MATLAB Price Barrier Up and Out Call

Question 1

Simulate M trajectories in [t0,t0+T] in N time-steps of size delta T
Number of trajectories and time step for the simulation

Answer 1

M = 4
deltaT = T/N

Question 2

Generate M x N samples from N(0,1)

Answer 2

X = randn (M,N)

Question 3

Generate the matrix of initial stock price in the first column and multipliers in other columns

Answer 3

Y = [S0*ones(M,1)

exp((r-0.5sigma^2)deltaT+sigmasqrt(deltaT)X)]

Question 4

Generate M trajectories with N monitoring times

Answer 4

S = cumprod(Y,2)

Question 5

Check whether the trajectory is below the barrier:
1 if the price did not cross the barrier at some monitoring time;
0 otherwise

Answer 5

indexUnderBarrier = all(S kleiner als B,2)

Question 6

Compute the payoff for each trajectory

value of S at maturity

Answer 6

ST = S(:,end)

Question 7

Compute the payoff for the European Call Option

Answer 7

payoff = max(ST-K,0)

Question 8

Assign payoff equal to zero if the price crossed the barrier

Answer 8

payoff = payoff*indexUnderBarrier

Question 9

MC estimate of the price of the option and standard deviation of this estimate

Answer 9

discountFactor = exp(-r*T)

price_MC_UO = discountFactor*mean(payoff)
stdev_MC_UO = discountFactor*std(payoff)/sqrt(M)