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.


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