Black–Scholes

One formula for a call: C = S_0 (d_1) - Ke^-rT (d_2), from a single Gaussian integral.

The idea

Theorem (Black–Scholes formula).

A European call gives its holder the right, at time $T$, to buy one share for the strike $K$, so it pays $\max(S_{T} - K, 0)$. If the stock is geometric Brownian motion with volatility $\sigma$ and money earns the risk-free rate $r$, that right is worth today $C = S_{0}\,\Phi(d_{1}) - K e^{-rT}\,\Phi(d_{2}), \qquad d_{2} = \frac{\ln(S_{0}/K) + \left(r - \sigma^{2}/2\right)T}{\sigma\sqrt{T}}, \qquad d_{1} = d_{2} + \sigma\sqrt{T},$ where $\Phi$ is the standard normal cumulative distribution function.

The formula takes five inputs — spot, strike, horizon, rate, volatility — and none is a forecast: the stock's expected return does not appear. It is a risk-neutral price, computed under the measure $q$ of Risk-Neutral Pricing, in which the stock drifts at the risk-free rate rather than at its true expected return.

Each term prices one leg of the exchange the call makes at exercise. The holder pays the strike, a fixed $K$ worth $Ke^{-rT}$ today, only on the exercise event, whose risk-neutral probability is $\Phi(d_{2})$. The holder receives the share, and that leg is worth $S_{0}\,\Phi(d_{1})$ — more than $S_{0}\,\Phi(d_{2})$, because on the exercise event the stock is conditionally large, and this conditioning is what separates $d_{1}$ from $d_{2}$.

Ways to work on it

• Walkthrough. Price a call by risk-neutral valuation on a lognormal stock.
• Proof. See why — derive both terms of the formula from the risk-neutral expectation.
• Practice. Compute call and put prices and the probabilities inside the formula.
• Hardest. Price an at-the-money call in closed form and find its high-volatility limit.

Not sure where to start? Take the ten-question placement test.