Discovered in the seventies Black-Scholes formula continues to play a central role in
Mathematical Finance. We recall this formula. Let (B t? 0 F t? 0 P) - t t note a standard
Brownian motion with B = 0 (F t? 0) being its natural ?ltra- 0 t t tion. Let E := exp B? t?
0 denote the exponential martingale associated t t 2 to (B t? 0). This martingale also called
geometric Brownian motion is a model t to describe the evolution of prices of a risky asset.
Let for every K? 0: + ? (t) :=E (K?E ) (0.1) K t and + C (t) :=E (E?K) (0.2) K t denote
respectively the price of a European put resp. of a European call associated with this
martingale. Let N be the cumulative distribution function of a reduced Gaussian variable: x 2 y
1 ? 2 ? N (x) := e dy. (0.3) 2? ?? The celebrated Black-Scholes formula gives an explicit
expression of? (t) and K C (t) in terms ofN : K ? ? log(K) t log(K) t ? (t)= KN ? + ?N ? ?
(0.4) K t 2 t 2 and ? ?
