Swaption Pricing (Black's Formula)¶

Background¶

swaption_pricing_black.py

This module implements Swaption Pricing.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" swaption_pricing_black.py

This module implements Swaption Pricing.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

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

def swaption_pricing_black(): """ Swaption Pricing.

This function demonstrates the key concepts and computational techniques
for swaption pricing.

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

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Swaption Pricing"
}

return results

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

# 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("Swaption Pricing")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. A European payer swaption gives the right to enter a payer swap (pay fixed, receive floating) at expiry $T$. Using Black's formula, write the swaption price in terms of the forward swap rate $S_0$, strike $K$, annuity $A_0$, and Black volatility $\sigma_{\text{Black}}$.

Solution to Exercise 1

The payer swaption price is

\[ V_{\text{payer}} = A_0\bigl[S_0\,\mathcal{N}(d_1) - K\,\mathcal{N}(d_2)\bigr], \]

where

\[ d_1 = \frac{\ln(S_0/K) + \frac{1}{2}\sigma_{\text{Black}}^2 T}{\sigma_{\text{Black}}\sqrt{T}}, \qquad d_2 = d_1 - \sigma_{\text{Black}}\sqrt{T}, \]

and $A_0 = \sum_{i=1}^n \tau_i\,P(0, T_i)$ is the present value of the swap annuity.

Exercise 2. Compute the price of a 2-year into 5-year payer swaption with $S_0 = 4\%$, $K = 4\%$ (ATM), $\sigma = 25\%$, $A_0 = 4.5$, and notional $10,000,000.

Solution to Exercise 2

Since $S_0 = K$ (ATM), $\ln(S_0/K) = 0$:

\[ d_1 = \frac{0 + 0.5 \times 0.0625 \times 2}{0.25\sqrt{2}} = \frac{0.0625}{0.3536} = 0.1768. \]

\[ d_2 = 0.1768 - 0.3536 = -0.1768. \]

\[ \Phi(0.1768) = 0.5701, \quad \Phi(-0.1768) = 0.4299. \]

\[ V = 10{,}000{,}000 \times 4.5 \times [0.04 \times 0.5701 - 0.04 \times 0.4299] = 45{,}000{,}000 \times 0.04 \times 0.1402. \]

\[ V = 45{,}000{,}000 \times 0.005608 = \$252{,}360. \]

Exercise 3. Explain payer-receiver swaption parity. If the payer swaption costs $252,360 and the forward swap rate equals the strike, what is the receiver swaption price?

Solution to Exercise 3

Payer-receiver parity states

\[ V_{\text{payer}}(K) - V_{\text{receiver}}(K) = A_0 \times (S_0 - K) \times N. \]

When $S_0 = K$ (ATM), $V_{\text{payer}} = V_{\text{receiver}}$. Therefore the receiver swaption also costs $\$252{,}360$.

Exercise 4. The swaption volatility cube is indexed by expiry, swap tenor, and strike. Explain what the volatility smile/skew looks like for swaptions and how it differs from equity options.

Solution to Exercise 4

For swaptions, the implied volatility surface typically shows:

Normal vol (basis point vol): More commonly used than lognormal vol for interest rates. The normal vol smile is often relatively flat, with a slight asymmetry.
Lognormal vol (Black vol): Shows a pronounced downward skew at low strikes (reflecting the possibility of near-zero or negative rates) and can exhibit a smile for very out-of-the-money options.

Unlike equity options (which show a persistent downward skew due to leverage effects and crash risk), swaption skews are more symmetric and depend heavily on the rate environment. In low-rate environments, Black vol skews are steep because lognormal vol diverges as rates approach zero, while normal vol remains well-behaved.