Stock Index Options Pricing under Jump Patterns Driven by Market States

This article reports that both jump amplitudes and arrival rates are related to the economic states in the DJX and the SPX markets. It then proposes a jump-diffusion process model with modulated frequency and amplitude (JD-MF-MA) to depict these patterns. Using this model, we also derive a closed-form formula for the European index option through the characteristic function pricing approach. The empirical results show that the model with modulated jumps not only captures the characteristics of returns but also improves pricing performance. Overall, the modulated jump should be the default modeling choice for derivatives pricing models.