylab = expression(paste(lambda[t], " is a square-root diffusion")),
main = expression(paste(
"Square-Root Diffusion; Parameter ",
bar(theta),
" Changes Value at Time 50 and 100")))
for(k in 2:100) # plot the other paths
lines(0:(numTime*3), cir_paths3[,k], col = adjustcolor("black", alpha.f = 1/(1+100/14)))
mtext(expression(paste(lambda[0], " = 0.01, ",
kappa, " = 0.6, ",
bar(theta)[0-50], " = 0.02, ",
bar(theta)[50-100], " = 0.8, ",
bar(theta)[100-150], " = 0.02, ",
sigma, " = 0.14, npath = 100")),
side = 3)
abline(v=50, col="red")
abline(v=100, col="red")
# end of plotting figure 4.3
# use calculate the default probability using closed-form solution
# first, between time 0 to 50
a1 = 1
timestep = seq(from=t0,to=t,by=(t-t0)/numTime)
C_tT1 = C_tT.fun2(t_i = timestep,
a1 = a1,
kappa = kappa2,
theta = theta2,
sigma = sigma2)
A_tT1 = A_tT.fun2(t_i = timestep,
a1 = a1,
kappa = kappa2,
theta = theta2,
sigma = sigma2)
DefaultProb_01 <- 1-exp(lambda0*C_tT1 + A_tT1)
# second, between time 51 to 100
C_tT2 = C_tT.fun2(t_i = timestep,
a1 = a1,
kappa = kappa2,
theta = theta3,
sigma = sigma2)
A_tT2 = A_tT.fun2(t_i = timestep,
a1 = a1,
kappa = kappa2,
theta = theta3,
sigma = sigma2)
phi = phi.fun(q = -C_tT2,
a1 = a1,
t = 1,
kappa = kappa2,
theta = theta2,
sigma = sigma2)
psi = psi.fun(q = -C_tT2,
a1 = a1,
t = 1,
kappa = kappa2,
theta = theta2,
sigma = sigma2)
DefaultProb_12 <- 1-exp(A_tT2)*exp(lambda0*psi + phi)
DefaultProb_02 = c(DefaultProb_01[1:numTime],DefaultProb_12)
# use calculate the default probability by averaging simulations
Bond_tT_Euler_path2 = exp(apply(
-cir_paths2[1:(numTime*2),]*((t-t0)/numTime),2,cumsum))
Bond_tT_Euler_path_avg2 = apply(Bond_tT_Euler_path2,1,mean,na.rm=T)
DefaultProb_02_icir = 1-Bond_tT_Euler_path_avg2
DefaultProb_02_icir = c(0,DefaultProb_02_icir)
# plot figure 4.4
yran <- range(DefaultProb_02) # plot range
plot(0:(numTime*2),
DefaultProb_02,
type = "l",
ylim = yran, # plot the first path
xlab = "Time",
ylab = expression("Probability of Default"),
main = expression(paste(
"Probability of Default; Square-Root Diffusion Parameter ",
bar(theta),
" Changes Value at Time 50")))
lines(x=0:(numTime*2), y=DefaultProb_02_icir[1:101],col="green")
abline(v=50, col="red")
mtext(expression(paste(lambda[0], " = 0.01, ",
kappa, " = 0.6, ",
bar(theta), " = 0.02, ",
bar(theta)["new"], " = 0.8, ",
sigma, " = 0.14, npath = 1000")),
side = 3)
legend("topleft",inset=0.01,legend=c("Closed-form solution", "Mean of simulations"),
col=c("black","green"), lty=1:1,cex=0.9,box.lty=0, y.intersp=0.8)
# end of plotting figure 4.4