r"""
Stoll's put-call parity
=======================

Hans Stoll (1969) showed that a European call, a put and the stock are
tied together with no model at all: buying a call and selling a put
with the same strike and maturity replicates a forward, so

.. math:: c - p = S e^{-qT} - K e^{-rT}.

A pair that breaks the relation is an arbitrage: the conversion (long
stock, long put, short call) locks in the residual.
"""

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

from financialkit.options import black_scholes_price, parity_residual, put_call_parity

# %%
# Parity holds strike by strike, whatever the volatility
# ------------------------------------------------------

S, T, r, q = 100.0, 0.5, 0.05, 0.02
K = np.linspace(70, 130, 7)
for sigma in (0.1, 0.3, 0.8):
    c = black_scholes_price(S, K, T, r, sigma, q)
    p = black_scholes_price(S, K, T, r, sigma, q, option_type="put")
    print(f"sigma = {sigma:.1f}: max |c - p - (S e^-qT - K e^-rT)| = {np.max(np.abs(parity_residual(c, p, S, K, T, r, q))):.1e}")

# %%
# Exploiting a mispriced put
# --------------------------
#
# Quote the put 0.40 too cheap and run a reversal: buy the put, sell the
# call, buy the stock and borrow the present value of the strike. At
# expiry the position is worth exactly zero in every state, so its
# initial cash flow is pure profit.

c = black_scholes_price(S, 100.0, T, r, 0.25, q)
p_fair = put_call_parity(c, S, 100.0, T, r, q)
p_quoted = p_fair - 0.40
profit = c - p_quoted - S * np.exp(-q * T) + 100.0 * np.exp(-r * T)
print(f"fair put {p_fair:.4f}, quoted {p_quoted:.4f}, locked-in profit {profit:.4f}")

S_T = np.linspace(50, 150, 201)
legs = {
    "long put": np.maximum(100.0 - S_T, 0.0),
    "short call": -np.maximum(S_T - 100.0, 0.0),
    "long stock": S_T,
    "repay loan": -100.0 * np.ones_like(S_T),
}
fig, ax = plt.subplots(figsize=(7, 4))
for name, value in legs.items():
    ax.plot(S_T, value, lw=1, label=name)
ax.plot(S_T, sum(legs.values()), "k", lw=2.5, label="total: zero in every state")
ax.set_xlabel("stock price at expiry")
ax.set_ylabel("value at expiry")
ax.legend()
fig.tight_layout()

plt.show()