Dynamic Conic Finance via Backward Stochastic Difference Equations

Source abstract

We present an arbitrage free theoretical framework for modeling bid and ask prices of dividend paying securities in a discrete time setup using the 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 filtered probability space. We prove a representation theorem for such measures in terms of a family of dynamic convex risk measures, and we provide a representation of dynamic risk measures in terms of $g$-expectations, and solutions of backward stochastic difference equations with convex drivers. 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 we study 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.