Option pricing and ARCH processes

Recent progresses in option pricing using ARCH processes for the underlying are summarized. The stylized facts are multiscale heteroscedasticity, fat-tailed distributions, time reversal asymmetry, and leverage. The process equations are based on a finite time increment, relative returns, fat-tailed innovations, and multiscale ARCH volatility. The European option price is the expected payoff in the physical measure P weighted by the change of measure dQ/dP, and an expansion in the process increment δt allows for numerical evaluations. A cross-product decomposition of the implied volatility surface allows to compute efficiently option prices, Greeks, replication cost, replication risk, and real option prices. The theoretical implied volatility surface and the empirical mean surface for options on the SP500 index are in excellent agreement.