European option valuation under mixed rough Bergomi model equipped with jump factor: a pattern search calibration study

This paper extends the rough Bergomi model to enhance its accuracy in modeling financial market's dynamics. The proposed model incorporates a jump component into the price process to capture sudden market events. Additionally, the volatility process is enriched by combining two fractional Brownian motions with a common Hurst parameter, which allows for a more precise reflection of short-term volatility dependencies. To ensure arbitrage-free pricing, a risk-neutral probability measure is introduced. Under this measure, the martingale property of the price is verified, and the existence of a solution for the model is established. A combined formula for pricing a European call option is subsequently derived, and a pattern search algorithm is proposed for the model's calibration. The results demonstrate that the proposed model is capable of accurately reproducing market prices across both short- and long-term maturities. Furthermore, the model successfully reproduces the observed volatility smile and skew in the market. This capability affirms the model's effectiveness in capturing and understanding the empirical dynamics of financial markets.