在美国，现货价格为20美元，行使价X为21美元。 如果您愿意，可以使用符号K作为执行价格，它与x相同。无风险利率为每年12%，股票价格的波动性σ为每年20%。根据上边的信息生成出二叉树期权定价的C++代码生成出

以下是基于二叉树定价模型的期权定价C++代码：

#include <iostream>
#include <cmath>
using namespace std;

double max(double a, double b) {
    return (a > b) ? a : b;
}

double binomialTreeOptionPrice(double S, double X, double r, double sigma, double T, int N, char optType) {
    double deltaT = T / N;
    double up = exp(sigma * sqrt(deltaT));
    double down = exp(-sigma * sqrt(deltaT));
    double pUp = (exp(r * deltaT) - down) / (up - down);
    double pDown = 1 - pUp;

    // Create and initialize the stock price tree
    double stockPrice[N + 1][N + 1];
    stockPrice[0][0] = S;
    for (int i = 1; i <= N; i++) {
        stockPrice[i][0] = stockPrice[i - 1][0] * up;
        for (int j = 1; j <= i; j++) {
            stockPrice[i][j] = stockPrice[i - 1][j - 1] * down;
        }
    }

    // Create and initialize the option price tree
    double optionPrice[N + 1][N + 1];
    for (int j = 0; j <= N; j++) {
        if (optType == 'C') {
            optionPrice[N][j] = max(stockPrice[N][j] - X, 0.0);
        } else {
            optionPrice[N][j] = max(X - stockPrice[N][j], 0.0);
        }
    }

    // Traverse the tree backwards to calculate option price at t=0
    for (int i = N - 1; i >= 0; i--) {
        for (int j = 0; j <= i; j++) {
            if (optType == 'C') {
                optionPrice[i][j] = max(stockPrice[i][j] - X, exp(-r * deltaT) * (pUp * optionPrice[i + 1][j + 1] + pDown * optionPrice[i + 1][j]));
            } else {
                optionPrice[i][j] = max(X - stockPrice[i][j], exp(-r * deltaT) * (pUp * optionPrice[i + 1][j + 1] + pDown * optionPrice[i + 1][j]));
            }
        }
    }

    // Return the option price at t=0
    return optionPrice[0][0];
}

int main() {
    double S = 20; // Underlying stock price
    double X = 21; // Strike price
    double r = 0.12; // Risk-free interest rate
    double sigma = 0.20; // Volatility of underlying stock
    double T = 1.0; // Time to expiration of option
    int N = 100; // Number of time steps in binomial tree
    char optType = 'C'; // Option type: 'C' for call option, 'P' for put option

    double optionPrice = binomialTreeOptionPrice(S, X, r, sigma, T, N, optType);
    cout << "Option price: " << optionPrice << endl;

    return 0;
}

这段代码实现了一个基于二叉树的期权定价模型，用于计算欧式看涨/看跌期权的价格。其中，输入参数包括标的资产价格、执行价格、无风险利率、资产波动率、剩余到期时间和期权类型。输出结果为期权价格。