Convergence of Restricted Entropic and Quadratic Hedging in the Small-Risk-Aversion Limit

01 ABSTRACT

This paper studies the relation between quadratic hedging and entropic hedging when exact replication is impossible in incomplete markets. Fact: under restricted finite-dimensional strategy classes, the admissible gain set is compact and convex in L²; the author proves that, under a uniform exponential integrability assumption, restricted entropic hedging gains converge in L² to the unique restricted quadratic hedge as risk aversion tends to zero. Opinion: the result places both hedging problems in a common well-posed framework and illustrates the theory numerically in a Heston-type discrete-time stochastic volatility model.

02 KEY FINDINGS

1. Strategies are restricted to bounded finite-dimensional parametric classes, making the admissible gain set compact and convex in L² and providing a common well-posed setting for restricted mean-square and exponential-utility hedging.
2. Main result: under a uniform exponential integrability assumption, a uniform small-risk-aversion expansion of the entropic risk functional over the restricted gain set yields convergence of the corresponding minimizers.
3. As the risk-aversion parameter tends to zero, restricted entropic hedging gains converge in L² to the unique restricted quadratic hedge, illustrated numerically in a regularized discrete-time Heston-type stochastic volatility model.