Option Prices as Probabilities: A New Look at Generalized Black-Scholes Formulae

di Christophe Profeta (Autore)

Bernard Roynette (Autore)

Marc Yor (Autore)

Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, 2010

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 ? ?

• Autore: Christophe Profeta, Bernard Roynette, Marc Yor
• Editore: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
• Collana: Springer Finance Lecture Notes
• Anno: 2010
• Rilegatura: Paperback / softback
• Pagine: 270 p.
• Dimensioni: 235 x 155 mm
• EAN: 9783642103940

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