Paper: arXiv 2603.29994

Authors: Jules Arzel, Noureddine Lehdili

Abstract

This paper studies the problem of hedging and pricing a European call option under proportional transaction costs, from two complementary perspectives. We first derive the optimal hedging strategy under CARA utility, following the stochastic control framework of Davis et al. (1993), characterising the no-transaction band via the Hamilton-Jacobi-Bellman Quasi-Variational Inequality (HJBQVI) and the Whalley-Wilmott asymptotic approximation. We then adopt a deep hedging approach, proposing two architectures that build on the No-Transaction Band Network of Imaki et al. (2023): NTBN-Delta, which makes delta-centring explicit, and WW-NTBN, which incorporates the Whalley-Wilmott formula as a structural prior on the bandwidth and replaces the hard clamp with a differentiable soft clamp. Numerical experiments show that WW-NTBN converges faster, matches the stochastic control no-transaction bands more closely, and generalises well across transaction cost regimes. We further apply both frameworks to the bull call spread, documenting the breakdown of price linearity under transaction costs.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper effectively bridges classical stochastic control theory (HJBQVI, Whalley-Wilmott) with modern deep hedging architectures by incorporating domain-specific structural priors into neural networks. The numerical experiments rigorously demonstrate improved convergence and generalization over existing deep hedging baselines, making it a strong contribution to quantitative risk management.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimize Deep Hedging under Proportional Transaction Costs] --> B[Inputs: Option Parameters & Price Paths]
    
    subgraph Methodology & Processing
        B --> C[Stochastic Control Baseline: HJBQVI & Whalley-Wilmott Asymptotics]
        B --> D[Deep Hedging Architectures: NTBN-Delta & WW-NTBN with Soft Clamp]
        C -->|Structural Prior| D
        D --> E[Neural Network Training & Numerical Experiments]
    end

    E --> F[Outcomes: WW-NTBN Converges Faster & Matches HJB Bands]
    E --> G[Outcomes: Demonstrates Breakdown of Price Linearity in Bull Spreads]

Cited References (19)

  1. The Pricing of Options and Corporate Liabilities (1973)
  2. Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case (1969)
  3. European Option Pricing with Transaction Costs (1993)
  4. An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs (1997)
  5. Option Pricing and Replication with Transaction Costs (1985)
  6. Option Replication in Discrete Time with Transaction Costs (1992)
  7. Optimal Replication of Options with Transactions Costs and Trading Restrictions (1993)
  8. Option Pricing and Hedging with Small Transaction Costs (2015)
and 11 more
  1. Option pricing for incomplete markets via stochastic optimization: transaction costs, adaptive control and forecast (2001)
  2. Option Pricing in the Presence of Transactions Costs (2000)
  3. Hedging options with transaction costs (2006)
  4. Optimal replication of contingent claims under transaction costs (1989)
  5. Utility indifference pricing: an overview (2004)
  6. Deep Hedging (2019)
  7. No-Transaction Band Network: A Neural Network Architecture for Efficient Deep Hedging (2023)
  8. Deep Hedging under Rough Volatility (2021)
  9. Stochastic Finance: An Introduction in Discrete Time (2002)
  10. Approximation Capabilities of Multilayer Feedforward Networks (1991)
  11. Adam: A Method for Stochastic Optimization (2015)

0 of 19 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.