In this article we introduce a linear–quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply GMM and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear–quadratic volatility process and a Lévy-driven Ornstein–Uhlenbeck process. We compare the quality of fit for several models, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normally distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. Monte Carlo experiments are employed to confirm this.