We review the general class of analytically tractable asset-price models that was introduced by Brigo and Mercurio (2001a Mathematical Finance—Bachelier Congr. 2000 (Springer Finance) ed H Geman, D B Madan, S R Pliska and A C F Vorst (Berlin: Springer) pp 151-74), where the considered asset can be an exchange rate, a stock index or even a forward Libor rate. The class is based on an explicit SDE under a given forward measure and includes models featuring (i) explicit asset-price dynamics, (ii) a virtually unlimited number of parameters and (iii) analytical formulae for European options.

We also review the fundamental case where the asset-price density is given, at every time, by a mixture of log-normal densities with equal means. We then introduce two other cases: the first is still based on log-normal densities, but it allows for different means in the distributions; the second is based on processes of hyperbolic-sine type.

Finally, we test the goodness of calibration to real market data of the considered models, choosing a particularly asymmetric volatility surface. As expected, the model based on hyperbolic-sine density mixtures achieves the lowest calibration error.