Symmetric martingales and symmetric smiles

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.