Option pricing with realistic ARCH processes

This paper presents a complete computation of option prices based on a realistic process for the underlying and on the construction of a risk-neutral measure as induced by a no-arbitrage replication strategy. The underlying is modelled with a long-memory ARCH process, with relative returns, fat-tailed innovations and multi-scale leverage. The process parameters are estimated on the SP500 stock index (in the physical measure). The change of measure from to the risk-neutral measure is derived rigorously along each path drawn from the process, yielding a Radon--Nikodym derivative for a given choice of a risk aversion function. A small expansion allows to compute explicitly this change of measure. Finally, a given European option's price is obtained as the expectation in of the discounted payoff with a weight given by the change of measure . This procedure is implemented in a Monte Carlo simulation, and allows to compute the option prices, without further adjustable parameters. The computed implied volatility surfaces are compared with empirical surfaces based on European put and call options on the SP500 from 1996 to 2010. Our pricing scheme is able to reproduce the level, the smile, the smirk and the term structure of the surfaces, without any calibration on the observed option prices. We discuss the respective roles of the and measures, the distribution of the terminal prices in both measures, the small impact of the risk aversion and drift premium, and finally we suggest simplifications of our pricing scheme for practical purposes.