Black–Scholes Option Pricing
The textbook formula for European options in an idealised world with continuous hedging and constant volatility.
Black–Scholes assumes the underlying follows geometric Brownian motion, trading is frictionless, and positions can be re-hedged continuously.
Under these assumptions, European option prices can be written in closed form and all standard Greeks (Δ, Γ, Vega, Θ, Rho) are available analytically.
In practice, Black–Scholes is more a language than a belief: markets quote implied volatilities from observed prices rather than believing volatility is constant.
Control-style view: the Black–Scholes hedge is a continuous feedback law (delta hedging) that keeps the portfolio locally insensitive to small price movements.
Black–Scholes price and Greeks with continuous dividend yield. The chart shows price vs spot S; the gap to the intrinsic line is the time value. The shaded region is where the option finishes in the money at expiry.
The orange dashed line is the intrinsic value you would get by exercising now. The blue line is the Black–Scholes price. The red bracket at the current S is the time value: the extra premium for uncertainty and time to expiry.
The shaded band shows where the option ends up in the money at expiry. Under the risk-neutral measure, d₂ encodes both the forward moneyness and volatility; N(d₂) (or N(−d₂) for puts) is the probability of finishing in the money.
Practitioners read this as: price = intrinsic + time value, with delta as the hedge ratio and gamma/vega telling you how fragile that hedge is. Put–call parity is the consistency check tying it all together: if it fails badly in real markets, there is either a data problem or an arbitrage story.