Internship - Local Volatility Models

The options market plays a central role in modern quantitative finance, as option prices provide valuable information about market expectations of future returns and volatility. Extracting this information from observed option prices is therefore an important problem. 

The main objective of this internship is to investigate optimal-transport-based methods for calibration of local volatility models. This involves studying the formulation and numerical implementation of local volatility calibration as a martingale optimal transport problem, with the aim of constructing a diffusion process whose marginal distributions match those implied by observed option prices. 

The performance of the optimal-transport-based approach will be compared with a benchmark (the Dupire and regularized calibration methods) using the evaluation framework developed during the preliminary stages. Finally, you will investigate whether the resulting local volatility surfaces can provide meaningful features for forecasting tasks. 

You will build a local volatility model benchmark to better understand and compare different approaches, with the aim of finding the best model in terms of accuracy and robustness. Quality metrics will be implemented to define quantitative comparison criteria. 

· Local Volatility Calibration (explicit formula): implement a Dupire-based calibration as a benchmark. This approach provides an explicit expression for local volatility in terms of partial derivatives of option prices. It will also allow us to define suitable tests and metrics for assessing calibration quality. 

· Local Volatility Calibration (regularized inverse problems): investigate the direct calibration of a local volatility function by minimizing a weighted option-pricing error, following inverse-problem approaches. The objective is, once again, to obtain a stable local volatility surface while maintaining a good fit to observed option prices. 

· Local Volatility Calibration by optimal transport: focus on recent methods for calibrating local volatility models using optimal transport techniques, in particular the approach introduced by Guo, Loeper and Wang. The calibration problem will be formulated as a continuous-time martingale optimal transport problem, then applied to local volatility calibration and compared with the previous approaches. 

· You are pursuing a Master's degree in Quantitative Finance, Financial Mathematics, Applied Mathematics, Statistics or related fields. 

· You are interested in optimization, volatility modeling, numerical methods and quantitative research. 

· You have strong Python skills. 

· You have knowledge of optimization and stochastic processes. 

· You are familiar with some of the following: local volatility and the Dupire equation; numerical methods; optimal control and HJB equations. 

· You have a rigorous and analytical mindset, with strong programming and problem-solving skills. 

· You enjoy comparing mathematical models through both theory and experiments. 

· You are autonomous, curious and attentive to numerical details, with clear communication skills and a collaborative approach.