 Black-Scholes model

The Black-Scholes model is a frictionless market with a positive bank account $d{B}_t=r{B}_{t}dt$ and a non-dividend-paying stock with constant volatility $\sigma >0$ and initial price $S_0>0$. Under its risk-neutral measure,Thus the conditional stock price has a lognormal distribution. A physical-measure drift $\mu$ may differ from $r$; the pricing dynamics express the martingale condition on discounted traded prices, not a forecast of actual returns.

In the usual natural Brownian filtration, nonzero volatility permits claim replication by trading the stock and bank. For an integrable replicable terminal financial payoff $g(S_T)$, the price is $e^{-r(T-t)}{\mathbb{E}}_Q[g(S_T)\mid {\mathcal{F}}_t]$. The delta hedge holds the stock-price derivative of this value, with the remainder in the bank. The Black-Scholes formula evaluates this expectation for a European call option.^[ref]