Zero-Coupon Bond Pricing¶

Background¶

hull_white_zcb_pricing.py

This module implements Zero-Coupon Bond Pricing.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" hull_white_zcb_pricing.py

This module implements Zero-Coupon Bond Pricing.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

======================================================================¶

def hull_white_zcb_pricing(): """ Zero-Coupon Bond Pricing.

This function demonstrates the key concepts and computational techniques
for zero-coupon bond pricing.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Zero-Coupon Bond Pricing
print(f"Computing Zero-Coupon Bond Pricing...")

# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Zero-Coupon Bond Pricing"
}

return results

def main(): """Main execution function.""" results = hull_white_zcb_pricing()

# Create visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(results["time_points"], "b-", linewidth=2)
ax.set_xlabel("Time")
ax.set_ylabel("Value")
ax.set_title("Zero-Coupon Bond Pricing")
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("/tmp/hull_white_zcb_pricing.png", dpi=150)
print(f"Figure saved to /tmp/hull_white_zcb_pricing.png")
plt.close()

return results

if name == "main": main() ```

Exercises¶

Exercise 1. The Hull-White ZCB price is \(P(t, T) = A(t,T)\,e^{-B(t,T)\,r(t)}\). Derive the expression for \(B(t,T)\) from the ODE \(B'(\tau) = 1 - \lambda B(\tau)\), \(B(0) = 0\).

Exercise 2. For \(\lambda = 0.1\) and \(\eta = 0.015\), compute \(B(0, 5)\) and \(B(0, 30)\). Interpret the saturation of \(B\) for long maturities.

Exercise 3. Compute the ZCB price \(P(0, 10)\) if \(r(0) = 3\%\), \(A(0,10) = 0.72\), and \(B(0,10) = 6.32\).

Exercise 4. Explain why the \(A(t,T)\) coefficient depends on the entire initial yield curve while \(B(t,T)\) depends only on \(\lambda\).