Scaling Limits for Exponential Hedging in Trinomial Models

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.

Theorems & Definitions (27)

• Theorem 1.1
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Lemma 3.1