Understanding option prices

Abstract

This paper follows the insights of Black and Scholes (1973 J. Political Economy 81 637–54) and Merton (1973 Bell J. Economics Management Sci. 4 141–83) in contexts where their conclusions cannot be exactly obtained. Specifically, we consider an infinite activity Lévy process with no continuous martingale component driving the stock and address the issue of determining capital requirements permitting sufficient time diversification to the option writer. A new framework for pricing options results. Model prices are shown to be potentially consistent with market observations on observed premia. We note that the advance by Rogers (2000 J. Appl. Probability 37 1173–80) on first passage probabilities for spectrally one-sided processes makes the current analysis tractable.