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Option market making with hedging-induced market impact

Paulin Aubert, Etienne Chevalier, Vathana Ly Vath

TL;DR

This work develops a coupled market-making model for European options in which the hedging activity of the market maker induces permanent and transient price impact on the underlying, captured by a dynamic mid-price and resilience framework with Hawkes-driven exogenous flow. Option order flow is modeled by Cox processes whose intensities depend on the market-maker’s quotes and the state, while hedging the resulting option exposure generates impulse trades in the underlying that feed back into price formation and liquidity via a time-invariant order-book with cumulative liquidity functions. The authors establish model-consistency by ruling out instantaneous and dynamic arbitrage, address terminal manipulation in the coupled setting, and prove well-posedness with quadratic-growth bounds for the value function, providing a solid theoretical foundation. A neural-policy, simulation-based approach inspired by Deep Hedging is developed to approximate optimal quoting and hedging strategies, and extensive numerical experiments illustrate how liquidity, inventory risk, and hedging-induced market impact interact across symmetric baseline, asymmetric scenarios, and low-liquidity regimes, offering practical insights for option market-makers.

Abstract

This paper develops a model for option market making in which the hedging activity of the market maker generates price impact on the underlying asset. The option order flow is modeled by Cox processes, with intensities depending on the state of the underlying and on the market maker's quoted prices. The resulting dynamics combine stochastic option demand with both permanent and transient impact on the underlying, leading to a coupled evolution of inventory and price. We first study market manipulation and arbitrage phenomena that may arise from the feedback between option trading and underlying impact. We then establish the well-posedness of the mixed control problem, which involves continuous quoting decisions and impulsive hedging actions. Finally, we implement a numerical method based on policy optimization to approximate optimal strategies and illustrate the interplay between option market liquidity, inventory risk, and underlying impact.

Option market making with hedging-induced market impact

TL;DR

This work develops a coupled market-making model for European options in which the hedging activity of the market maker induces permanent and transient price impact on the underlying, captured by a dynamic mid-price and resilience framework with Hawkes-driven exogenous flow. Option order flow is modeled by Cox processes whose intensities depend on the market-maker’s quotes and the state, while hedging the resulting option exposure generates impulse trades in the underlying that feed back into price formation and liquidity via a time-invariant order-book with cumulative liquidity functions. The authors establish model-consistency by ruling out instantaneous and dynamic arbitrage, address terminal manipulation in the coupled setting, and prove well-posedness with quadratic-growth bounds for the value function, providing a solid theoretical foundation. A neural-policy, simulation-based approach inspired by Deep Hedging is developed to approximate optimal quoting and hedging strategies, and extensive numerical experiments illustrate how liquidity, inventory risk, and hedging-induced market impact interact across symmetric baseline, asymmetric scenarios, and low-liquidity regimes, offering practical insights for option market-makers.

Abstract

This paper develops a model for option market making in which the hedging activity of the market maker generates price impact on the underlying asset. The option order flow is modeled by Cox processes, with intensities depending on the state of the underlying and on the market maker's quoted prices. The resulting dynamics combine stochastic option demand with both permanent and transient impact on the underlying, leading to a coupled evolution of inventory and price. We first study market manipulation and arbitrage phenomena that may arise from the feedback between option trading and underlying impact. We then establish the well-posedness of the mixed control problem, which involves continuous quoting decisions and impulsive hedging actions. Finally, we implement a numerical method based on policy optimization to approximate optimal strategies and illustrate the interplay between option market liquidity, inventory risk, and underlying impact.

Paper Structure

This paper contains 26 sections, 14 theorems, 174 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Market modeling and the market maker problem
  3. Modeling the underlying market
  4. Order book representation
  5. Execution price and cost functions
  6. Dynamics of the underlying
  7. Modeling the options market
  8. The market maker’s optimization problem
  9. Market maker inventories
  10. Admissible strategies
  11. Performance criterion and value function
  12. Model consistency and well-posedness
  13. Underlying market: absence of arbitrage and manipulation
  14. Coupled markets and terminal manipulation
  15. Well-posedness of the control problem
  16. ...and 11 more sections

Key Result

Lemma 2.1

Let $b>0$ the best bid price and $a>0$ denote the best ask price. For every admissible order size the cost functionals EQN:underyling_bid_price and EQN:underyling_ask_price admit the representations Consequently, $q\mapsto P_B(b,q)$ is concave and $q\mapsto P_A(a,q)$ is convex on their domains. Moreover, both maps are absolutely continuous and $\mathcal{C}^1$ a.e., with a.e. derivatives In part

Figures (13)

  • Figure 1: Learning metrics for linear order book with $I_0 = 0$.
  • Figure 2: Average quoting strategy ($I_0 = 0$).
  • Figure 3: Average inventories and P&L along trajectory.
  • Figure 4: Sample path: underlying price and corresponding quoting behavior of the agent ($I_0=0$).
  • Figure 5: Learning metrics for linear order book with $I_0 = -100$.
  • ...and 8 more figures

Theorems & Definitions (28)

  • Definition 2.1: Generalized inverse of cumulative liquidity functions
  • Lemma 2.1: Analytical structure of execution costs
  • proof
  • Definition 2.2: Admissible strategies
  • Lemma 2.2: Non-negativity preservation of bid–ask quotes
  • proof
  • Proposition 2.1: Reduction by cash additivity
  • proof
  • Lemma 3.3: No instantaneous round-trip
  • proof
  • ...and 18 more