Joint Calibration of Local Volatility Models with Stochastic Interest Rates using Semimartingale Optimal Transport

ArXiv ID: 2308.14473 “View on arXiv”

Authors: Unknown

Abstract

We develop and implement a non-parametric method for joint exact calibration of a local volatility model and a correlated stochastic short rate model using semimartingale optimal transport. The method relies on the duality results established in Joseph, Loeper, and Obloj, 2023 and jointly calibrates the whole equity-rate dynamics. It uses an iterative approach which starts with a parametric model and tries to stay close to it, until a perfect calibration is obtained. We demonstrate the performance of our approach on market data using European SPX options and European cap interest rate options. Finally, we compare the joint calibration approach with the sequential calibration, in which the short rate model is calibrated first and frozen.

Keywords: Local Volatility Model, Stochastic Short Rate Model, Semimartingale Optimal Transport, Joint Calibration, Equity-Rate Dynamics

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematical machinery including semimartingale optimal transport, duality theorems, and convex analysis with non-trivial derivations, while also demonstrating empirical implementation on real market data (SPX options and caps) with explicit calibration results and comparison of methodologies.

  flowchart TD
    A["Research Goal: Joint Calibration of Local Volatility & Stochastic Rates"] --> B{"Key Methodology: Semimartingale Optimal Transport"}
    
    B --> C["Data Inputs: SPX Options & Cap Rates"]
    
    C --> D["Iterative Process: Parametric Model to Perfect Calibration"]
    
    D --> E["Computational Process: Duality Solving via SOCP"]
    
    E --> F["Joint Calibration: Equity-Rate Dynamics"]
    
    F --> G["Key Finding: Superior vs Sequential Calibration"]
    
    style A fill:#e1f5e1
    style G fill:#fff2cc