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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]

Fischer Black developed the Black-Scholes model with Myron Scholes. Their 1973 analysis shows how continuous adjustment of a replicating portfolio can eliminate the local exposure to the underlying random price change, leading to a no-arbitrage equation for the value of a contingent claim.^[ref]

This approach makes the hedge part of the mathematical explanation of the price. It also makes the assumptions visible: ideal continuous adjustment and the specified price dynamics are ingredients of the derivation, so changing the trading rules or dynamics can change the valuation problem. The Black-Scholes equation is therefore a useful starting point for studying both a tractable model and the effects of relaxing its assumptions.^[ref]

Myron Scholes, with Fischer Black, developed the Black-Scholes model and its option pricing equation. Their 1973 paper connects a contingent claim's value to the cost of a dynamically adjusted replicating portfolio, under explicit assumptions about trading and the underlying price process.^[ref]

The mathematical insight is that a carefully chosen hedge can cancel the instantaneous random component of the claim's value. No-arbitrage then constrains the remaining locally riskless return, producing the Black-Scholes equation together with a final payoff condition. The result is a conditional conclusion within a model: it does not require an ability to predict the actual stock-price path.^[ref]

Institutional profile: www.gsb.stanford.edu/faculty-research/faculty/myron-s-scholes