Option pricing and ARCH processes

► Option pricing with general ARCH processes for the underlying is reviewed. ► Process for the underlying: long memory ARCH with leverage and fat-tail innovations. ► In discrete time, construction of the equivalent martingale measures and option prices. ► Separation of the implied volatility surface into a static mean shape and a dynamical backbone. ► Implied volatility surfaces for many processes, with the cost and risk of replication.