Calibration to Cap Volatilities¶

Background¶

hull_white_calibration_caps.py

This module implements Calibration to Cap Vols.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" hull_white_calibration_caps.py

This module implements Calibration to Cap Vols.

Author: Financial Math Library """

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

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

def hull_white_calibration_caps(): """ Calibration to Cap Vols.

This function demonstrates the key concepts and computational techniques
for calibration to cap vols.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Calibration to Cap Vols
print(f"Computing Calibration to Cap Vols...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Calibration to Cap Vols"
}

return results

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

# 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("Calibration to Cap Vols")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. Calibrating Hull-White to cap volatilities involves finding \(\eta\) and \(\lambda\) that minimize the difference between model and market cap prices. Write the objective function.

Solution to Exercise 1

The objective function is

\[ f(\eta, \lambda) = \sum_{i=1}^{N} w_i\bigl[V_i^{\text{model}}(\eta, \lambda) - V_i^{\text{market}}\bigr]^2, \]

where \(V_i^{\text{model}}\) is the Hull-White cap (or caplet) price for the \(i\)-th instrument, \(V_i^{\text{market}}\) is the market price, and \(w_i\) are weights (often \(1/V_i^{\text{market}}\) for relative fitting or \(1/(V_i^{\text{market}})^2\) for percentage error minimization). The model prices are computed using the Hull-White caplet formula derived from bond options.

Exercise 2. Explain the trade-off between fitting \(\eta\) as a constant versus a piecewise constant (time-dependent) function.

Solution to Exercise 2

Constant \(\eta\): Two parameters (\(\eta, \lambda\)). Simple, stable calibration but may not fit the entire cap volatility term structure accurately. Best when the term structure is relatively flat.
Piecewise constant \(\eta(t)\): One \(\eta\) value per caplet interval, plus \(\lambda\). Can exactly fit all cap prices (as many parameters as instruments). But may produce non-smooth volatility functions, unstable hedges, and overfit to market noise.

A common compromise is to use 3-5 piecewise constant segments for \(\eta(t)\) while keeping \(\lambda\) constant, balancing accuracy and stability.

Exercise 3. If the calibration produces \(\lambda = 0.03\) and \(\eta = 0.008\), compute the long-run standard deviation of the short rate and the implied cap volatility for a 1-year caplet.

Solution to Exercise 3

Long-run std: \(\sigma_\infty = \eta/\sqrt{2\lambda} = 0.008/\sqrt{0.06} = 0.008/0.2449 = 0.03266 = 3.27\%\).

For a 1-year caplet on \([T_1, T_2] = [1, 1.25]\): \(\sigma_P = \eta \cdot |B(1, 1.25)| \cdot \sqrt{(1 - e^{-0.06})/0.06}\). With \(B = (e^{-0.0075} - 1)/0.03 \approx -0.2498\):

\[ \sigma_P = 0.008 \times 0.2498 \times \sqrt{0.9709} = 0.001998 \times 0.9854 \approx 0.001969. \]

Converting to Black implied volatility requires additional computation, but the bond volatility is approximately \(0.20\%\), corresponding to a caplet implied vol of roughly \(15-20\%\) depending on the forward rate level.

Exercise 4. Describe common issues in Hull-White calibration to caps and how they are resolved.

Solution to Exercise 4

Non-uniqueness: Multiple \((\eta, \lambda)\) pairs can produce similar cap prices, especially when only ATM caps are used. Resolved by adding swaption prices as additional calibration targets, or by fixing \(\lambda\) from historical data.
Negative rates: If the calibrated \(\eta\) is large relative to the rate level, the model produces many negative rate scenarios. Resolved by using shifted Hull-White or checking that implied negative rate probabilities are reasonable.
Term structure mismatch: Constant \(\eta\) cannot fit both short-dated and long-dated caps simultaneously if the volatility term structure is non-monotonic. Resolved by using piecewise constant \(\eta(t)\).
Numerical instability: The cap pricing formula involves products of exponentials that can overflow for extreme parameters. Resolved by working in log-space and using bounded optimization.