r"""
Goldman-Sosin-Gatto floating lookbacks
======================================

"Buy at the low, sell at the high": Barry Goldman, Howard Sosin and
Mary Ann Gatto (1979) priced the option whose strike is the lowest
price over its life, :math:`S_T - \min_t S_t` (or the highest, for the
put). It is never out of the money, so it costs much more than a
vanilla call. The joint law of a Brownian motion and its minimum gives
the closed form; Monte Carlo can match it exactly only by sampling each
step's minimum from the Brownian bridge, because checking the price at
discrete dates misses the true low.
"""

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

from financialkit.exotic_options import floating_lookback_price, lookback_option_mc
from financialkit.options import black_scholes_price

S, T, r, sigma = 100.0, 1.0, 0.05, 0.25
print(f"Haug (2007), 4.15.1: {floating_lookback_price(120.0, 0.5, 0.10, 0.30, q=0.06, S_extreme=100.0):.4f}")

# %%
# Lookback against at-the-money vanilla options
# ---------------------------------------------

vols = np.linspace(0.05, 0.6, 40)
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].plot(vols, floating_lookback_price(S, T, r, vols), label="floating lookback call")
axes[0].plot(vols, floating_lookback_price(S, T, r, vols, option_type="put"), label="floating lookback put")
axes[0].plot(vols, black_scholes_price(S, S, T, r, vols), "k--", label="ATM call")
axes[0].set_xlabel(r"volatility $\sigma$")
axes[0].set_ylabel("value")
axes[0].legend()

# %%
# Discrete monitoring misses the minimum; the bridge does not
# -----------------------------------------------------------

steps = np.array([2, 4, 8, 16, 32, 64])
exact = floating_lookback_price(S, T, r, sigma)
for continuous, label in ((False, "minimum over the grid"), (True, "bridge-sampled minimum")):
    res = [lookback_option_mc(S, T, r, sigma, n_steps=int(m), n_paths=20_000, continuous=continuous, rng=int(m)) for m in steps]
    axes[1].errorbar(steps, [x.estimate for x in res], yerr=[2 * x.std_error for x in res], fmt="o-", capsize=3, label=label)
axes[1].axhline(exact, color="k", lw=1, label="closed form")
axes[1].set_xscale("log", base=2)
axes[1].set_xlabel("time steps")
axes[1].set_ylabel("floating lookback call")
axes[1].legend()
fig.tight_layout()
plt.show()