Option Replication with Price Impact and Execution Costs

Summary

This study examines how hedging trades can alter the payoff of the option they are meant to replicate when the underlying market has price impact and trading incurs execution costs. In a binomial setting, replication is characterized by a fixed-point equation; in continuous time, it leads to a nonlinear pricing partial differential equation with an implicit terminal condition that reflects the hedger's moving target.

For monotone, convex, Lipschitz payoffs such as calls and puts, the authors establish exact replication under midpoint execution costs. Numerical experiments show how price impact shifts the effective strike, option prices depend nonlinearly on contract count, and execution costs smooth terminal holdings. They also illustrate cases where hedging trades can move an otherwise out-of-the-money option into the money, and relate option-market spreads and order-book shape to those of the underlying. The results are model-based; the document provides no empirical market validation or quantitative detail on the experiments.

Key ideas

• Hedging trades can change the payoff that option replication is intended to match when they affect the underlying price.\nThe binomial replication problem is represented by a fixed-point equation.\nIn continuous time, pricing follows a nonlinear PDE with an implicit terminal condition.\nExact replication is established for monotone convex Lipschitz payoffs under midpoint execution costs.\nThe numerical analysis links option spreads and order-book shape to price impact and liquidity in the underlying.

Tags

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


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









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.