Scaling Limits for Exponential Hedging in Trinomial Models

2026-03-30

AI Summary

This study investigates the asymptotic behavior of exponential utility-based no-arbitrage pricing and hedging strategies for path-dependent European options in a trinomial model, where risk aversion scales linearly with trading frequency. Employing a purely probabilistic approach that integrates duality theory, martingale methods, and weak convergence techniques, the authors establish—for the first time—a nontrivial continuous-time limit under linear scaling of the risk-aversion parameter. This limit transforms the discrete exponential utility hedging problem into a volatility control problem with a penalization term. In the Markovian payoff setting, they prove the asymptotic optimality of the corresponding delta hedging strategies and derive the convergence of the exponential certainty equivalent price to its Black–Scholes counterpart.

Abstract

We study scaled trinomial models converging to the Black--Scholes model, and analyze exponential certainty-equivalent prices for path-dependent European options. As the number of trading dates $n$ tends to infinity and the risk aversion is scaled as $nl$ for a fixed constant $l>0$, we derive a nontrivial scaling limit. Our analysis is purely probabilistic. Using a duality argument for the certainty equivalent, together with martingale and weak-convergence techniques, we show that the limiting problem takes the form of a volatility control problem with a specific penalty. For European options with Markovian payoffs, we analyze the optimal control problem and show that the corresponding delta-hedging strategy is asymptotically optimal for the primal problem.

Problem

• scaling limits
• exponential hedging
• trinomial models
• certainty-equivalent pricing
• path-dependent options

Innovation

• trinomial models
• exponential hedging
• scaling limits
• volatility control
• certainty-equivalent pricing