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

Springer Science & Business Media, Jan 26, 2010 - Mathematics - 270 pages
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 ? ?

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### Contents

 Reading the BlackScholes Formula in Terms of First and Last Passage Times 1 Generalized BlackScholes Formulae for Martingales in Terms of Last Passage Times 21 Representation of some particular Azéma supermartingales 65 An Interesting Family of BlackScholes Perpetuities 89 Study of Last Passage Times up to a Finite Horizon 115 Put Option as Joint Distribution Function in Strike and Maturity 143 Existence and Properties of PseudoInverses for Bessel and Related Processes 161
 Existence of PseudoInverses for Diffusions 203 Complements 239 Bessel Functions and Bessel Processes 251 References 259 Further Readings 264 Index 268 Copyright