Diffusion-limit convergence of calibrated binomial trees

Determine whether, when option data are consistent with a continuous-time local-volatility model, a sequence of calibrated recombining binomial trees with increasingly fine time grids converges to that continuous-time local-volatility model as the number of tree steps tends to infinity, under suitable scaling of tree-design parameters such as the asset-price bounds and admissibility margin.

Background

The paper proves existence of global minimizers for the finite-dimensional penalized calibration problem on compact subsets of a class of strictly admissible recombining binomial trees. The authors distinguish this finite-dimensional well-posedness result from a possible asymptotic connection with continuous-time local-volatility models.

The unresolved question concerns a sequence of calibrated trees whose number of time steps increases and whose design parameters may depend on the discretization level. The remark identifies potential sufficient conditions—uniformly vanishing log-price increments, convergence of conditional first and second moments, a negligible third-moment remainder, tightness of the induced processes, and well-posedness of the limiting martingale problem—that could support weak convergence to the corresponding diffusion and convergence of discounted expectations for suitable European payoffs.