Risk-Sensitive Option Market Making with Arbitrage-Free eSSVI Surfaces: A Constrained RL and Stochastic Control Bridge

Paper: arXiv 2510.04569

Authors: Jian’an Zhang

Abstract

We formulate option market making as a constrained, risk-sensitive control problem that unifies execution, hedging, and arbitrage-free implied-volatility surfaces inside a single learning loop. A fully differentiable eSSVI layer enforces static no-arbitrage conditions (butterfly and calendar) while the policy controls half-spreads, hedge intensity, and structured surface deformations (state-dependent rho-shift and psi-scale). Executions are intensity-driven and respond monotonically to spreads and relative mispricing; tail risk is shaped with a differentiable CVaR objective via the Rockafellar–Uryasev program. We provide theory for (i) grid-consistency and rates for butterfly/calendar surrogates, (ii) a primal–dual grounding of a learnable dual action acting as a state-dependent Lagrange multiplier, (iii) differentiable CVaR estimators with mixed pathwise and likelihood-ratio gradients and epi-convergence to the nonsmooth objective, (iv) an eSSVI wing-growth bound aligned with Lee’s moment constraints, and (v) policy-gradient validity under smooth surrogates. In simulation (Heston fallback; ABIDES-ready), the agent attains positive adjusted P&L on most intraday segments while keeping calendar violations at numerical zero and butterfly violations at the numerical floor; ex-post tails remain realistic and can be tuned through the CVaR weight. The five control heads admit clear economic semantics and analytic sensitivities, yielding a white-box learner that unifies pricing consistency and execution control in a reproducible pipeline.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 7.8/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly advanced mathematical framework integrating stochastic control, constrained reinforcement learning, and differential geometry of arbitrage-free eSSVI surfaces, with rigorous theoretical proofs (e.g., convergence rates, primal-dual theory, CVaR gradient estimators). Empirically, it uses a sophisticated agent-based simulator (ABIDES-ready) with Heston fallback, reporting concrete metrics like P&L, arbitrage violations, and tunable tail risk, indicating substantial implementation and backtesting effort.

Research Flowchart

  flowchart TD
  A["Research Goal: Unify execution, hedging, & arbitrage-free surfaces in risk-sensitive market making"] --> B{"Key Methodology: Constrained RL & Stochastic Control Bridge"}
  
  B --> C["Policy Control Heads<br/>Half-spreads, Hedge intensity, Surface deformations"]
  B --> D["eSSVI Layer<br/>Enforces no-arbitrage constraints<br/>(Butterfly & Calendar)"]
  B --> E["Objective Function<br/>Differentiable CVaR via Rockafellar-Uryasev"]
  
  C --> F["Computational Process<br/>Constrained Policy Optimization<br/>Primal-Dual & Gradient Flow"]
  D --> F
  E --> F
  
  F --> G["Simulations: Heston Fallback & ABIDES"]
  
  G --> H["Key Outcomes:<br/>Positive P&L, Zero Calendar Violations, Tunable Tail Risk"]

Cited References (66)

1. Optimal Dealer Pricing under Transactions and Return Uncertainty (1981)
2. Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders (1985)
3. Continuous Auctions and Insider Trading (1985)
4. High-frequency trading in a limit order book (2008)
5. Dealing with the Inventory Risk: A Solution to the Market Making Problem (2013)
6. Algorithmic and High-Frequency Trading (2015)
7. Buy Low, Sell High: A High Frequency Trading Perspective (2014)
8. The Price Impact of Order Book Events: An Empirical Study (2014)

2 of 66 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.