Scaling Limits for Exponential Hedging in Trinomial Models

TL;DR

This paper investigates the asymptotic behavior of exponential hedging in trinomial models converging to Black-Scholes, revealing a limit characterized as a volatility control problem with implications for delta-hedging strategies.

Contribution

It introduces a probabilistic approach to analyze the scaling limits of exponential hedging in trinomial models, deriving a novel volatility control limit problem.

Findings

The scaled trinomial models converge to a volatility control problem.

The limiting delta-hedging strategy is asymptotically optimal.

A duality and weak-convergence approach is used for analysis.

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.