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 problemand it is numerically solved ] via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm  .
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 .
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classical formula by Dupire, despite being heavily used by industry practitioners , can potentially produce unstable or even singular volatility surfaces
most common approach among industry practitioners is based on the celebrated Dupire's formula
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, 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
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Inspired by the work of Benamou and Brenier, we formulate a
We formulate a time continuous
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martingale
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and minimizes
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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 problemand it is numerically solved
Inspired by the seminal work of Benamou and Brenier
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1
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, we formulate the problem as a convex optimization problem, derive its dual formulation, and solve it numerically
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. 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