Python 選擇權入門：用 Black-Scholes 計算理論價格與 Greeks（附完整程式碼）

import numpy as np
from scipy.stats import norm

def black_scholes_price(S, K, T, r, sigma, option_type="call"):
    """
    S: 標的價格（假設範例）
    K: 履約價（假設範例）
    T: 到期時間（年），假設範例
    r: 無風險利率（假設範例）
    sigma: 年化波動率（假設範例）
    option_type: "call" 或 "put"
    """
    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)
    elif option_type == "put":
        price = K * np.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1)
    else:
        raise ValueError("option_type 必須是 'call' 或 'put'")

    return price

# 假設範例：標的價格 100、履約價 100、剩餘 0.5 年、無風險利率 2%、波動率 20%
S, K, T, r, sigma = 100, 100, 0.5, 0.02, 0.20

call_price = black_scholes_price(S, K, T, r, sigma, "call")
put_price = black_scholes_price(S, K, T, r, sigma, "put")

print(f"假設範例 Call 理論價格: {call_price:.4f}")
print(f"假設範例 Put 理論價格: {put_price:.4f}")

import numpy as np
from scipy.stats import norm

def black_scholes_greeks(S, K, T, r, sigma, option_type="call"):
    d1 = (np.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    # Gamma 與 Vega 對 call/put 是相同的
    gamma = norm.pdf(d1) / (S * sigma * np.sqrt(T))
    vega = S * norm.pdf(d1) * np.sqrt(T) / 100  # 每 1% 波動率變動的影響

    if option_type == "call":
        delta = norm.cdf(d1)
        theta = (
            -S * norm.pdf(d1) * sigma / (2 * np.sqrt(T))
            - r * K * np.exp(-r * T) * norm.cdf(d2)
        ) / 365  # 轉換成每日 theta
    elif option_type == "put":
        delta = norm.cdf(d1) - 1
        theta = (
            -S * norm.pdf(d1) * sigma / (2 * np.sqrt(T))
            + r * K * np.exp(-r * T) * norm.cdf(-d2)
        ) / 365
    else:
        raise ValueError("option_type 必須是 'call' 或 'put'")

    return {"delta": delta, "gamma": gamma, "theta": theta, "vega": vega}

# 承上假設範例參數
S, K, T, r, sigma = 100, 100, 0.5, 0.02, 0.20
greeks_call = black_scholes_greeks(S, K, T, r, sigma, "call")

for name, value in greeks_call.items():
    print(f"假設範例 Call {name}: {value:.6f}")

import numpy as np
from scipy.stats import norm
from scipy.optimize import brentq

def bs_price_for_iv(sigma, S, K, T, r, market_price, option_type="call"):
    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 - market_price

# 假設範例：市場上這張 call 的報價是 6.5
S, K, T, r, market_price = 100, 100, 0.5, 0.02, 6.5

implied_vol = brentq(
    bs_price_for_iv, 0.001, 3.0,
    args=(S, K, T, r, market_price, "call")
)
print(f"假設範例隱含波動率: {implied_vol:.4%}")

import numpy as np
import matplotlib.pyplot as plt

K = 100           # 假設範例履約價
premium = 6.5     # 假設範例權利金
spot_range = np.linspace(70, 130, 200)

payoff_call_buy = np.maximum(spot_range - K, 0) - premium

plt.figure(figsize=(8, 5))
plt.plot(spot_range, payoff_call_buy, label="買進 Call 到期損益（假設範例）")
plt.axhline(0, color="black", linewidth=0.8)
plt.xlabel("到期標的價格（假設範例）")
plt.ylabel("損益")
plt.legend()
plt.title("選擇權到期損益圖（假設範例參數）")
plt.show()