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I am trying to calculate an american option price via the simulation of the early exercise boundary using the method presented in this document: Monte Carlo Method For pricing a put Option.

I have inmplemented the boundary calculation algorithm (p15) but I only find this kind of results:

enter image description here

Which differ greatly from what is expected.

Can you take a look at my code ? (I have used an OptPayoff mtrix so that I won't have to look at the optimal payoff at each step).

r =0.05
sigma= 0.15
T = 1
K = 100
S0 = 100
N = 1000
m = 100

Normal<-matrix(0,N,m+1)

#simulation of normal(0,1) 

for (i in 2:(m+1)){
  Normal[,i]= rnorm(N)
}

#Black-Scholes trajectories

BS <-matrix(100,N,m+1)
for (i in 2:(m+1)){
  BS[,i]= BS[,i-1] * exp((r-sigma*sigma/2)* T/m + sigma * Normal[,i]*sqrt(T/m))
}

# Instantaneous payoff actualised to 0
Payoff <-matrix(K-S0,N,m+1)
for (i in 2:(m+1)){
  Payoff[,i]= (K - BS[,i])*exp ( -i*r*T/m) 
}

for (i in 1:N){
  for (j in 2:(m+1)){
    Payoff[i,j] = max(Payoff[i,j],0)
  }
}


#Optimal payoff after j, actualised to 0
OptPayoff <-matrix(0,N,m)

for (i in 1:N){
  for (j in 1:m){
    variable = Payoff[i,(j+1):m]
    OptPayoff[i,j] = max(variable)
  }
}

theta = matrix(K,m+1)

#Boundary algorithm

for (j  in (m-1):0 ){
  theta[j+1,1]= theta[j+2,1]
  thetahat = theta[j+1,1]
  Phat = 0 
  for (i  in 1:N){
    if (BS[i,j+1] < theta [j+1,1]) { Phat = Phat + Payoff[i,j+1] /N}
    else { Phat = Phat + OptPayoff[i,j+1] / N }
  }
    for (i  in 1:N ){
    if(BS[i,j+1] < theta [j+1,1]) {theta[j+1,1] =BS[i,j+1]
    P= 0 
    for (ii  in 1:N ){
      if (BS[ii,j+1] < theta [j+1,1]) { P = P + Payoff[ii,j+1] /N}
      else { P =  P + OptPayoff[ii,j+1] / N }
    }
    if (P>Phat){ Phat = P 
                 thetahat= theta[j+1,1]}
    }
    }
  theta[j+1,1] = thetahat
}

plot(theta,type='l')
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