Hull-White Calibration & Swaption Pricing

Calibration of the Hull-White one-factor short-rate model on €STR data, and multi-method swaption pricing under three equivalent measures.

Overview

This project implements the full pipeline from model calibration to derivative pricing within the Hull-White (HW) framework:

1. Calibration of the HW parameters $(a, \sigma)$ from historical €STR data under the real-world measure $\mathbb{P}$, using two methods: AR(1) regression and Maximum Likelihood Estimation (MLE).
2. Yield curve fitting via a Nelson-Siegel parametrization to extract the initial term structure $\theta(t)$.
3. Swaption pricing under three equivalent measures — risk-neutral $\mathbb{Q}$, forward measure, and annuity measure — using Monte Carlo simulations and a Trinomial Tree.
4. Convergence & sensitivity analysis of both numerical methods.

Model

The Hull-White one-factor model under the risk-neutral measure $\mathbb{Q}$:

$$dr_t = (\theta(t) - a r_t),dt + \sigma,dW_t$$

• $a$ : mean-reversion speed (constant)
• $\sigma$ : volatility (constant)
• $\theta(t)$ : time-dependent drift calibrated to fit the initial yield curve

Results

Method	Swaption Price
Monte Carlo — Risk-neutral $\mathbb{Q}$	9.887 %
Monte Carlo — Forward measure	9.887 %
Monte Carlo — Annuity measure	9.886 %
Trinomial Tree	9.888 %

All four methods converge to ~9.89%, confirming the consistency of the implementation across measures and numerical schemes.

Repository Structure

rates-models/
├── notebooks/
│   └── hull_white_swaption_pricing.ipynb   # Full pipeline: calibration → pricing
├── data/
│   └── ESTR.csv                            # ECB Euro Short-Term Rate (daily)
└── README.md

Getting Started

Prerequisites

pip install numpy pandas scipy matplotlib

Run the notebook

jupyter notebook notebooks/hull_white_swaption_pricing.ipynb

The notebook reads data from ../data/ESTR.csv — keep the folder structure intact.

References

• Hull, J. & White, A. (1990). Pricing Interest-Rate Derivative Securities. The Review of Financial Studies.
• Brigo, D. & Mercurio, F. (2006). Interest Rate Models — Theory and Practice. Springer.
• ECB Statistical Data Warehouse — Euro Short-Term Rate (€STR): EST.B.EU000A2X2A25.WT