C++ 几何布朗运动的模拟

背景信息：

考虑伪代码：

问题：

我似乎对算法1伪代码的S上标j部分有问题，我的教授说，由于N=10000和N=10，我计算的是10。以下是我的代码中有问题的部分：

double Z[n][N];
    for(int j = 0; j < N; j++){
        for(int i = 0; i < n/2; i++){
            Z[2*i][j] = Box_MullerX(n,N,shifted_hs[i],shifted_hs[i+1]);
            Z[2*i+1][j] = Box_MullerY(n,N,shifted_hs[i],shifted_hs[i+1]);
        }
    }

    /*for(int j = 0; j < N; j++){
        for(int i = 0; i < n; i++){
            cout << Z[j][i] << " ";
        }
        cout << endl;
    }*/

    double S[N][n];
    S[0][0] = S0;
    double t[n+1];
    for(int i = 0; i <= n; i++){
        t[i] = (double)i*T/(n-1);
    }
    for(int j = 1; j <= N; j++){
        for(int i = 1; i <= n; i++){
            S[j][i] = S[j-1][i-1]*exp((mu - sigma/2)*(t[i] - t[i-1]) + sqrt(sigma)*sqrt(t[i] - t[i-1])*Z[j][i]);
        }
    }

double Z[n][n]；
对于（int j=0；j请解释你期望得到什么和得到什么。这有助于理解可能的变异

我注意到你在代码中混合了整数值和双数值，这不是好的做法。
另外-标记Setchell右-检查你的sigma值。我认为它应该是平方。并尝试正确设置
S
的初始值

我试图以我理解的方式修复代码中的错误。由于编译器版本较旧，代码发生了一些变化

#include <iostream>
#include <cmath>
#include <math.h>
#include <random>

using namespace std;

const double M_PI = 3.1415926535897932384626433832795;
const double M_SQRT1_2 = 0.70710678118654752440084436210485;

double phi(long double x);
double Black_Scholes(double T, double K, double S0, double r, double sigma);
double Halton_Seq(int index, double base);
double Box_MullerX(double u_1, double u_2);
double Box_MullerY(double u_1, double u_2);
double Random_Shift_Halton(int n,int N,double mu, double sigma, double T, double S0);


double erfc(double v) {
    double y = 1.0 / ( 1.0 + 0.3275911 * (v >= 0 ? v : -v));
    y = 1 - (((((
        + 1.061405429  * y
        - 1.453152027) * y
        + 1.421413741) * y
        - 0.284496736) * y
        + 0.254829592) * y)
        * exp (-v * v);
    return (v >= 0 ? y : -y);
}

double phi(double x){
    return 0.5 * erfc(-x * M_SQRT1_2);
}

double Black_Scholes(double T, double K, double S0, double r, double sigma){
    double d_1 = (log(S0/K) + (r+sigma*sigma/2.)*(T))/(sigma*sqrt(T));
    double d_2 = (log(S0/K) + (r-sigma*sigma/2.)*(T))/(sigma*sqrt(T));
    double C = S0*phi(d_1) - phi(d_2)*K*exp(-r*T);
    return C;
}

double Halton_Seq(int index, double base){
    double f = 1.0, r = 0.0;
    double indice = (double)index;
    while(indice >= base){
        f = f/base;
        r = r + f*(fmod(indice,base));
        indice = ceil(indice/base);
    }
    return r;
}

double Box_MullerX(double u_1, double u_2){
    double R = -2.0*log(u_1);
    double theta = 2.0*M_PI*u_2;
    double X = sqrt(R)*cos(theta);

    return X;
}

double Box_MullerY(double u_1, double u_2){
    double R = -2.0*log(u_1);
    double theta = 2.0*M_PI*u_2;
    double Y = sqrt(R)*sin(theta);

    return Y;
}

double Random_Shift_Halton(int n,int N,double mu, double sigma, double T, double S0){
    //  Generate the Halton_Sequence
    double* hs = new double[N+1];
    for(int i = 0; i <= N; i++){
        hs[i] = Halton_Seq(i,2.0);
    }
    // Generate two random numbers from U(0,1)
    double* u = new double[N+1];
    random_device rd;
    mt19937 e2(rd());
    uniform_real_distribution<double> dist(0.0, 1.0);
    for(int i = 0; i <= N; i++){
        u[i] = dist(e2);
    }
    // Add the vector u to each vector hs and take fraction part of the sum
    double* shifted_hs = new double[N+1];
    for(int i = 0; i <= N; i++){
        shifted_hs[i] = hs[i] + u[i];
        if(shifted_hs[i] > 1){
            shifted_hs[i] = shifted_hs[i] - 1.0;
        }
    }

    // Use Geometric Brownian Motion
    double** Z = new double*[N];    
    for(int i = 0; i < N; ++i)
       Z[i] = new double[n];
    for(int j = 0; j < N; j++){
        for(int i = 0; i < n/2; i++){
            Z[j][2*i] = Box_MullerX(shifted_hs[j],shifted_hs[j+1]);
            Z[j][2*i+1] = Box_MullerY(shifted_hs[j],shifted_hs[j+1]);
        }
    }

    double** S = new double*[N+1];  
    for(int i = 0; i <= N; ++i) 
       S[i] = new double[n+1];
    for (int i = 0; i <= N; i++) S[i][0] = S0;
    for (int j = 0; j <= n; j++) S[0][j] = S0;
    double* t = new double[n+1];
    for(int i = 0; i <= n; i++){
        t[i] = (double)i*T/(double)(n-1);
    }
    for(int j = 1; j <= N; j++){
        for(int i = 1; i <= n; i++){
            S[j][i] = S[j-1][i-1]*exp((mu - sigma/2.0)*(t[i] - t[i-1]) + sqrt(sigma)*sqrt(t[i] - t[i-1])*Z[j-1][i-1]);
        }
    }

    // Use random-shift halton sequence to obtain 40 independent estimates for the price of the option
    //....

    delete(hs);
    delete(u);
    delete(shifted_hs);
    for(int i = 0; i < N; ++i) {
       delete(Z[i]);
    }
    for(int i = 0; i <= N; ++i) delete(S[i]);
    delete(S);
    delete(Z);
    delete(t);

    return 0.0;
}

int main()
{
    int n = 10;
    int N = 10000;

    // Calculate BS
    double T = 1.0;
    double K = 50.0;
    double r = 0.1;
    double sigma = 0.3;
    sigma *= sigma;
    double S0 = 50.0;
    double price = Black_Scholes(T,K,S0,r,sigma);
    //cout << price << endl;

    // Random-shift Halton Sequence
    Random_Shift_Halton(n,N,r,sigma,T, S0);
    return 0;
}

#包括
#包括
#包括
#包括
使用名称空间std；
常数双M_PI=3.1415926535897932384626433832795；
常数双平方米=0.70710678118654752440084436210485；
双φ（长双x）；
双Black_Scholes（双T、双K、双S0、双r、双西格玛）；
双Halton_Seq（整数指数，双基）；
双箱磨浆机（双箱磨浆机1，双箱磨浆机2）；
双盒磨粉机（双盒磨粉机1、双盒磨粉机2）；
双随机移位（int n，int n，双μ，双西格玛，双T，双S0）；
双电流变电容器（双v）{
双y=1.0/（1.0+0.3275911*（v>=0？v:-v））；
y=1-(((((
+1.061405429*y
-1.453152027）*y
+1.421413741）*y
-0.284496736）*y
+0.254829592）*y）
*exp（-v*v）；
返回值（v>=0？y:-y）；
}
双φ（双x）{
返回0.5*erfc（-x*M_SQRT1_2）；
}
双Black_Scholes（双T、双K、双S0、双r、双西格玛）{
双d_1=（对数（S0/K）+（r+sigma*sigma/2.）*（T））/（sigma*sqrt（T））；
双d_2=（对数（S0/K）+（r-sigma*sigma/2.）*（T））/（sigma*sqrt（T））；
双C=S0*phi（d_1）-phi（d_2）*K*exp（-r*T）；
返回C；
}
双Halton_Seq（整数指数，双基）{
双f=1.0，r=0.0；
双指数=（双）指数；
while（标记>=基）{
f=f/基；
r=r+f*（fmod（标记，基础））；
籼=细胞（籼/基）；
}
返回r；
}
双箱磨浆机（双箱磨浆机1、双箱磨浆机2）{
双R=-2.0*log（u_1）；
双θ=2.0*M_π*u_2；
双X=sqrt（R）*cos（θ）；
返回X；
}
双箱研磨机（双u 1，双u 2）{
双R=-2.0*log（u_1）；
双θ=2.0*M_π*u_2；
双Y=sqrt（R）*sin（θ）；
返回Y；
}
双随机移位Halton（整数n，整数n，双mu，双西格玛，双T，双S0）{
//生成Halton_序列
双精度*hs=新双精度[N+1]；
对于（int i=0；i）你似乎有
mu signa/2
而不是
mu sigmaSQUARED/2
我将sigma定义为sigma squared，这就是为什么我在需要时将sigma设置为
0.09
而不是
0.3
，如果它真的是sigma squared？
#include <iostream>
#include <cmath>
#include <math.h>
#include <random>

using namespace std;

const double M_PI = 3.1415926535897932384626433832795;
const double M_SQRT1_2 = 0.70710678118654752440084436210485;

double phi(long double x);
double Black_Scholes(double T, double K, double S0, double r, double sigma);
double Halton_Seq(int index, double base);
double Box_MullerX(double u_1, double u_2);
double Box_MullerY(double u_1, double u_2);
double Random_Shift_Halton(int n,int N,double mu, double sigma, double T, double S0);


double erfc(double v) {
    double y = 1.0 / ( 1.0 + 0.3275911 * (v >= 0 ? v : -v));
    y = 1 - (((((
        + 1.061405429  * y
        - 1.453152027) * y
        + 1.421413741) * y
        - 0.284496736) * y
        + 0.254829592) * y)
        * exp (-v * v);
    return (v >= 0 ? y : -y);
}

double phi(double x){
    return 0.5 * erfc(-x * M_SQRT1_2);
}

double Black_Scholes(double T, double K, double S0, double r, double sigma){
    double d_1 = (log(S0/K) + (r+sigma*sigma/2.)*(T))/(sigma*sqrt(T));
    double d_2 = (log(S0/K) + (r-sigma*sigma/2.)*(T))/(sigma*sqrt(T));
    double C = S0*phi(d_1) - phi(d_2)*K*exp(-r*T);
    return C;
}

double Halton_Seq(int index, double base){
    double f = 1.0, r = 0.0;
    double indice = (double)index;
    while(indice >= base){
        f = f/base;
        r = r + f*(fmod(indice,base));
        indice = ceil(indice/base);
    }
    return r;
}

double Box_MullerX(double u_1, double u_2){
    double R = -2.0*log(u_1);
    double theta = 2.0*M_PI*u_2;
    double X = sqrt(R)*cos(theta);

    return X;
}

double Box_MullerY(double u_1, double u_2){
    double R = -2.0*log(u_1);
    double theta = 2.0*M_PI*u_2;
    double Y = sqrt(R)*sin(theta);

    return Y;
}

double Random_Shift_Halton(int n,int N,double mu, double sigma, double T, double S0){
    //  Generate the Halton_Sequence
    double* hs = new double[N+1];
    for(int i = 0; i <= N; i++){
        hs[i] = Halton_Seq(i,2.0);
    }
    // Generate two random numbers from U(0,1)
    double* u = new double[N+1];
    random_device rd;
    mt19937 e2(rd());
    uniform_real_distribution<double> dist(0.0, 1.0);
    for(int i = 0; i <= N; i++){
        u[i] = dist(e2);
    }
    // Add the vector u to each vector hs and take fraction part of the sum
    double* shifted_hs = new double[N+1];
    for(int i = 0; i <= N; i++){
        shifted_hs[i] = hs[i] + u[i];
        if(shifted_hs[i] > 1){
            shifted_hs[i] = shifted_hs[i] - 1.0;
        }
    }

    // Use Geometric Brownian Motion
    double** Z = new double*[N];    
    for(int i = 0; i < N; ++i)
       Z[i] = new double[n];
    for(int j = 0; j < N; j++){
        for(int i = 0; i < n/2; i++){
            Z[j][2*i] = Box_MullerX(shifted_hs[j],shifted_hs[j+1]);
            Z[j][2*i+1] = Box_MullerY(shifted_hs[j],shifted_hs[j+1]);
        }
    }

    double** S = new double*[N+1];  
    for(int i = 0; i <= N; ++i) 
       S[i] = new double[n+1];
    for (int i = 0; i <= N; i++) S[i][0] = S0;
    for (int j = 0; j <= n; j++) S[0][j] = S0;
    double* t = new double[n+1];
    for(int i = 0; i <= n; i++){
        t[i] = (double)i*T/(double)(n-1);
    }
    for(int j = 1; j <= N; j++){
        for(int i = 1; i <= n; i++){
            S[j][i] = S[j-1][i-1]*exp((mu - sigma/2.0)*(t[i] - t[i-1]) + sqrt(sigma)*sqrt(t[i] - t[i-1])*Z[j-1][i-1]);
        }
    }

    // Use random-shift halton sequence to obtain 40 independent estimates for the price of the option
    //....

    delete(hs);
    delete(u);
    delete(shifted_hs);
    for(int i = 0; i < N; ++i) {
       delete(Z[i]);
    }
    for(int i = 0; i <= N; ++i) delete(S[i]);
    delete(S);
    delete(Z);
    delete(t);

    return 0.0;
}

int main()
{
    int n = 10;
    int N = 10000;

    // Calculate BS
    double T = 1.0;
    double K = 50.0;
    double r = 0.1;
    double sigma = 0.3;
    sigma *= sigma;
    double S0 = 50.0;
    double price = Black_Scholes(T,K,S0,r,sigma);
    //cout << price << endl;

    // Random-shift Halton Sequence
    Random_Shift_Halton(n,N,r,sigma,T, S0);
    return 0;
}