Symmetric martingales and symmetric smiles

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:119

Article

Tehranchi, Michael R.

关键词: Ocone martingales;    Symmetric increments;    Implied volatility;   

DOI  :  10.1016/j.spa.2009.07.007

来源: Elsevier

【 摘 要 】

A local martingale X is called arithmetically symmetric if the conditional distribution of X-T - X-t is symmetric given F-t, for all 0 <= t <= T. Letting F-t(T) = F-t V sigma(< X >(T)), the main result of this note is that for a continuous local martingale X the following are equivalent: (1) X is arithmetically symmetric. (2) The conditional distribution of X-T given F-t(T) is N(X-t, < X >(t) - < X >(t)) for all 0 <= t <= T. (3) X is a local martingale for the enlarged filtration (F-t(T))(t >= 0) for each T > 0. The notion of a geometrically symmetric martingale is also defined and characterized as the Doleans-Dade exponential of an arithmetically symmetric local martingale. As an application of these results, we show that a market model of the implied volatility surface that is initially flat and that remains symmetric for all future times must be the Black-Scholes model. (C) 2009 Elsevier BN. All rights reserved.

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