A deep learning approach for pricing convertible bonds with path-dependent reset and call provisions

Abstract

This paper develops a deep learning framework for pricing convertible bonds with path-dependent downward reset and issuer call provisions governed by rolling-window triggers. We formulate the valuation problem as a path-dependent partial differential equation (PPDE) that captures both the historical stock-price path and the evolution of the conversion price. Model-specific PPDEs are derived under GBM, CEV, and Heston dynamics. Under suitable conditions, we establish the existence and uniqueness of a piecewise viscosity solution linked by contractual transmission conditions at monitoring dates. For computation, we construct a fixed-grid backward dynamic programming scheme and approximate its conditional expectations using neural networks, with $L^2$ convergence to the exact fixed-grid recursion as the approximation errors vanish. An application to the China CITIC Bank Convertible Bond produces stable prices across the three models and close agreement with the LSMC benchmark, but outperforms in dimensional scaling. The results show that contractual provisions have a greater valuation effect than the choice of underlying dynamics. The call provision reduces the bond value by truncating upside gains, whereas the downward reset provision increases it under the benchmark specification because improved conversion terms dominate the effect of earlier redemption. Delta and Gamma obtained by automatic differentiation of smooth local network approximations closely agree with central finite-difference estimates. The framework provides a flexible approach to pricing and sensitivity analysis for convertible bonds with complex path-dependent provisions.