Valuation in Illiquid Markets

What is the value of a financial instrument in an illiquid market? The classical valuation theory which is based on the law of one price assumes implicitly that market participants can trade freely in both directions at the same price. In the absence of perfect liquidity the law of one price should be replaced by a two price theory where the terms of trade depend on the direction of the trade. A static as well as a continuous time theory for two price economies is discussed. The two prices are termed bid and ask or lower and upper price but they should not be confused with the vast literature relating bid-ask spreads to transaction costs or other frictions involved in modeling financial markets. The bid price arises as the infimum of test valuations given by certain market scenarios whereas the ask price is the supremum of such valuations. The two prices correspond to nonlinear expectation operators. Specific dynamic models which are driven by purely discontinuous Lévy processes are considered. This article emerged from papers written jointly with D. Madan, M. Pistorius, W. Schoutens and M. Yor (2014) [9, 10].