mu = 0.1
sigma = 0.3
nu = mu - 0.5 * sigma^2

delta_t = 1/12

## Part (a):
##
n_years = 4000
n_samples = n_years * 12

## Our random epsilons:
##
eps = rnorm(n_samples)

## Create the random component in Eq.~11.20:
##
Z = exp(nu * delta_t + sigma * eps * sqrt(delta_t))
S0 = 1.0

ST = cumprod(Z * S0) ## the value of S(t_{k+1})
one_over_log_S = log(ST)/(1:n_samples)

##postscript('../../WriteUp/Graphics/Chapter11/chap_11_ex_10_convergence.eps', onefile=FALSE, horizontal=FALSE)
plot((1:n_samples)/12, one_over_log_S, ylim=c(0, 0.01), type='l', xlab='year')
grid()
##dev.off()