r"""
Merton's structural model
=========================

Merton (1974) treated a firm's equity as a European call on its assets
:math:`V`, struck at the face value :math:`D` of its zero-coupon debt.
Debt is then a riskless bond minus a put, the risk-neutral default
probability is :math:`N(-d_2)`, and the credit spread follows from
leverage and asset volatility alone. Hull's Example 24.3 backs out the
unobserved asset value and volatility from equity (3) and its
volatility (80%).
"""

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

from financialkit.credit import merton_calibrate, merton_model

m = merton_calibrate(E=3.0, sigma_E=0.80, D=10.0, T=1.0, r=0.05)
print(f"assets V = {m.asset_value:.2f}, sigma_V = {m.asset_volatility:.4f}")
print(f"debt = {m.debt:.2f}, PD = {m.default_probability:.1%}, spread = {1e4 * m.credit_spread:.0f} bp, recovery = {m.recovery_rate:.1%}")

# %%
# Payoffs at maturity
# -------------------

VT = np.linspace(0, 25, 300)
fig, ax = plt.subplots(figsize=(6, 4))
ax.plot(VT, np.maximum(VT - 10, 0), label="equity: max(V - D, 0)")
ax.plot(VT, np.minimum(VT, 10), label="debt: min(V, D)")
ax.axvline(10, color="0.5", ls=":")
ax.set_xlabel("asset value at maturity")
ax.legend()

# %%
# The term structure of credit spreads
# ------------------------------------
#
# Merton's spreads start at zero for very short maturities (assets cannot
# jump below the debt), rise, and for highly levered firms fall again
# because surviving firms drift away from default.

T = np.linspace(0.05, 10, 120)
fig, ax = plt.subplots(figsize=(7, 4))
for lev in (0.5, 0.7, 0.9, 1.1):
    s = [merton_model(100.0, 100 * lev * np.exp(0.05 * t), t, 0.05, 0.25).credit_spread for t in T]
    ax.plot(T, 1e4 * np.array(s), label=f"quasi-leverage {lev:.1f}")
ax.set_xlabel("maturity (years)")
ax.set_ylabel("credit spread (bp)")
ax.legend()

plt.show()