# Monte Carlo Computation for path-dependent option price

Given a risk-neutral stock price in the form of geometric brownian motion (GBM),

$\displaystyle d S_t = S_t (r dt + \sigma d W_t),$

we will demonstrate the computation of the discretely monitored lookback put premium. Let the option maturity be ${T}$, and the discretely monitored dates be ${m+1}$. Then, the discretely monitored lookback put premium is given by the formula

$\displaystyle P_{m} = e^{-rT}\mathbb{E}[\max_{0\le i \le m} S(i T/m) - S(T)].$

We would compute option price computed by Page 70 of the paper [BGK99],

$\displaystyle S_{0} = 100, r = 0.1, \sigma = 0.3, T = 0.2, m = 4.$

The exact answer should be ${P_{4} = 6.574365}$ according to the paper above.

Matlab code is here. The method is simple:

step1. Use the standard Euler method with ${n}$ intervals in ${[0, T]}$ to simulate the path of ${S_{t}}$, compute discounted payoff of the sample path;

step 2. Repeat the path simulation of step 1 ${k}$ times, and take the average.

The result shows that if ${n = 4000}$ and ${k = 10000}$, then the option price is ${P_{4} = 6.4670}$ with ${95\%}$ confidence interval ${[6.3327, 6.6013]}$. The total running time on my desktop PC is ${3.6094}$ seconds.