General valuation principles for arbitrary payoffs and applications to power options under stochastic volatility

Abstract

For plain vanilla options the martingale pricing formulae are usually expressed in terms of artificial probabilities using different numeraires. Firstly, this allows fur a concise and intuitive representation of the pricing equation. Secondly, it is of major importance for computational purposes especially in complex setups. In this paper we propose a more general technique to derive a quasi-closed form pricing equation for (arbitrary) payoffs using equivalent measures. This yields a general model-independent form of the pricing equation for arbitrary claims in terms of artificial probabilities. To illustrate the method it is applied to several exotic options in the Black Scholes framework as well as in a stochastic volatility model. In particular, quasi-closed form solutions are explicitly derived fur power options and powered options under stochastic volatility.