O’Kane and Turnbull: the CDS par spread

A credit default swap exchanges a running premium for a payment of \(1 - R\) at default. O’Kane and Turnbull (2003) wrote both legs as integrals over the default density and evaluated them on a grid with defaults at mid-step, including the premium accrued between the last payment date and default. The par spread equates the legs; for a flat hazard and a continuously paid premium it is exactly \(\lambda(1 - R)\).

import matplotlib.pyplot as plt
import numpy as np

from financialkit.credit import HazardCurve, cds_legs, cds_par_spread

res = cds_legs(HazardCurve.flat(0.02), 5.0, 0.05, 0.4, frequency=1, steps_per_year=1)
print("Hull's annual setting (hazard 2%, R = 40%, r = 5%):")
print(f"  premium annuity {res.rpv01 - res.accrued_on_default:.4f}, accrual {res.accrued_on_default:.4f}, protection {res.protection_leg:.4f}")
print(f"  par spread {1e4 * res.par_spread:.1f} bp  vs  credit triangle {1e4 * 0.02 * 0.6:.0f} bp")

Hull's annual setting (hazard 2%, R = 40%, r = 5%):
  premium annuity 4.0728, accrual 0.0422, protection 0.0506
  par spread 123.0 bp  vs  credit triangle 120 bp

Payment frequency and the credit triangle

More frequent premiums and accrual on default converge to the triangle.

freqs = [1, 2, 4, 12, 52]
fig, ax = plt.subplots(figsize=(7, 4))
for h in (0.01, 0.03, 0.06):
    s = [cds_par_spread(HazardCurve.flat(h), 5.0, 0.05, 0.4, frequency=f, steps_per_year=max(f, 12)) for f in freqs]
    ax.semilogx(freqs, 1e4 * np.array(s) / (1e4 * h * 0.6), "o-", label=f"hazard {h:.0%}")
ax.axhline(1.0, color="k", ls=":")
ax.set_xlabel("premium payments per year")
ax.set_ylabel("par spread / lambda (1 - R)")
ax.legend()

<matplotlib.legend.Legend object at 0x11612ba10>

The term structure of par spreads

curve = HazardCurve([1, 3, 5, 10], [0.005, 0.012, 0.02, 0.03])
mats = np.linspace(0.5, 10, 20)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(mats, [1e4 * cds_par_spread(curve, T, 0.03) for T in mats], "o-", label="par spread")
ax.step([0, 1, 3, 5, 10], 1e4 * 0.6 * np.r_[0.005, curve.hazards], where="pre", label="hazard x (1 - R)")
ax.set_xlabel("CDS maturity")
ax.set_ylabel("bp")
ax.legend()

plt.show()