Pricing inflation products with stochastic volatility and stochastic interest rates

We consider a Heston type inflation model in combination with a Hull-White model for nominal and real interest rates, in which all the correlations can be non-zero. Due to the presence of the Heston dynamics our derived inflation model is able to capture the implied volatility skew/smile, which is present in the inflation option market data. We derive an efficient approximate semi-closed pricing formula for two types of inflation dependent options: index and year-on-year inflation options. The derived pricing formulas allow for an efficient calibration of the inflation model. We also illustrate our approach using a real-life pension fund example, where the Heston Hull-White model is used to determine the value of conditional future indexations.