Convex duality in optimal investment and contingent claim valuation in illiquid markets

TL;DR

This paper develops a convex duality framework for optimal investment and contingent claim valuation in illiquid markets with nonlinear costs and constraints, generalizing classical models and deriving optimality conditions.

Contribution

It introduces a general convex duality approach that decomposes into risk preferences, trading costs, and constraints, extending classical pricing formulas to illiquid markets.

Findings

Dual expressions decompose into risk, costs, and constraints

Valid dual representations under generalized no-arbitrage and asymptotic elasticity conditions

Derivation of optimality conditions using extended shadow prices

Abstract

This paper studies convex duality in optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into tree terms, corresponding to the agent's risk preferences, trading costs and portfolio constraints, respectively. The dual representations are shown to be valid when the market model satisfies an appropriate generalization of the no-arbitrage condition and the agent's utility function satisfies an appropriate generalization of asymptotic elasticity conditions. When applied to classical liquid market models or models with bid-ask spreads, we recover well-known pricing formulas in terms of martingale measures and consistent price systems. Building on the general theory of convex stochastic optimization, we also derive optimality conditions in terms of an extended notion of a "shadow price".