The Black-Scholes Model

The Black-Scholes model revolutionized mathematical finance by providing a closed-form formula for pricing European options using stochastic calculus.

Model setup

Assume:

Stock price follows geometric Brownian motion: $dS_t = \mu S_t dt + \sigma S_t dB_t$ .
Risk-free rate $r$ is constant.
No arbitrage, perfect markets, continuous trading.

A European call option pays $(S_T - K)^+$ at maturity $T$ , where $K$ is the strike price.

Black-Scholes formula

Theorem8.1Black-Scholes formula

The fair price of a European call option is:

$C(t, S_t) = S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2),$

where

$d_1 = \frac{\log(S_t/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma\sqrt{T-t},$

and $\Phi$ is the standard normal CDF.

Derivation: Under the risk-neutral measure (via Girsanov), $S_t$ has drift $r$ , and the option price is $e^{-rT}\mathbb{E}_\mathbb{Q}[(S_T - K)^+]$ . Computing this expectation (using the lognormal distribution of $S_T$ ) yields the formula.

ExampleAt-the-money call

For $S_0 = K = 100$ , $r = 0.05$ , $\sigma = 0.2$ , $T = 1$ , the Black-Scholes price is approximately $C \approx 10.45$ . The option has significant value due to volatility.

Greeks

The Greeks measure sensitivities of the option price:

Delta: $\Delta = \frac{\partial C}{\partial S} = \Phi(d_1)$ (hedge ratio).
Gamma: $\Gamma = \frac{\partial^2 C}{\partial S^2}$ (convexity).
Vega: $\frac{\partial C}{\partial \sigma}$ (sensitivity to volatility).
Theta: $\frac{\partial C}{\partial t}$ (time decay).

Delta-hedging (holding $\Delta$ shares) replicates the option payoff, forming the basis of dynamic hedging strategies.

Summary

Black-Scholes uses stochastic calculus (Itô, Girsanov, Feynman-Kac) to derive a closed-form option pricing formula, with applications to hedging, risk management, and derivatives trading.