r"""
Kyle's insider and the price impact lambda
==========================================

Albert Kyle (1985) placed an insider who knows the liquidation value
:math:`v \sim N(p_0, \sigma_v^2)` in a market with noise traders whose
orders :math:`u \sim N(0, \sigma_u^2)` camouflage it. A competitive
market maker sees only the total flow :math:`y = x + u` and sets
:math:`p = E[v \mid y] = p_0 + \lambda y`. In equilibrium

.. math::

   \lambda = \frac{\sigma_v}{2\sigma_u}, \qquad x = \frac{\sigma_u}{\sigma_v}(v - p_0),

so the market's depth :math:`1/\lambda` rises with noise trading, the
insider earns :math:`\sigma_v\sigma_u/2` at the noise traders' expense,
and the price reveals exactly half of the insider's information.
"""

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

from financialkit.market_microstructure import kyle_equilibrium, kyle_lambda, simulate_kyle

eq = kyle_equilibrium(sigma_v=2.0, sigma_u=1.0, p0=50.0)
print(f"lambda {eq.lam}, beta {eq.beta}, insider profit {eq.insider_profit}, Var(v | p) {eq.posterior_variance}")

# %%
# Order flow moves the price
# --------------------------
#
# Regressing the value on order flow across many auctions recovers the
# market maker's lambda: the price is the best linear forecast of the
# value.

v, y, p = simulate_kyle(eq, 3000, rng=7)
fit = kyle_lambda(v - eq.p0, y)
fig, ax = plt.subplots(figsize=(6, 4))
ax.plot(y, v, ".", ms=2, color="0.6", label="value $v$")
ax.plot(np.sort(y), eq.p0 + eq.lam * np.sort(y), "C3", label=f"price, $\\lambda$ = {eq.lam}")
ax.set_xlabel("total order flow $y$")
ax.set_ylabel("price / value")
ax.set_title(f"fitted slope {fit.coef[1]:.3f} ± {fit.std_errors[1]:.3f}")
ax.legend()
fig.tight_layout()

# %%
# The insider's best response
# ---------------------------
#
# Given lambda, trading :math:`\beta(v - p_0)` earns
# :math:`\beta\sigma_v^2(1 - \lambda\beta)` on average, maximised at the
# equilibrium :math:`\beta = 1/(2\lambda)`: trading harder moves the price
# against itself.

betas = np.linspace(0.05, 1.0, 20)
sim_profit = [np.mean(b * (vv - eq.p0) * (vv - pp)) for b in betas for vv, _, pp in [simulate_kyle(eq, 4000, beta=b, rng=8)]]
fig, ax = plt.subplots(figsize=(6, 3.5))
ax.plot(betas, betas * eq.sigma_v**2 * (1 - eq.lam * betas), "k-", label="expected profit")
ax.plot(betas, sim_profit, "o", ms=4, color="C0", label="simulated")
ax.axvline(eq.beta, color="C3", ls="--", label="equilibrium $\\beta$")
ax.set_xlabel(r"insider intensity $\beta$")
ax.set_ylabel("insider profit")
ax.legend()
fig.tight_layout()

# %%
# More noise, deeper market
# -------------------------

sigma_u = np.linspace(0.25, 4, 50)
fig, ax = plt.subplots(figsize=(6, 3.5))
ax.plot(sigma_u, [kyle_equilibrium(2.0, s).lam for s in sigma_u], label=r"$\lambda$")
ax.plot(sigma_u, [kyle_equilibrium(2.0, s).insider_profit for s in sigma_u], label="insider profit")
ax.set_xlabel(r"noise trading $\sigma_u$")
ax.legend()
fig.tight_layout()

plt.show()