# 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.