Black–Scholes

The idea: replicate, don’t predict

A European call pays $\max(S_T - K, 0)$ at expiry $T$. Black, Scholes and Merton showed that, under geometric Brownian motion and frictionless trading, you can replicate this payoff exactly by continuously holding a changing amount of the stock and cash. If a portfolio replicates the option, it must cost the same as the option, or there is an arbitrage.

The remarkable consequence: the price does not depend on the expected return $\mu$. Hedging removes the stock’s directional risk, so only volatility matters. You can price as if every asset grew at the risk-free rate $r$ — risk-neutral pricing.

The PDE

The hedging argument gives the Black–Scholes partial differential equation for the option value $V(S, t)$:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$

The formula

Solving it for a European call on a non-dividend stock:

$$C = S\,N(d_1) - K e^{-rT} N(d_2)$$ $$d_1 = \frac{\ln(S/K) + (r + \tfrac{\sigma^2}{2})T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$$

where $N$ is the standard normal CDF. The put follows from put–call parity:

$$C - P = S - K e^{-rT}$$

Reading the formula

• $N(d_2)$ is the risk-neutral probability that the call finishes in the money.
• $N(d_1)$ is the call’s delta: the number of shares in the replicating portfolio.
• $K e^{-rT} N(d_2)$ is the present value of paying the strike, weighted by the chance you’ll pay it.
• $\sigma\sqrt{T}$ is the total volatility over the option’s life, the single most important input.

Assumptions, and what breaks

Assumption	Reality
Constant volatility	Volatility moves and differs by strike — the smile
Continuous hedging, no costs	Discrete hedging leaves P&L noise; costs are real
Lognormal prices, no jumps	Gaps and fat tails make OTM options dearer than BS says
Constant rate, no dividends	Easily extended (Merton for dividends, Black-76 for futures)

Despite all this, Black–Scholes is used everywhere — not as a model of truth, but as a quoting convention. Traders quote options in implied volatility, and the formula translates it to and from a price.

Try every input in the Black–Scholes calculator.