Why Unrestricted Arbitrage Makes Utility Maximization Ill-Posed
Summary
The post asks whether expected utility portfolio optimization can be formulated when market models permit arbitrage. It references work on pricing under a weakened no-arbitrage assumption, while noting that the cited approach does not resolve utility maximization. The reply gives a basic limitation: optimization needs constraints on arbitrage opportunities.
Without such limits, a trader could take a position with no initial cost and a positive future payoff, or reverse the cash flows. This makes the objective unbounded, described in the reply as an infinite Sharpe ratio, so a well-defined optimum is unavailable. The exchange offers this conceptual argument rather than a formal model or a specific method for imposing limits. It does not discuss which constraints are appropriate in practice or how they affect an optimization solution.
Key ideas
- The post distinguishes pricing under arbitrage from expected utility maximization.
- Unrestricted arbitrage can create a zero-cost position with a positive future payoff.
- Such an opportunity makes the utility optimization problem unbounded.
- Limits on arbitrage are needed to define a meaningful optimization problem.
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Full text
# Has there been any research that allows utility maximization with arbitrage? # Has there been any research that allows utility maximization with arbitrage? I have read the paper "Pricing without no-arbitrage condition in discrete time" by Carassus and Lépinette (2022) that explains how to price under arbitrage. They introduced a weak assumption of the No Arbitrage condition. However, they mention that while this solves the pricing issue, the expected utility maximization is still a problem. Just wondering if any one here has heard of some study/paper that allows to solve this when there is arbitrage in the market. IRL it is difficult to expect no-arbitrage to be fulfilled everywhere, subsequently sometimes models used in practice do not even satisfy this assumption. I was wondering if it is possible from a mathematical point of view to perform portfolio optimization with such models. Every input is helpful! ## Answer by phdstudent (score 3) https://quant.stackexchange.com/a/83786 You need limits to arbitrage. If there are no limits to arbitrage, you can make a trade that has zero cashflows today and positive cashflows tomorrow (or vice-versa). Any utility maximization problem would then deliver an infinite Sharpe Ratio, and not be well defined.
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