Paper: arXiv 2605.12189

Authors: Qinwen Zhu, Wen Chen, Nicolas Langrené

Abstract

This paper develops a deep learning-based framework for pricing convertible bonds with path-dependent contractual features, namely downward conversion price reset and issuer call clauses under rolling-window trigger rules, which are widespread in the convertible bond market. We formulate the valuation problem as a path-dependent partial differential equation (PPDE), which explicitly captures the dependence of the convertible bond value on the historical path of the underlying asset and the dynamic evolution of the conversion price. We derive consistent PPDE formulations for three canonical underlying dynamics: geometric Brownian motion (GBM), constant elasticity of variance (CEV) and Heston stochastic volatility. We then construct a discrete-time dynamic programming scheme in which conditional expectations are approximated by neural networks, which remains tractable in such high-dimensional path-dependent setting. Empirical tests on China CITIC Bank Convertible Bond show that our framework produces stable and accurate prices and sensitivity patterns across all model specifications. Three key economic insights emerge: 1. Contractual features dominate underlying dynamics in determining convertible bond values. 2. The call provision decreases convertible bonds prices by truncating upside gains. 3. Counterintuitively, despite improving conversion terms, the downward reset provision further decreases the price of convertible bonds by lowering the effective call threshold and making early redemption more likely. The proposed PPDE-deep learning approach provides an efficient, flexible tool for pricing convertible bonds with complex path-dependent structures.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper demonstrates high mathematical sophistication by formulating path-dependent convertible bond valuation as a PPDE and utilizing neural-network-based dynamic programming, while supporting these derivations with concrete empirical testing on real-world China CITIC Bank bond data.

Research Flowchart

  flowchart TD
    A[<b>Goal:</b> Price convertible bonds with path-dependent features] --> B[<b>Framework:</b> Formulate as Path-Dependent PDE (PPDE)]
    B --> C[<b>Models:</b> GBM, CEV, and Heston Stochastic Volatility]
    C --> D[<b>Method:</b> Deep learning-based dynamic programming]
    D --> E[<b>Inputs:</b> Historical asset paths & Reset/Call provisions]
    E --> F[<b>Verification:</b> Empirical test on China CITIC Bank Bond]
    F --> G[<b>Key Insights:</b> Contract dominance, Call-truncation, Reset-price reduction]