Local Volatility Calibration by Optimal Transport

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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, 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, 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.
Original languageEnglish
Title of host publication2017 MATRIX Annals
EditorsDavid Wood, Jan de Gier, Cheryl E Praeger, Terence Tao
PublisherSpringer
Chapter5
Pages51-64
Number of pages13
Volume2
Edition1
ISBN (Electronic)9783030041618
ISBN (Print)9783030041601
DOIs
Publication statusPublished - 2019

Publication series

NameMATRIX Book Series
PublisherSpringer
Volume2
ISSN (Print)2523-3041
ISSN (Electronic)2523-305X

Cite this

Guo, I., Loeper, G., & Wang, S. (2019). Local Volatility Calibration by Optimal Transport. In D. Wood, J. de Gier, C. E. Praeger, & T. Tao (Eds.), 2017 MATRIX Annals (1 ed., Vol. 2, pp. 51-64). (MATRIX Book Series; Vol. 2). Springer. https://doi.org/10.1007/978-3-030-04161-8_5