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

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 Lévy-driven pure jump process (i.e. the driving Lévy 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 backward stochastic differential equation (BSDE) characterizing the value function when using an exponential utility function. First we analyze 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.

Keywords:

• BSDE
• cross hedging
• exponential utility
• Lévy process
• stationary spread

AMS Subject Classifications:

• Primary: 91G80
• Secondary: 60G51
• 60H10
• 60J75
• 93E20

Acknowledgements

C. Mereu gratefully acknowledges support by the Deutsche Forschungsgemeinschaft within the Research Training Group “Modelling, analysis and simulation in economathematics” (DFG RTG 1100).

Notes

No potential conflict of interest was reported by the authors.

Dedicated to Bernt Øksendal on the occasion of his 70th birthday.

Additional information

Funding

This work was supported by the Deutsche Forschungsgemeinschaft within the Research Training Group “Modelling, analysis and simulation in economathematics” (DFG RTG 1100).