Risk-Neutral Pricing & Black–Scholes

Pricing by replication

In the Black–Scholes model the stock follows $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ and cash grows at the risk-free rate $r$. A European option paying $h(S_T)$ can be replicated by continuously trading the stock and cash. If a self-financing portfolio reproduces the payoff exactly, the option must cost the same as the portfolio — otherwise there is an arbitrage.

Holding $\Delta_t = \partial V/\partial S$ shares hedges away the $dW_t$ risk. Requiring the hedged portfolio to earn the risk-free rate yields the Black–Scholes PDE:

$$\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac12\sigma^2S^2\frac{\partial^2 V}{\partial S^2} - rV = 0 .$$

Notice that $\mu$ does not appear. The option price does not depend on the stock's expected return.

Risk-neutral pricing

The PDE solution can be written as a discounted expectation under the risk-neutral measure $\mathbb{Q}$, under which the stock drifts at $r$:

$$V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[h(S_T)], \qquad S_T = S_0\exp\!\Big(\big(r - \tfrac12\sigma^2\big)T + \sigma\sqrt T Z\Big).$$

$\mathbb{Q}$ is a pricing device, not a forecast. It is the unique measure that makes all discounted traded prices martingales.

The Black–Scholes formula

For a call with strike $K$ and maturity $T$:

$$C = S_0\,N(d_1) - K e^{-rT} N(d_2), \qquad d_{1,2} = \frac{\ln(S_0/K) + \left(r \pm \tfrac12\sigma^2\right)T}{\sigma\sqrt T}.$$

• $N(d_2)$ is the risk-neutral probability that the call finishes in the money.
• $N(d_1)$ is the call's delta, $\partial C / \partial S$.
• $Ke^{-rT}N(d_2)$ is the present value of the strike you expect to pay; $S_0N(d_1)$ is the present value of the stock you expect to receive, weighted by the stock measure.

Puts follow from put–call parity, $C - P = S_0 - Ke^{-rT}$.