The book delves into the mathematical concepts of peacocks and martingales, emphasizing their significance in finance. It employs diverse techniques, including time inversion and Skorokhod embeddings, to analyze these phenomena, providing a comprehensive exploration of their applications and implications within mathematical finance.
A New Look at Generalized Black-Scholes Formulae
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 ? ?