Dynamic Conic Finance via Backward Stochastic Difference Equations

TL;DR

This paper develops a comprehensive discrete-time framework for modeling bid and ask prices of dividend-paying securities using dynamic acceptability indices, convex risk measures, and backward stochastic difference equations, ensuring arbitrage-free pricing.

Contribution

It introduces a novel arbitrage-free market model based on dynamic acceptability indices and BSΔEs, extending the theory of dynamic risk measures to discrete-time securities pricing.

Findings

Market model is arbitrage free.

Bid and ask prices are characterized by dynamic acceptability indices.

Properties of pricing operators are established.

Abstract

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.