Local Volatility Calibration by Optimal Transport

TL;DR

This paper introduces a novel local volatility calibration method using optimal transport theory, formulating it as a convex optimization problem that reconstructs asset dynamics without the need for price interpolation.

Contribution

It develops a new local volatility calibration technique based on martingale optimal transport, avoiding traditional interpolation methods and providing a convex optimization framework.

Findings

Successfully reconstructs asset price dynamics between two dates

Replaces interpolation with a convex optimization approach

Uses augmented Lagrangian and ADMM algorithms for numerical solution

Abstract

The calibration of volatility models from observable option prices is a fundamental problem in quantitative finance. The most common approach among industry practitioners is based on the celebrated Dupire's formula [6], which requires the knowledge of vanilla option prices for a continuum of strikes and maturities that can only be obtained via some form of price interpolation. In this paper, we propose a new local volatility calibration technique using the theory of optimal transport. We formulate a time continuous martingale optimal transport problem, which seeks a martingale diffusion process that matches the known densities of an asset price at two different dates, while minimizing a chosen cost function. Inspired by the seminal work of Benamou and Brenier [1], we formulate the problem as a convex optimization problem, derive its dual formulation, and solve it numerically via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm. The solution effectively reconstructs the dynamic of the asset price between the two dates by recovering the optimal local volatility function, without requiring any time interpolation of the option prices.