Mimicking Volatilities under Marginal Constraints via $q_x$-Bass Martingales: A Parallel Computing Approach

Abstract

We study the calibration of continuous-time martingale models to observed derivative prices through a dynamic martingale optimal transport problem motivated by minimizing deviations from the volatility of a prescribed reference model. Using $q_x$-Bass martingales, we approximate this problem through a sequence of discrete-time problems and prove that the corresponding continuous-time interpolations are asymptotically optimal. Building on the computational advantages of $q_x$-Bass martingales, we develop numerical methods for the discrete-time problems, including a convergent projected subgradient algorithm. The resulting framework does not require full marginal distributions at calibration maturities and accommodates general state-dependent reference dynamics, while avoiding PDE-based methods and allowing for parallelization across time intervals.