Credit Default Swap Pricing and Bootstrapping¶

Background¶

cds_pricing_bootstrapping.py

This module implements CDS Pricing and Bootstrapping.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" cds_pricing_bootstrapping.py

This module implements CDS Pricing and Bootstrapping.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

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

def cds_pricing_bootstrapping(): """ CDS Pricing and Bootstrapping.

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

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

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "CDS Pricing and Bootstrapping"
}

return results

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

# 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("CDS Pricing and Bootstrapping")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. A CDS has a notional of $10,000,000, a spread of 200 bps, quarterly payments, and a recovery rate of $40\%$. Compute the annual premium payment.

Solution to Exercise 1

The annual premium is $\text{Notional} \times \text{Spread} = 10{,}000{,}000 \times 0.02 = \$200{,}000$. With quarterly payments, each payment is $200{,}000/4 = \$50{,}000$ (approximately, ignoring day count conventions and accrual). The protection buyer pays $50,000 per quarter. In the event of default, the protection seller pays $(1 - R) \times N = 0.60 \times 10{,}000{,}000 = \$6{,}000{,}000$.

Exercise 2. CDS bootstrapping extracts hazard rates from market CDS spreads. For a 1-year CDS with spread $s_1 = 100$ bps, annual payment, risk-free rate $3\%$, and recovery $40\%$, compute the 1-year survival probability.

Solution to Exercise 2

The CDS par condition equates premium and protection legs. For the 1-year case:

\[ s_1 \cdot P(0,1) \cdot Q(1) = (1 - R) \cdot P(0,1) \cdot [1 - Q(1)], \]

where $Q(1)$ is the survival probability. Solving:

\[ 0.01 \times Q(1) = 0.60 \times [1 - Q(1)] \implies 0.01\,Q + 0.60\,Q = 0.60 \implies Q(1) = \frac{0.60}{0.61} \approx 0.9836. \]

The 1-year survival probability is approximately $98.36\%$, and the implied hazard rate is $h \approx -\ln(0.9836) \approx 1.66\%$.

Exercise 3. Explain why the recovery rate assumption significantly affects bootstrapped hazard rates and CDS prices.

Solution to Exercise 3

The protection leg payment is $(1-R) \times N$, so a higher recovery rate reduces the protection leg value for the same default probability. To match a given CDS spread, higher $R$ requires higher hazard rates (more likely default). Specifically, hazard rates scale approximately as $h \approx s/(1-R)$, so changing $R$ from $40\%$ to $20\%$ reduces implied hazard rates by roughly a factor of $0.75$. Since market CDS spreads are directly observable but recovery rates are estimated, the hazard rates are model-dependent on the recovery assumption.

Exercise 4. After bootstrapping hazard rates $h_1, h_2, h_3$ at years 1, 2, 3 from CDS spreads, how would you price a 2-year CDS with a non-standard spread?

Solution to Exercise 4

Using the bootstrapped survival curve $Q(t) = e^{-\int_0^t h(s)\,ds}$, compute:

Premium leg: $\text{PV}_{\text{prem}} = s \sum_{i=1}^{n} \tau_i P(0, T_i) Q(T_i)$
Protection leg: $\text{PV}_{\text{prot}} = (1-R) \sum_{i=1}^{n} P(0, T_i)[Q(T_{i-1}) - Q(T_i)]$

The mark-to-market value of the CDS is $\text{PV}_{\text{prot}} - \text{PV}_{\text{prem}}$. For a par CDS, $s$ is chosen so the value is zero. For a non-standard spread, the value is nonzero and represents the cost of entering the contract at off-market terms.