[Paper Review] From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models

Pierre Henry‐Labordère|arXiv (Cornell University)|Apr 9, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance3 references3 citations

TL;DR

This paper introduces a novel class of stochastic volatility models (SVMs) that are exactly calibrated to market vanillas, VIX options, and realized variance derivatives by modifying the drift of the volatility process via a martingale Schrödinger bridge approach. Unlike local-stochastic volatility models, the instantaneous volatility-of-volatility remains unchanged post-calibration, and the method uses a dynamic Sinkhorn algorithm for efficient numerical calibration, revealing a deep link to Dyson non-colliding Brownian motions.

ABSTRACT

Following closely the construction of the Schrodinger bridge, we build a new class of Stochastic Volatility Models exactly calibrated to market instruments such as for example Vanillas, options on realized variance or VIX options. These models differ strongly from the well-known local stochastic volatility models, in particular the instantaneous volatility-of-volatility of the associated naked SVMs is not modified, once calibrated to market instruments. They can be interpreted as a martingale version of the Schrodinger bridge. The numerical calibration is performed using a dynamic-like version of the Sinkhorn algorithm. We finally highlight a striking relation with Dyson non-colliding Brownian motions.

Motivation & Objective

• To develop a new class of stochastic volatility models that exactly match market prices of vanillas, VIX options, and options on realized variance.
• To preserve the instantaneous volatility-of-volatility structure of the original model, avoiding the distortions introduced by local-stochastic volatility models.
• To ensure the calibrated model remains a martingale under the risk-neutral measure, thus avoiding arbitrage.
• To provide a numerically efficient calibration method using a dynamic variant of the Sinkhorn algorithm.
• To establish a theoretical connection between the resulting model dynamics and Dyson non-colliding Brownian motions.

Proposed method

• The model is constructed via a martingale version of the Schrödinger bridge problem, minimizing relative entropy under the constraint that the stock price is a martingale.
• The calibration is achieved by solving a stochastic control problem that introduces a time- and state-dependent drift term to the volatility process, preserving the original volatility-of-volatility structure.
• The dynamics are derived using a dual formulation involving the solution of a Burgers-like PDE with a nonlinear term involving the correlation and volatility function.
• The Radon-Nikodym derivative for the change of measure is expressed as an exponential martingale involving the solution to the dual control problem.
• A dynamic Sinkhorn algorithm is employed for numerical calibration, iteratively adjusting the drift to match target marginal distributions.
• The method is extended to include VIX-like constraints by introducing an additional control term for the realized variance, ensuring consistency with VIX options.

Experimental results

Research questions

• RQ1 Can a stochastic volatility model be calibrated to vanilla options, VIX options, and options on realized variance without altering the instantaneous volatility-of-volatility?
• RQ2 How can the Schrödinger bridge framework be adapted to enforce martingale constraints in the context of stochastic volatility models?
• RQ3 What is the impact of adding a state-dependent drift to the volatility process on the model's dynamics and calibration properties?
• RQ4 Can the resulting model dynamics be linked to known stochastic processes such as Dyson non-colliding Brownian motions?
• RQ5 How can the calibration be efficiently computed in practice, especially for high-dimensional or path-dependent derivatives?

Key findings

• The proposed model achieves exact calibration to market vanillas, VIX options, and options on realized variance by modifying only the drift of the volatility process, without altering the original volatility-of-volatility structure.
• The calibrated model remains a true martingale under the risk-neutral measure, ensuring no arbitrage, in contrast to previous Schrödinger bridge approaches that failed to preserve martingality.
• The numerical calibration is performed efficiently using a dynamic Sinkhorn algorithm, enabling practical implementation for complex derivatives pricing.
• The solution to the dual control problem leads to a Burgers-like PDE whose solution governs the optimal drift and Radon-Nikodym derivative.
• A striking connection is established between the resulting model dynamics and Dyson non-colliding Brownian motions, suggesting deeper mathematical structure.
• The method provides a consistent framework for incorporating multiple market constraints (marginals and realized variance) within a single, unified stochastic volatility model.