Option Prices as Probabilities
A New Look at Generalized Black-Scholes Formulae
Seiten
2010
|
2010
Springer Berlin (Verlag)
978-3-642-10394-0 (ISBN)
Springer Berlin (Verlag)
978-3-642-10394-0 (ISBN)
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 ? ?
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.
| Erscheint lt. Verlag | 12.2.2010 |
|---|---|
| Reihe/Serie | Springer Finance | Springer Finance Lecture Notes |
| Zusatzinfo | XXI, 270 p. 3 illus. |
| Verlagsort | Berlin |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 438 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik |
| Wirtschaft ► Betriebswirtschaft / Management | |
| Schlagworte | 60G44, 60J65, 60H99, 60J60 • Azéma supermartingale • Black-Scholes • Black-Scholes-Formel • Black-Scholes Formulae • Finite Horizon • Last passages times • Martingale • Optionspreisberechnung • Pseudo-inverses • Quantitative Finance |
| ISBN-10 | 3-642-10394-4 / 3642103944 |
| ISBN-13 | 978-3-642-10394-0 / 9783642103940 |
| Zustand | Neuware |
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