# Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives

## TL;DR

This paper develops a new theory of semimartingale optimal transport with path dependent constraints, introduces duality via PPDEs, and applies it to calibrate volatility models to exotic derivatives.

## Contribution

It extends optimal transport theory to path dependent settings, introduces semifiltrations for dimension reduction, and applies these methods to model calibration.

## Key findings

- Established duality with PPDE representation
- Developed dimension reduction via semifiltrations
- Successfully calibrated volatility models to exotic derivatives

## Abstract

In this paper, we introduce and develop the theory of semimartingale optimal transport in a path dependent setting. Instead of the classical constraints on marginal distributions, we consider a general framework of path dependent constraints. Duality results are established, representing the solution in terms of path dependent partial differential equations (PPDEs). Moreover, we provide a dimension reduction result based on the new notion of "semifiltrations", which identifies appropriate Markovian state variables based on the constraints and the cost function. Our technique is then applied to the exact calibration of volatility models to the prices of general path dependent derivatives.