OPTION PRICING UNDER THE LIFTED HESTON MODEL WITH JUMPS

Abstract

This study develops and evaluates the Lifted Heston with Jumps (LHJ) model, a finite-dimensional Markovian option-pricing framework that augments the Lifted Heston approximation of rough volatility with asymmetric double-exponential jumps. The moment generating function of the log price is derived in closed transform form by combining the lifted Riccati system with the Kou jump transform, and option prices are recovered through Laplace inversion. Six nested specifications, namely Heston, Bates, rough Heston, rough Heston with jumps, Lifted Heston and Lifted Heston with jumps, are calibrated in implied-volatility space to S&P 500 index options spanning January 2017 to December 2023, with parameters estimated each Wednesday and evaluated out of sample on the following trading days. Errors are disaggregated jointly by moneyness and maturity and by bull, bear, calm and turbulent market regimes. The results show that the advanced models earn their accuracy on the put surface, where the heavy left tail of the index carries pronounced skew. The proposed model attains the lowest out-of-sample implied-volatility root mean squared error of the six models on both surfaces and holds the put lead across the bull, bear and turbulent regimes, because roughness corrects the at-the-money skew while the asymmetric jump supplies the short-maturity crash tail. It reaches this accuracy in roughly one quarter of the calibration time of the exact rough jump model, so the lifting delivers the accuracy of rough volatility with jumps at close to conventional cost.