Hedge and Speculate: Replicating Option Payoffs with Limit and Market Orders

Abstract

We consider an agent who takes a short position in a contingent claim and employs limit orders (LOs) and market orders (MOs) to trade in the underlying asset to maximize expected utility of terminal wealth. The agent solves a combined optimal stopping and control problem where trading has frictions: MOs (executed by the agent and other traders) have permanent price impact and pay exchange fees, and LOs earn the spread (relative to the midprice of the asset) and pay no exchange fees. We show how the agent replicates the payoff of the claim and also speculates in the asset to maximize expected utility of terminal wealth. In the strategy, MOs are used to keep the inventory on target, to replicate the payoff, and LOs are employed to build the inventory at favorable prices and boost expected terminal wealth by executing roundtrip trades that earn the spread. We calibrate the model to the E-mini contract that tracks the S&P500 index, provide numerical examples of the performance of the strategy, and proof the convergence of the numerical scheme we employ.

Keywords: Optimal hedging, market impact, impulse control, algorithmic trading, high-frequency trading

JEL Classification: C6, C61, D81, G1, G13