Clube da Matemática

Black-Scholes Model (1973)

Hypotheses

The price $S_t$ of the underlying asset follows a geometric Brownian motion under the risk-neutral measure $\Q$ :

(1)

Complete market, frictionless: no transaction costs, unlimited short positions, infinite divisibility.
$r$ and $\sigma$ constant along $[0, T]$ .
No dividends (relaxable: replace $r$ with $r - q$ ).
Absence of arbitrage — fundamental principle.

The Black-Scholes PDE

By Itô's lemma applied to $V(S, t)$ and construction of the replicating portfolio (long $\Delta$ shares, short 1 option), risk is eliminated and the portfolio must earn $r$ . Thus:

(2)

with final condition $V(S, T) = \max(S - K, 0)$ for a European call.

Closed-form solution

(3)

where

(4)

and $N(\cdot)$ is the cumulative distribution function of the standard normal:

(5)

Put-call parity

(6)

The Greeks

Partial sensitivities of the option price. Every derivatives desk in the world monitors these numbers.

(7)

(8)

(9)

(10)

Black-Scholes Equation