Smile at the Uncertainty

The success of the Black-Scholes (BS) formula is mainly due to the possibility of synthesizing option prices through a unique parameter, the implied volatility, which is so crucial for traders to be directly quoted in many financial markets. This is because the BS formula allows one to immediately convert a volatility into the price at which the related option can be exchanged. The BS model, however, can not be used to price simultaneously all options in a given market. In fact, the assumption of a deterministic underlying-asset volatility leads to constant implied volatilities for any fixed maturity, in contrast with the smile/skew effect commonly observed in practice. Moreover, historical analysis shows that volatilities are indeed stochastic. Stochastic volatility models, therefore, seem to be a more realistic choice when modelling asset price dynamics for valuing derivative securities. However, only few examples, see Hull and White (1987) or Heston (1993), retain enough analytical tractability so as to be relevant in practice. In general, the calibration to market option prices, and the consequent book re-evaluation, can be extremely burdensome and time consuming. Stochastic volatility models can also be problematic as far as hedging is concerned: hedging volatility changes is less straightforward than in the BS case where we just have one volatility parameter. In general, we can only calculate the sensitivity with respect to the model parameters, which may have an economic meaning but are likely not to have a clear impact on implied volatility surfaces. The purpose of this paper is to propose a stochastic-volatility model that is analytically tractable as much as BS’ and for which Vega hedging can be defined in a natural way. The model is based on an uncertain volatility whose random value is drawn, on an infinitesimal future time, from a finite distribution. The model is similar in spirit to (and developed independently from) that of Alexander et al. (2003).