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Bridging Stochastic Control and Deep Hedging: Structural Priors for No-Transaction Band Networks

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.

Bridging Stochastic Control and Deep Hedging: Structural Priors for No-Transaction Band Networks

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.

Paper Structure

This paper contains 55 sections, 1 theorem, 79 equations, 12 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Optimal Hedging via Stochastic Control
  2. Market Model and the Frictionless Problem
  3. Price Dynamics and Portfolio Construction
  4. The Optimal Investment Problem and the HJB Equation
  5. Solution under CARA Utility
  6. Adding a Short Option Position
  7. Transaction Costs: Problem Formulation
  8. HJB Quasi-Variational Inequality and the No-Transaction Band
  9. CARA Utility and Dimension Reduction
  10. Indifference Pricing
  11. The Whalley-Wilmott Asymptotic Band
  12. The asymptotic formula.
  13. Interpretation.
  14. Validity.
  15. Numerical Scheme
  16. ...and 40 more sections

Key Result

Proposition 1.3

Under CARA utility, the value function takes the form where $\delta(t, T) = e^{-r(T-t)}$ is the discount factor and $Q(t, y, S)$ solves the reduced quasi-variational inequality with terminal condition $Q(T, y, S) = \exp\bigl(-\gamma(yS - \lambda S|y| - \varphi(S))\bigr)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (12)

  • Figure 1: Comparison of the MLP and NTBN-$\Delta$ architectures. $\ell_t = \Delta_{\mathrm{BS}} - \mathrm{LeakyReLU}(\delta_\ell)$ and $u_t = \Delta_{\mathrm{BS}} + \mathrm{LeakyReLU}(\delta_u)$.
  • Figure 2: Forward pass of the WW-NTBN.
  • Figure 3: Writer indifference prices vs transaction cost level for all models. The BS reference price at $\lambda = 0$ is shown as a horizontal dashed line.
  • Figure 4: Writer and buyer indifference prices vs transaction cost level for SC (left) and WW-NTBN (right), illustrating the widening bid-ask spread.
  • Figure 5: No-transaction band boundaries at $t = 0.1$ for $\lambda = 0.1\%$ and $\lambda = 1\%$.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Remark 1.1
  • Definition 1.2: Trading regions
  • Proposition 1.3: Dimension reduction
  • proof
  • Remark 1.4
  • Remark 2.1
  • Remark B.1