Closed-form solutions for option pricing in the presence of volatility smiles: a density-function approach

ABSTRACT A new approach is proposed in this paper, by means of which an equity price or an interest or FX rate is modeled in such a way that its terminal distribution is assumed to have a particular four-parameter functional form that encompasses the lognormal distribution as a special case. For each expiry, the best combination of parameters that gives rise to an optimal (in a sense to be described) match to market call and put prices can be found using a very efficient and rapid procedure. The approach can prove useful in the marking-to-model of out-of-the-money options and in the creation of the smooth strike/expiry smile volatility surface needed as input for all process-driven pricing models. Further desirable features of the method stem from the fact that closed-form solutions are presented, for the first time, not only for call and put prices consistent with this distribution but also for the cumulative distribution arising from the chosen density. Thanks to these analytic solutions, the search procedure needed to calibrate the model to market prices can be rendered extremely fast.