Option Pricing with Realistic Arch Processes

Abstract

Option pricing should be 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. This paper presents a realistic and complete, "first principles,'' computation of option prices. The underlying is modeled by 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 P measure) and allows to reproduce all empirical statistics, from 1 day to 1 year. For a given risk aversion function, the change of measure from P to the risk-neutral measure Q can be derived rigorously along each path drawn from the process, namely the Radon-Nikodym derivative dQ/dP can be constructed. A small DT expansion allows to compute explicitly the change of measure. Finally, the European option price is obtained as the expectation in P of the discounted payoff with a weight given by the change of measure dQ/dP. This procedure is implemented in a Monte Carlo simulation, and allows to compute the option prices, without further adjustable parameters. The empirical study uses European put and call options on the SP500 from 1996 to 2010. The computed implied volatility surfaces are compared with the empirical surfaces, in particular with respect to the level, smile, smirk and term structure. All the main characteristics of the implied volatility surfaces are correctly reproduced. Further points concern the respective role of the P and Q measures, the distribution of the terminal prices in both measures, the negligible effect of the risk aversion and drift premium, and the empirical validity of the no-arbitrage argument. Finally, simplifications of the present exact scheme are suggested.

Keywords: Option pricing, implied volatility, ARCH process, Student innovations, long memory volatility, SP500 European options

JEL Classification: C22, C15, G12