  We present an arbitrage free theoretical framework for modeling bid and ask
prices of dividend paying securities in a discrete time setup using theory of
dynamic acceptability indices. In the first part of the paper we develop the
theory of dynamic subscale invariant performance measures, on a general
probability space, and discrete time setup. We prove a representation theorem
of such measures in terms of a family of dynamic convex risk measures, and
provide a representation of dynamic risk measures in terms of g-expectations,
and solutions of BS$\Delta$Es with convex drivers. We study the existence and
uniqueness of the solutions, and derive a comparison theorem for corresponding
BS$\Delta$Es.
  In the second part of the paper we discuss a market model for dividend paying
securities by introducing the pricing operators that are defined in terms of
dynamic acceptability indices, and find various properties of these operators.
Using these pricing operators, we define the bid and ask prices for the
underlying securities and then for derivatives in this market. We show that the
obtained market model is arbitrage free, and we also prove a series of
properties of these prices.
