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

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