Supply-Demand Symmetry

We study what properties a liquidity surface (LS) must possess when the supply and demand of a given market are in equilibrium. To formalize this concept, we first analyze foreign exchange markets, where symmetry between the buy and sell sides just amounts to the invariance of market impact with respect to a change of base currency. We speak in this case of supply-demand symmetry, and call the corresponding LS symmetrical. We then extend the idea to general securities, applying similar change of numeraire arguments. We give necessary and sufficient conditions for characterizing symmetrical LS's and we classify all possible solutions to these conditions, providing some stylized examples. Symmetrical LS's are interpreted as equilibria around which liquidity imbalances fluctuate. We show that market impact functions which are even with respect to order size-usually adopted by market impact models in the literature to represent bid-ask symmetry-are not symmetrical LS's and are always associated to an excess of supply; they are approximations of symmetrical LS's only in the limit of highly liquid regimes (no bid-ask spread, small orders, small market impact).