Article ID Journal Published Year Pages File Type
1156119 Stochastic Processes and their Applications 2009 13 Pages PDF
Abstract

A local martingale XX is called arithmetically symmetric if the conditional distribution of XT−XtXT−Xt is symmetric given FtFt, for all 0≤t≤T0≤t≤T. Letting FtT=Ft∨σ(〈X〉T), the main result of this note is that for a continuous local martingale XX the following are equivalent: (1)XX is arithmetically symmetric.(2)The conditional distribution of XTXT given FtT is N(Xt,〈X〉T−〈X〉t)N(Xt,〈X〉T−〈X〉t) for all 0≤t≤T0≤t≤T.(3)XX is a local martingale for the enlarged filtration (FtT)t≥0 for each T≥0T≥0. The notion of a geometrically symmetric martingale is also defined and characterized as the Doléans–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.

Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
Authors
,