Risk-neutral compatibility with option prices

A common problem is to choose a “risk neutral” measure in an incomplete market in asset pricing models. We show in this paper that in some circumstances it is possible to choose a unique “equivalent local martingale measure” by completing the market with option prices. We do this by modeling the behavior of the stock price X , together with the behavior of the option prices for a relevant family of options which are (or can theoretically be) effectively traded. In doing so, we need to ensure a kind of ’‘compatibility” between X and the prices of our options, and this poses some significant mathematical difficulties.