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- Author: Cristophe Profeta
- Publisher: Springer
- Year: 2010
- Pages: 270
- ISBN-10: 3642103944
- ISBN-13: 9783642103940
- Format: 15.2 x 22.9 x 1.8 cm, minkšti viršeliai
- Language: English
- SAVE -10% with code: EXTRA
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Description
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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- Author: Cristophe Profeta
- Publisher: Springer
- Year: 2010
- Pages: 270
- ISBN-10: 3642103944
- ISBN-13: 9783642103940
- Format: 15.2 x 22.9 x 1.8 cm, minkšti viršeliai
- Language: English English
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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