77.2 波动率策略设计

import numpy as np
from scipy.stats import norm

def black_scholes(S, K, T, r, sigma, option_type='call'):
    """
    Black-Scholes期权定价
    S: 标的资产价格
    K: 行权价
    T: 到期时间（年）
    r: 无风险利率
    sigma: 波动率
    """
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)

    if option_type == 'call':
        price = S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
    else:
        price = K*np.exp(-r*T)*norm.cdf(-d2) - S*norm.cdf(-d1)

    return price

def calculate_greeks(S, K, T, r, sigma):
    """
    计算Greeks
    """
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    N_d1 = norm.cdf(d1)
    N_d1_prime = norm.pdf(d1)

    # Delta
    delta = N_d1

    # Gamma
    gamma = N_d1_prime / (S * sigma * np.sqrt(T))

    # Theta
    theta = -(S * N_d1_prime * sigma) / (2 * np.sqrt(T)) - r * K * np.exp(-r*T) * norm.cdf(d2)

    # Vega
    vega = S * N_d1_prime * np.sqrt(T)

    # Rho
    rho = K * T * np.exp(-r*T) * norm.cdf(d2)

    return {'delta': delta, 'gamma': gamma, 'theta': theta, 'vega': vega, 'rho': rho}