The calibration of local volatility models from observable option prices has always been an important problem in quantitative finance. The classical formula by Dupire, despite being heavily used by industry practitioners, can potentially produce unstable or even singular volatility surfaces. In this paper, we propose a new local volatility calibration technique using the theory of optimal transport. Inspired by the work of Benamou and Brenier, we formulate a martingale optimal transport problem which seeks a diffusion process that matches the known densities of an asset price at two different dates and minimizes a chosen cost function. The process effectively interpolates the dynamic of the asset price between the two dates while recovering the local volatility function. This approach leads to a convex optimisation problem and it is numerically solved via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm.
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