Monte Carlo computation for European call 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 call premium. Let the option maturity be {T} and strike {K} Then, the discretely monitored lookback put premium is given by the formula

\displaystyle P_{m} = e^{-rT}\mathbb{E}[(S_{T} - K)^{+}].

Matlab code is here. Let

\displaystyle S_{0} = 100, r = 0.02, \sigma = 0.2, T = 1.

The exact answer should be {C_{0} = 14.8065} according to the BS formula.

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 = 10000} and {k = 10000}, then the option price is {14.6734} with {95\%} confidence interval {[14.3444, 15.0023]}. The total running time on my desktop PC is {6.57} seconds.

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