Local Volatility Calibration by Optimal Transport

Research output: Chapter in Book/Report/Conference proceedingChapter (Book)Researchpeer-review

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
Guo, Ivan ; Loeper, Gregoire ; Wang, Shiyi. / Local Volatility Calibration by Optimal Transport. 2017 MATRIX Annals. editor / David Wood ; Jan de Gier ; Cheryl E Praeger ; Terence Tao. Vol. 2 1. ed. Springer, 2019. pp. 51-64 (MATRIX Book Series).
@inbook{7cb898c7261446af9f93dd8f0613077f,
title = "Local Volatility Calibration by Optimal Transport",
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.",
author = "Ivan Guo and Gregoire Loeper and Shiyi Wang",
year = "2019",
doi = "10.1007/978-3-030-04161-8_5",
language = "English",
isbn = "9783030041601",
volume = "2",
series = "MATRIX Book Series",
publisher = "Springer",
pages = "51--64",
editor = "David Wood and {de Gier}, Jan and Praeger, {Cheryl E} and Terence Tao",
booktitle = "2017 MATRIX Annals",
edition = "1",

}

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

Local Volatility Calibration by Optimal Transport. / Guo, Ivan; Loeper, Gregoire; Wang, Shiyi.

2017 MATRIX Annals. ed. / David Wood; Jan de Gier; Cheryl E Praeger; Terence Tao. Vol. 2 1. ed. Springer, 2019. p. 51-64 (MATRIX Book Series; Vol. 2).

Research output: Chapter in Book/Report/Conference proceedingChapter (Book)Researchpeer-review

TY - CHAP

T1 - Local Volatility Calibration by Optimal Transport

AU - Guo, Ivan

AU - Loeper, Gregoire

AU - Wang, Shiyi

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

U2 - 10.1007/978-3-030-04161-8_5

DO - 10.1007/978-3-030-04161-8_5

M3 - Chapter (Book)

SN - 9783030041601

VL - 2

T3 - MATRIX Book Series

SP - 51

EP - 64

BT - 2017 MATRIX Annals

A2 - Wood, David

A2 - de Gier, Jan

A2 - Praeger, Cheryl E

A2 - Tao, Terence

PB - Springer

ER -

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