Option Prices as Probabilities

A New Look at Generalized Black-Scholes Formulae
1. Auflage. 2010
ISBN: 978-3-642-10394-0
Verlag: Springer

Buch, Englisch, 270 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 446 g

Reihe: Springer Finance

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 ? ?

Zielgruppe

Research

Autoren/Hrsg.

Profeta, Christophe

Yor, Marc

Roynette, Bernard

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

Reading the Black-Scholes Formula in Terms of First and Last Passage Times.- Generalized Black-Scholes Formulae for Martingales, in Terms of Last Passage Times.- Representation of some particular Azéma supermartingales.- An Interesting Family of Black-Scholes Perpetuities.- Study of Last Passage Times up to a Finite Horizon.- Put Option as Joint Distribution Function in Strike and Maturity.- Existence and Properties of Pseudo-Inverses for Bessel and Related Processes.- Existence of Pseudo-Inverses for Diffusions.