CTM and the Delayed Heston Model: Pricing and Hedging of Variance and Volatility Swaps

Abstract

In this chapter, we apply the CTM for pricing and hedging of variance and volatility swaps for the delayed Heston model. We present a variance drift-adjusted version of the Heston model which leads to a significant improvement of the market volatility surface fitting (compared to the classical Heston model). 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 19 September to 17 October 2011 for the FOREX underlying EURUSD). Our model has two additional parameters compared to the Heston model and can be implemented very easily. It was initially introduced for the purpose of volatility derivative pricing. 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.