= Black-Scholes model
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{wiki=Black–Scholes_model}
The <Black-Scholes model> is a frictionless market with a positive <bank account> $dB_t=rB_tdt$ and a non-dividend-paying stock with constant <volatility> $\sigma>0$ and initial price $S_0>0$. Under its <risk-neutral measure>,
$$
dS_t=rS_tdt+\sigma S_tdW_t^Q,\qquad S_T=S_t\exp\!\left((r-\sigma^2/2)(T-t)+\sigma(W_T^Q-W_t^Q)\right).
$$
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>.<The Pricing of Options and Corporate Liabilities by Fischer Black and Myron Scholes>[Model assumptions and option-hedging derivation]{ref}
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