Symmetric martingales and symmetric smiles

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 ∨ σ ( 〈 X 〉 T ) , the main result of this note is that for a continuous local martingale X the following are equivalent: (1) arithmetically symmetric. nditional distribution of X T given F t T is N ( X t , 〈 X 〉 T − 〈 X 〉 t ) for all 0 ≤ t ≤ T . a local martingale for the enlarged filtration ( F t T ) t ≥ 0 for each T ≥ 0 . otion 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.