VIX option pricing with exponentially decaying jumps

Spikes in the VIX index exhibits subsequent fading feature that may not be fully captured by the usual mean reversion in drift. In our VIX direct model, we incorporate an exponentially decaying jump term in the dynamics of log VIX so that jumps fade away over time and show mean reversion. We derive the analytical formulations of the characteristic function of log VIX and corresponding VIX derivatives. We use a large panel of VIX options data to calibrate our proposed model with exponentially decaying jumps through an iterative two-step procedure and compare it with a benchmark model with Poisson jumps. Our results show that the VIX direct model with exponentially decaying jumps achieves lower pricing errors than the benchmark model, both in-sample and out-of-sample. The inclusion of exponentially decaying jumps in the dynamics of log VIX allows for more frequent jumps of larger magnitude, representing jump behaviors that are closer to the observed spike-like patterns.