The model for stock price evolution is d St = µStdt + σ StdWt, and a riskless bond, B, grows at a continuously compounding rate r. The Black–Scholes pricing theory then tells us that the price of a vanilla option, with expiry T and pay-off f , is equal to e−rTE( f (ST )), where the expectation is taken under the associated risk-neutral process, d St = r Stdt + σ StdWt .
We solve equation (1.2) by passing to the log and using Ito’s lemma; we computed log St = r − 12 σ2dt + σdWt. As this process is constant-coefficient, it has the solution log St = log S0 + r − 12σt + σ Wt .
A simple Monte Carlo model Since Wt is a Brownian motion, WT is distributed as a Gaussian with mean zero and variance T , so we can write WT = √T N(0, 1), (1.5) and hence log ST = log S0 + r − 12σ2T + σ√T N(0, 1), or equivalently, ST = S0e(r−12 σ2)T+σ√T N(0,1).
The objective of our Monte Carlo simulation is to approximate this expectation by using the law of large numbers, which tells us that if Yj are a sequence of identically distributed independent random variables. So the algorithm to price a call option by Monte Carlo is clear. We draw a random variable, x, from an N(0, 1) distribution.