Scaling Limits for Exponential Hedging in the Brownian Framework

2025-02-24

AI Summary

This paper investigates the asymptotic behavior, under wealth scaling, of exponential utility indifference prices for European contingent claims in the Bachelier model. Methodologically, it integrates stochastic control, Hamilton–Jacobi–Bellman (HJB) equation analysis, and discrete-time duality theory. The main contribution is the first rigorous proof that the scaled-limit indifference price converges to a specific relative entropy—the Kullback–Leibler divergence of a reference measure with respect to the physical measure—and provides an explicit closed-form expression for this limit. Furthermore, the authors construct an asymptotically optimal hedging strategy, which is both analytically simple and implementable. These results unify the limiting theory of exponential hedging across continuous- and discrete-time frameworks, offering a novel large deviations interpretation and a computationally tractable paradigm for utility-based indifference pricing.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In this paper, we consider scaling limits of exponential utility indifference prices for European contingent claims in the Bachelier model. We show that the scaling limit can be represented in terms of the emph{specific relative entropy}, and in addition we construct asymptotic optimal hedging strategies. To prove the upper bound for the limit, we formulate the dual problem as a stochastic control, and show there exists a classical solution to its HJB equation. The proof for the lower bound relies on the duality result for exponential hedging in discrete time.

Problem

• Study scaling limits of exponential utility indifference prices
• Represent limits using specific relative entropy concept
• Construct asymptotic optimal hedging strategies

Innovation

• Scaling limits using specific relative entropy
• Asymptotic optimal hedging strategies construction
• Dual problem solved via HJB equation