Fast and realistic European ARCH option pricing and hedging

Abstract

The behavior of the implied volatility surface for European options was analysed in detail by Zumbach and Fernandez for prices computed with a new option pricing scheme based on the construction of the risk-neutral measure for realistic processes with a finite time increment. The resulting dynamics of the surface is static in the moneyness direction, and given by a volatility forecast in the time-to-maturity direction. This difference is the basis of a cross-product approximation of the surface. The subsequent speed-up for option pricing is large, allowing the computation of Greeks and the delta replication strategy in simulations with the cost of replication and the replication risk. The corresponding premia are added to the option arbitrage price in order to compute realistic implied volatility surfaces. Finally, the cross-product approximation for realistic prices can be used to analyse European options on the SP500 in depth. The cross-product approximation is used to compute a mean quotient implied volatility, which can be compared with the full theoretical computation. The comparison shows that the cost of hedging and the replication risk premium have contributions to the implied volatility smile that are of similar magnitude to the contribution from the process for the underlying asset.