A BSDE arising in an exponential utility maximization problem in a pure jump market model

TL;DR

This paper investigates a BSDE framework for exponential utility maximization in a pure jump Lévy market, providing conditions for well-definedness, bounds on strategies, explicit solutions for specific problems, and links to market no-arbitrage conditions.

Contribution

It introduces a novel BSDE approach for utility maximization in pure jump markets, analyzing generator conditions, and deriving explicit solutions for certain claims.

Findings

Conditions for market models to admit no free lunches.

Explicit solutions for cross-hedging problems under specific assumptions.

Bounds on the candidate optimal trading strategies.

Abstract

We consider the problem of utility maximization with exponential preferences in a market where the traded stock/risky asset price is modelled as a L\'evy-driven pure jump process (i.e. the driving L\'evy process has no Brownian component). In this setting, we study the terminal utility optimization problem in the presence of a European contingent claim. We consider in detail the BSDE (backward stochastic differential equation) characterising the value function when using an exponential utility function. First we analyse the well-definedness of the generator. This leads to some conditions on the market model related to conditions for the market to admit no free lunches. Then we give bounds on the candidate optimal strategy. Thereafter, we discuss the example of a cross-hedging problem and, under severe assumptions on the structure of the claim, we give explicit solutions. Finally, we establish an explicit solution for a related BSDE with a suitable terminal condition but a simpler generator.