Smiling for the Delayed Volatility Swaps

We present a variance drift‐adjusted version of the Heston model which leads to a significant improvement of the market volatility surface fitting (compared with Heston). The numerical example we performed with recent market data shows a significant reduction of the average absolute calibration error (calibration on 12 dates ranging from September 19 to October 17, 2011 for the FOREX underlying EURUSD). Our model has two additional parameters compared with the Heston model, can be implemented very easily, and was initially introduced for volatility derivative pricing purposes. The main idea behind our model is to take into account some past history of the variance process in its (risk‐neutral) diffusion. Using a change of time method for continuous local martingales, we derive a closed formula for the Brockhaus and Long approximation of the volatility swap price in this model. We also consider dynamic hedging of volatility swaps using a portfolio of variance swaps.