python中Montecarlo雪球定价的希腊字母计算

在Python中，可以使用NumPy和SciPy等库来进行Monte Carlo雪球定价的希腊字母计算。以下是一些常见希腊字母的计算示例：

1. Delta：

import numpy as np
from scipy.stats import norm

# 计算股票价格路径
def stock_price_path(S, r, sigma, T, N, M):
    dt = T / N
    prices = np.zeros((M, N+1))
    prices[:, 0] = S
    for i in range(1, N+1):
        eps = np.random.normal(0, 1, M)
        prices[:, i] = prices[:, i-1] * np.exp((r - 0.5*sigma**2)*dt + sigma*np.sqrt(dt)*eps)
    return prices

# 计算期权Delta
def option_delta(S, K, r, sigma, T, N, M):
    dt = T / N
    prices = stock_price_path(S, r, sigma, T, N, M)
    option_price = np.exp(-r*T) * np.maximum(prices[:, -1] - K, 0)
    delta = np.mean(option_price * (prices[:, 1] - prices[:, 0]) / (S * sigma * np.sqrt(dt)))
    return delta

# 示例
S = 100
K = 110
r = 0.05
sigma = 0.2
T = 1
N = 100
M = 10000
delta = option_delta(S, K, r, sigma, T, N, M)
print("Delta: ", delta)

2. Gamma：

# 计算期权Gamma
def option_gamma(S, K, r, sigma, T, N, M):
    dt = T / N
    prices = stock_price_path(S, r, sigma, T, N, M)
    option_price = np.exp(-r*T) * np.maximum(prices[:, -1] - K, 0)
    delta1 = np.mean(option_price * (prices[:, 1] - prices[:, 0]) / (S * sigma * np.sqrt(dt)))
    delta2 = np.mean(option_price * (prices[:, 2] - 2*prices[:, 1] + prices[:, 0]) / (S**2 * (dt**2)))
    gamma = (delta2 / delta1) * (1 / S)
    return gamma

# 示例
S = 100
K = 110
r = 0.05
sigma = 0.2
T = 1
N = 100
M = 10000
gamma = option_gamma(S, K, r, sigma, T, N, M)
print("Gamma: ", gamma)

3. Vega：

# 计算期权Vega
def option_vega(S, K, r, sigma, T, N, M):
    dt = T / N
    prices = stock_price_path(S, r, sigma, T, N, M)
    option_price = np.exp(-r*T) * np.maximum(prices[:, -1] - K, 0)
    vega = np.mean(option_price * prices[:, 1] * np.sqrt(dt) * norm.pdf((np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))))
    return vega

# 示例
S = 100
K = 110
r = 0.05
sigma = 0.2
T = 1
N = 100
M = 10000
vega = option_vega(S, K, r, sigma, T, N, M)
print("Vega: ", vega)

4. Theta：

# 计算期权Theta
def option_theta(S, K, r, sigma, T, N, M):
    dt = T / N
    prices = stock_price_path(S, r, sigma, T, N, M)
    option_price1 = np.exp(-r*T) * np.maximum(prices[:, -1] - K, 0)
    option_price2 = np.exp(-r*(T-dt)) * np.maximum(prices[:, -2] - K, 0)
    theta = np.mean((option_price2 - option_price1) / dt)
    return theta

# 示例
S = 100
K = 110
r = 0.05
sigma = 0.2
T = 1
N = 100
M = 10000
theta = option_theta(S, K, r, sigma, T, N, M)
print("Theta: ", theta)

以上仅为示例，实际应用中需要根据具体模型和期权类型进行相应的计算。