期权定价模型：从随机过程到 Python 代码

import numpy as np
from scipy.stats import norm

def black_scholes_d1_d2(sigma, S0, K, T, r):
    """Return the two standardized distances used in the BSM formula."""
    d1 = (np.log(S0 / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)
    return d1, d2

def black_scholes_formula(sigma, S0, K, T, r, cp):
    """Return a no-dividend European call (cp=1) or put (cp=-1) value."""
    if cp not in (-1, 1):
        raise ValueError("cp must be 1 for a call or -1 for a put")
    d1, d2 = black_scholes_d1_d2(sigma, S0, K, T, r)
    discount = np.exp(-r * T)
    return cp * (S0 * norm.cdf(cp * d1) - K * discount * norm.cdf(cp * d2))

# 页面演示调用：cp=1 表示看涨，cp=-1 表示看跌
for sigma in (0.10, 0.20, 0.40):
    price = black_scholes_formula(sigma, 100, 100, 1, 0.05, cp=1)
    print(f"σ={sigma:.0%}  C={price:.4f}")