When Hedging Changes the Payoff: Option Replication with Price Impact and Execution Costs

2026-09-28

AI Summary

This study addresses the challenge that price impact induced by delta-hedging causes drift in option replication targets, while execution costs further complicate pricing. Building upon a binomial framework and fixed-point equations, this work derives a nonlinear pricing partial differential equation (PDE) incorporating transaction costs, which is subsequently validated through numerical simulations. The primary contributions are threefold: it quantifies the displacement effect of price impact on strike prices; it reveals the mechanism whereby self-trading drives out-of-the-money options into the money, thereby explaining the origin of market spreads; and it elucidates the dynamic mechanism through which execution costs smooth position adjustments. Ultimately, these insights facilitate the precise replication of options with convex payoffs under realistic market frictions.

Abstract

Hedging a derivative by trading the underlying asset changes the payoff that the hedging intended to replicate. We study this phenomenon when trading generates price impact and execution costs. In a binomial model, we characterize replication through a fixed-point equation. In continuous time, we derive a nonlinear pricing PDE whose implicit terminal condition captures the nature of the moving target problem of the hedger. For monotone convex Lipschitz payoffs (such as calls and puts) we establish exact replication under midpoint execution costs. Numerical experiments illustrate: (i) how price impact shifts the effective strike, (ii) the non-linear dependence of the option price on the number of contracts, (iii) how execution costs smooth terminal holdings, (iv) the extent to which the hedger's own trading can bring an otherwise worthless option into the money, and (v) we explain the spread and the shape of the limit order book in the options market based on the price impact and the shape of the limit order book of the underlying.

Problem

• Option Replication
• Price Impact
• Execution Costs
• Hedging
• Derivative Pricing

Innovation

• Price Impact
• Execution Costs
• Option Replication
• Nonlinear PDE
• Limit Order Book