Black-Scholes model

In the Black-Scholes model with constant interest rate $r$ and volatility $\sigma$, a risky asset satisfiesunder the risk-neutral measure.

The price $V(t,s)$ of a sufficiently regular European claim satisfies

A delta hedge holds ${\Delta }_t={\partial }_{s}V(t,S_t)$ units of the risky asset. In the Black-Scholes model, this choice cancels the claim's Brownian exposure.

The Black-Scholes implied volatility is the volatility parameter that makes the Black-Scholes price equal an observed claim price.