A Convenient Way to Characterize Equivalent Martingale Measures in Incomplete Markets

Abstract

It is an empirical fact that the (empirically) relevant models for asset prices often describe markets that are incomplete in terms of their underlying assets, yielding many possible equivalent martingale measures under the no-arbitrage assumption. By using actual derivative prices, i.e., prices as observed in the market, additional information about the empirically relevant equivalent martingale measures might be obtained. In order to be able to process such information easily one needs a convenient way to represent all possible equivalent martingale measures in relation to derivative prices. In this paper we present such a convenient characterization. Conceptually, our characterization is not different from existing characterizations using, for example, Radon–Nikodym derivatives of martingale measures with respect to objective probabilities, but our characterization offers some advantages. The main advantage is that pricing derivatives is split up into two steps. The first step is solving a related complete markets pricing problem. This is a well-studied problem, so that it can easily be solved generally. In the second step a weighted average of the first step complete markets’ price must be calculated. Pricing under different equivalent martingale measures in the original market only differs with respect to the second step. The empirically relevant weighting can be determined by confronting the theoretical with the actually observed prices. As a byproduct we obtain a new and natural definition of idiosyncratic risk, which we show to be in line with the use of this term in the literature.