r"""
Hull and White: pricing CDO tranches without Monte Carlo
========================================================

A tranche of a synthetic CDO absorbs portfolio losses between its
attachment and detachment points. Hull and White (2004) priced tranches
in Li's one-factor Gaussian copula by conditioning on the common factor:
given it, defaults are independent and the pool loss is binomial, so the
expected tranche loss at each date is one integral over the factor.
Raising the correlation moves risk from the equity tranche, which loses
less, to the senior tranches, which lose more.
"""

# %%
import matplotlib.pyplot as plt
import numpy as np

from financialkit.credit import HazardCurve, expected_tranche_loss, tranche_spread
from financialkit.credit.visualizers import plot_tranche_losses

curve = HazardCurve.flat(0.012)
tranches = [(0.0, 0.03), (0.03, 0.07), (0.07, 0.10), (0.10, 0.15), (0.15, 0.30)]
res = [tranche_spread(a, d, curve, 0.25, 5.0, 0.03, n_names=125) for a, d in tranches]
ax = plot_tranche_losses(res)
ax.set_title("125 names, 5 years, copula correlation 0.25")

# %%
# Spreads against correlation
# ---------------------------

rhos = np.linspace(0.02, 0.9, 15)
fig, ax = plt.subplots(figsize=(7, 4))
for a, d in tranches:
    s = [tranche_spread(a, d, curve, rho, 5.0, 0.03).par_spread for rho in rhos]
    ax.semilogy(rhos, 1e4 * np.array(s), label=f"{100 * a:g}-{100 * d:g}%")
ax.set_xlabel("copula correlation")
ax.set_ylabel("par spread (bp, large pool)")
ax.legend()

# %%
# Finite pool versus the large-pool limit
# ---------------------------------------

p = float(curve.default_probability(5.0))
for n in (25, 125, 1000, None):
    el = expected_tranche_loss(0.03, 0.07, p, 0.25, n_names=n)
    print(f"{'LHP' if n is None else n:>5}: expected 3-7% tranche loss at 5y = {el:.4f}")

plt.show()