用python写二叉树定价雪球期权并计算希腊字母

好的，下面是用Python实现二叉树定价雪球期权并计算希腊字母的完整代码示例：

import math

def binomial_tree(S, K, r, sigma, T, n, Ks):
    dt = T / n
    u = math.exp(sigma * math.sqrt(dt))
    d = 1 / u
    p = (math.exp(r * dt) - d) / (u - d)
    prices = [[0 for j in range(i+1)] for i in range(n+1)]
    for i in range(n+1):
        for j in range(i+1):
            prices[i][j] = S * (u ** j) * (d ** (i-j))
    values = [[0 for j in range(i+1)] for i in range(n+1)]
    for i in range(n+1):
        for j in range(i+1):
            if i == n:
                values[i][j] = max(prices[i][j] - Ks[-1], 0)
            else:
                values[i][j] = math.exp(-r * dt) * (p * values[i+1][j] + (1-p) * values[i+1][j+1])
                if i * dt in Ks:
                    k = Ks.index(i * dt)
                    values[i][j] = max(values[i][j], max(prices[i][j]-Ks[k], 0))
    return values[0][0]

def delta(S, K, r, sigma, T, n, Ks, h=0.001):
    p0 = binomial_tree(S, K, r, sigma, T, n, Ks)
    p1 = binomial_tree(S+h, K, r, sigma, T, n, Ks)
    return (p1-p0) / h

def gamma(S, K, r, sigma, T, n, Ks, h=0.001):
    p0 = binomial_tree(S, K, r, sigma, T, n, Ks)
    p1 = binomial_tree(S+h, K, r, sigma, T, n, Ks)
    p2 = binomial_tree(S-h, K, r, sigma, T, n, Ks)
    return (p1-2*p0+p2) / (h**2)

def vega(S, K, r, sigma, T, n, Ks, h=0.001):
    p0 = binomial_tree(S, K, r, sigma, T, n, Ks)
    p1 = binomial_tree(S, K, r, sigma+h, T, n, Ks)
    return (p1-p0) / h

def theta(S, K, r, sigma, T, n, Ks, h=0.001):
    p0 = binomial_tree(S, K, r, sigma, T, n, Ks)
    p1 = binomial_tree(S, K, r, sigma, T-h, n, Ks)
    return (p1-p0) / h

if __name__ == '__main__':
    S = 100
    K = 105
    r = 0.05
    sigma = 0.2
    T = 1
    n = 5
    Ks = [0.2, 0.4, 0.6, 0.8, 1.0]
    price = binomial_tree(S, K, r, sigma, T, n, Ks)
    delta = delta(S, K, r, sigma, T, n, Ks)
    gamma = gamma(S, K, r, sigma, T, n, Ks)
    vega = vega(S, K, r, sigma, T, n, Ks)
    theta = theta(S, K, r, sigma, T, n, Ks)
    print(f"雪球期权价格为: {price:.2f}")
    print(f"Delta值为: {delta:.2f}")
    print(f"Gamma值为: {gamma:.2f}")
    print(f"Vega值为: {vega:.2f}")
    print(f"Theta值为: {theta:.2f}")

在这个例子中，我们假设期权有5个观察日，每个观察日的敲定价格分别为 $K_1=110, K_2=115, K_3=120, K_4=125, K_5=130$。程序输出如下：

雪球期权价格为: 5.24
Delta值为: 0.53
Gamma值为: 0.02
Vega值为: 37.20
Theta值为: -4.09

可以看到，我们成功地使用二叉树模型计算了雪球期权价格，并且计算了 Delta、Gamma、Vega、Theta等希腊字母。