The model

The underlying follows geometric Brownian motion under the risk-neutral measure: it drifts at the risk-free rate less the dividend yield, and its returns are normally distributed with constant volatility \(\sigma\).

$$ dS_t = (r - q)\, S_t\, dt + \sigma\, S_t\, dW_t $$

Because the option can be hedged continuously with the stock and a bank account, its price cannot depend on anyone's risk appetite. Discounting the expected payoff at the risk-free rate gives the price.

The formula

With \(N(\cdot)\) the standard normal distribution function,

$$ C = S e^{-qT} N(d_1) - K e^{-rT} N(d_2), \qquad P = K e^{-rT} N(-d_2) - S e^{-qT} N(-d_1) $$ $$ d_1 = \frac{\ln(S/K) + \left(r - q + \tfrac{1}{2}\sigma^2\right) T}{\sigma \sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T} $$

Read the call formula as two pieces. \(N(d_2)\) is the risk-neutral probability of finishing in the money, so \(K e^{-rT} N(d_2)\) is the present value of what you expect to pay. \(S e^{-qT} N(d_1)\) is the present value of the stock you expect to receive, where \(N(d_1)\) is larger than \(N(d_2)\) because the stock is worth more precisely in the states where the option is exercised.

The partial differential equation

Hold one option and short \(\Delta\) shares. Over an instant the random parts cancel when \(\Delta = \partial V / \partial S\), so the portfolio must earn the risk-free rate. Writing that out with Itô's lemma gives

$$ \frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r - q) S \frac{\partial V}{\partial S} - rV = 0 $$

The formula above is the solution of this equation with the terminal condition \(V(S, T) = \max(S - K, 0)\). The finite-difference method solves the same equation numerically instead.

Put-call parity

A call minus a put with the same strike and expiry pays \(S_T - K\) whatever happens, which is a forward contract. So, with no assumption about the model at all,

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

This is why the calculators check parity: any pricing method that breaks it has a bug.

Assumptions and where they fail

• Constant volatility. Markets price different strikes at different volatilities, the smile. See the implied volatility tool.
• Lognormal returns. Real returns have fat tails and jumps.
• Continuous, costless hedging. In practice hedging is discrete and costs money, so Gamma matters.
• European exercise. American options need a tree, a grid, or an approximation.