README.md

---
title: Hull-White Simulator
emoji: 📊
colorFrom: blue
colorTo: indigo
sdk: gradio
sdk_version: 5.31.0
app_file: app.py
pinned: false
license: mit
tags:
- actuarial
- finance
- stochastic-models
- monte-carlo
- interest-rates
- quantitative-finance
- gradio
- dashboard
- hull-white
- risk-management
short_description: Simulate Hull-White interest rate paths and dynamics.
---

# 📊 Hull-White Interest Rate Model Dashboard

An interactive web dashboard for exploring the Hull-White short rate model, designed specifically for actuaries and financial professionals.

[![Open in Spaces](https://huggingface.co/datasets/huggingface/badges/raw/main/open-in-hf-spaces-sm.svg)](https://huggingface.co/spaces/alidenewade/hull-white-simulator)

## 🎯 Overview

The Hull-White model is a widely-used short rate model in quantitative finance, particularly valuable for:
- **Interest rate derivatives pricing**
- **Risk management and ALM**
- **Solvency II capital calculations**
- **Insurance liability valuation**

This dashboard provides an intuitive interface to explore the model's behavior through Monte Carlo simulations.

## 📈 Model Description

The Hull-White model follows the stochastic differential equation:

$$dr(t) = (θ(t) - ar(t))dt + σdW$$


Where:
- `r(t)` = instantaneous short rate at time t
- `a` = mean reversion speed parameter
- `σ` = volatility parameter  
- `θ(t)` = time-dependent drift function
- `dW` = Wiener process increment

## 🚀 Features

### Interactive Visualizations
- **📊 Short Rate Paths**: Visualize multiple simulated interest rate trajectories
- **📉 Mean Convergence**: Compare Monte Carlo means against theoretical expectations
- **📈 Variance Analysis**: Examine variance convergence properties
- **💰 Discount Factors**: Analyze zero-coupon bond pricing convergence
- **🔍 Parameter Sensitivity**: Study the critical σ/a ratio effects
- **📋 Statistics Table**: Summary statistics at key time points

### Adjustable Parameters
| Parameter | Range | Description |
|-----------|-------|-------------|
| Scenarios | 100 - 10,000 | Number of Monte Carlo paths |
| Time Horizon | 5 - 50 years | Simulation time length |
| Time Steps | 100 - 500 | Discretization granularity |
| Mean Reversion (a) | 0.01 - 0.5 | Speed of mean reversion |
| Volatility (σ) | 0.01 - 0.3 | Interest rate volatility |
| Initial Rate (r₀) | 0.01 - 0.15 | Starting interest rate |

## 🎛️ How to Use

1. **Adjust Model Parameters**: Use the sliders in the left panel to modify Hull-White parameters
2. **Explore Visualizations**: Click through the tabs to see different aspects of the model
3. **Analyze Convergence**: Pay special attention to the σ/a ratio - values > 1 show poor convergence
4. **Compare Theory vs Practice**: Observe how simulated results converge to theoretical expectations
5. **Generate Statistics**: Review the summary table for quantitative analysis

## 📊 Key Insights

### Convergence Properties
- **σ/a < 1**: Good Monte Carlo convergence
- **σ/a ≈ 1**: Moderate convergence issues  
- **σ/a > 1**: Poor convergence, especially for discount factors

### Practical Considerations
- **More scenarios** improve convergence but increase computation time
- **Higher volatility** requires more scenarios for stable results
- **Longer time horizons** show more pronounced convergence issues

## 🔧 Technical Implementation

### Model Features
- **Gaussian Process**: Exploits Hull-White's analytical properties
- **Conditional Moments**: Uses exact conditional mean and variance formulas
- **Vector Operations**: Efficient numpy-based simulations
- **Reproducible Results**: Fixed random seed for consistency

### Performance Optimized
- Real-time parameter updates
- Efficient matrix operations
- Responsive visualization updates
- Memory-efficient data handling

## 📚 Educational Value

Perfect for:
- **University Finance Courses**: Teaching stochastic interest rate models
- **Actuarial Training**: Understanding ALM and risk management
- **Professional Development**: Exploring quantitative finance concepts
- **Model Validation**: Testing parameter sensitivity and convergence

## 🎓 Theoretical Background

The implementation follows established literature:
- **Brigo & Mercurio**: Interest Rate Models - Theory and Practice
- **Glasserman**: Monte Carlo Methods in Financial Engineering
- **Hull**: Options, Futures, and Other Derivatives

### Key Mathematical Properties
- **Mean**: E[r(t)|ℱₛ] = r(s)e^(-a(t-s)) + α(t) - α(s)e^(-a(t-s))
- **Variance**: Var[r(t)|ℱₛ] = (σ²/2a)(1 - e^(-2a(t-s)))
- **Alpha Function**: α(t) = f^M(0,t) + (σ²/2a²)(1-e^(-at))²

## 🛠️ Installation & Deployment

### Local Development
```bash
# Clone the repository
git clone https://github.com/alidenewade/hull-white-dashboard.git
cd hull-white-dashboard

# Install dependencies
pip install -r requirements.txt

# Run the application
python app.py