Dynamic Concave Performance Measures and Bid and Ask Prices with Stochastic Liquidity

In this article we provide a new definition of concave conditional performance measures. We prove a duality to conditional risk measures. New dynamic performance measures based on dynamic risk measures are established. We are able to derive dynamic bid and ask prices with respect to a stochastic liquidity process by modelling the market as a passive counterparty which accepts at zero cost a certain collection of cash flows. An economical abnormality is detected in a comparable definition of bid and ask prices existing in literature. Using the binomial model, analytic formulas of bid and ask prices of European options are given