Smile dynamics III

In two articles published in 2004 and 2005 in Risk, Lorenzo Bergomi assessed the structural limitations of existing models for equity derivatives and introduced a new model based on the direct modelling of the joint dynamics of the spot and the implied variance swap volatilities. Here he presents new work on an extension of this model, which, while remaining Markovian, provides control on the smile of forward variances and can be calibrated to Vix futures and options

To remedy some of the limitations of popular models used for equity derivatives hightlighted in previous work (Bergomi, 2004), we proposed a stochastic volatility model based on a specification of the joint dynamics of the underlying spot and its implied forward variance swap (VS) variances (Bergomi, 2005). The aim of this model was to afford better control of:

- the term structure of the volatility of volatility;

- the forward skew; and

- the correlation between spot and VS volatilities for the pricing of options such as reverse cliquets, accumulators and options on realised variance. In practice, as there were no traded instruments to hedge the volatility-of-volatility risk, the level of the corresponding parameter in our model was set to a conservative value and we concluded our article with the following statement: "It is the hope of the author that the liquidity of options on volatility and variance increases so that we will soon be able to trade the smile of the volatility of volatility!"

Smile dynamics III