# R Monte Carlo Simulation Price Path Converging Volatility
#number of simulation runs
nSims = 1000
S = 100
r = 0
q = 0
volatility = 0.25
#drift term
mu = r - q
#every trading day increment
dt = 1 / 365
vol_vec = vector()
for (tDays in 1:365) {
# standard normal distribution random number
z=rnorm(tDays*nSims, mean=0, sd=1)
# generate log-normal return matrix
return_matrix=matrix(exp((mu-0.5*volatility^2) *dt+volatility*sqrt(dt) *z), ncol=nSims)
# return value: price path matrix
path=rbind(matrix(rep(S,nSims),ncol=nSims), S*apply(return_matrix,2,cumprod))
# calculate the volatility of path
vol=mean(apply(path, 2, function(x){365*mean( (diff(log(x),n=1)-0.5*(volatility*dt)^2 )^2 ) } ) )
vol_vec=c(vol_vec, vol)
}
plot(vol_vec, type="l", col="blue")
abline(h=volatility, col="black", pch=22, lty=2)
