Utility Maximisation with Model-independent Constraints

TL;DR

This paper studies an agent's utility maximisation in financial markets with derivative contracts, incorporating model-independent constraints on portfolio valuation, and derives explicit solutions in specific models like Black-Scholes-Merton.

Contribution

It introduces a novel optimisation framework with pathwise constraints based on model-independent valuation bounds, providing explicit solutions in complete markets and the Black-Scholes-Merton model.

Findings

Explicit optimal terminal wealth expression using max-plus decomposition.

Derived the process form in Black-Scholes-Merton model.

Numerical analysis of portfolios with mispriced options.

Abstract

We consider an agent who has access to a financial market, including derivative contracts, who looks to maximise her utility. Whilst the agent looks to maximise utility over one probability measure, or class of probability measures, she must also ensure that the mark-to-market value of her portfolio remains above a given threshold. When the mark-to-market value is based on a more pessimistic valuation method, such as model-independent bounds, we recover a novel optimisation problem for the agent where the agents investment problem must satisfy a pathwise constraint. For complete markets, the expression of the optimal terminal wealth is given, using the max-plus decomposition for supermartingales. Moreover, for the Black-Scholes-Merton model the explicit form of the process involved in such decomposition is obtained, and we are able to investigate numerically optimal portfolios in the presence of options which are mispriced according to the agent's beliefs.