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SLV_calibration_WR_test_main.cpp
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SLV_calibration_WR_test_main.cpp
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#include <cmath>
#include <iostream>
#include <algorithm>
#include <memory>
#include <vector>
#include <boost/numeric/ublas/vector.hpp>
#include <boost/numeric/ublas/matrix.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/math/interpolators/cardinal_cubic_b_spline.hpp>
#include <boost/numeric/ublas/vector_proxy.hpp> //subrange,subslice
#include <boost/numeric/ublas/matrix_proxy.hpp> //w = row(A, i); w = column(A, j); // a row or column of matrix as a vector
#include <boost/numeric/ublas/banded.hpp>//ublas::banded_matrix<double> M(nx, nx, 1, 1);//size1,size2,lower,upper; 是否有求解三对角矩阵的线性方程组程序？
#include <boost/numeric/ublas/lu.hpp>//permutation_matrix,用于LU分解
#include <boost/numeric/ublas/triangular.hpp>//solve函数，用于求解上三角或下三角线性方程组：V = solve(M, f, ublas::lower_tag());
#include <boost/numeric/ublas/matrix_sparse.hpp>
#include <boost/numeric/ublas/storage.hpp>
#include <boost/numeric/ublas/assignment.hpp>//赋值符号<<==给matrix和vector赋值
#include <boost/numeric/ublas/io.hpp> //使用std::cout输出matrix和vector
#include "CumNorm.h"
//#include <math.h>
namespace ublas = boost::numeric::ublas;
namespace interp = boost::math::interpolators;
using namespace std;
#include<Core>
#include<SVD>
#include<Dense>
struct hestonParas
{
double theta;
double kappa;
double VoV; //xi
double rho;
double Y0; // v0
};
template<typename _Matrix_Type_>
_Matrix_Type_ pseudoInverse(const _Matrix_Type_ &a, double epsilon =
std::numeric_limits<double>::epsilon())
{
Eigen::JacobiSVD< _Matrix_Type_ > svd(a ,Eigen::ComputeThinU | Eigen::ComputeThinV);
double tolerance = epsilon * std::max(a.cols(), a.rows()) *svd.singularValues().array().abs()(0);
return svd.matrixV() * (svd.singularValues().array().abs() > tolerance).select(svd.singularValues().array().inverse(), 0).matrix().asDiagonal() * svd.matrixU().adjoint();
}
void fnexp(
ublas::vector<double> b,
ublas::vector<double>& d
)
{
for (size_t i = 0; i < b.size(); ++i)d(i) = exp(b(i));
}
void stackvector(
ublas::vector<double> Y,
double para,
ublas::matrix<double>& matrix0
)
{
//throw"The para must be set to 1 or 2.";
size_t n = Y.size();
matrix0.resize(n, n);
if (para == 1)
{
for (size_t i = 0; i < n; ++i)
for (size_t j = 0; j < n; ++j)matrix0(i, j) = Y[i];
}
else if (para == 2)
{
for (size_t i = 0; i < n; ++i)
for (size_t j = 0; j < n; ++j)matrix0(i, j) = Y[j];
}
matrix0 = ublas::trans(matrix0);
}
//create the initial prob density
void yutian_initial(
ublas::vector<double> S, //stock direction grid
ublas::vector<double> z, //v direction grid
hestonParas hp, //heston paramaters
double r, //interest rate
double q, //dividend
double sigma_initial, //sigma at t = 0
double dt, //
ublas::matrix<double>& initial_p //return: initial matrix of prob density
)
{
//S is a column vector, z is a row vector
//sigma_initial = sigma(S0, 0)
size_t rows = S.size();
size_t cols = z.size();
double mu_S = ((r - q) - 0.5 * pow(sigma_initial, 2.0) * hp.Y0) * dt;
double sigma_S = sigma_initial * sqrt(hp.Y0) * sqrt(dt);
double mu_z = ((hp.kappa * hp.theta - 0.5 * pow(hp.VoV, 2)) / hp.Y0 - hp.kappa) * dt;
double sigma_z = hp.VoV * sqrt(dt / hp.Y0);
initial_p.resize(rows, cols);
double PI = boost::math::constants::pi<double>();
for (size_t i = 0; i < rows; ++i)
{
for (size_t j = 0; j < z.size(); ++j)
{
initial_p(i, j) = 1 / (2 * PI * sigma_S * sigma_z * sqrt(1.0 - pow(hp.rho, 2))) *
exp(
-(
pow((S(i) - mu_S), 2) / pow(sigma_S, 2) +
(pow(z(j) - mu_z, 2) / pow(sigma_z, 2) - (2 * hp.rho * (S(i) - mu_S) * (z(j) - mu_z) / (sigma_S * sigma_z))) / (2 * (1 - pow(hp.rho, 2)))
)
);
}
}
}
void bisection_initial(
ublas::vector<double> S, //stock direction grid
ublas::vector<double> z, //v direction grid
hestonParas hp, //heston paramaters
double r, //interest rate
double q, //dividend
double sigma_initial, //sigma at t = 0
double& dt, //return: initial value: 1e-5
ublas::matrix<double>& initial_p //return: initial matrix of prob density
)
{
double dt1 = 1e-5;
double dt2 = 0.1;
double er = 1e-10;
double int_diff = 1.0;
size_t nS = S.size();
size_t nz = z.size();
double S_max = S[nS - 1], S_min = S[0]; // 最大最小值，s,z是顺序生成的？
double z_max = z[nz - 1], z_min = z[0];
double dS = (S_max - S_min) / (nS - 1);
double dz = (z_max - z_min) / (nz - 1);
dt = dt1;
double t = 1.0;
//这里如果条件不加<15就会一直循环下去
while ((int_diff) > er && t<15)
{
yutian_initial(S, z, hp, r, q, sigma_initial, dt, initial_p);
double int_p = 0;
for (size_t i = 0; i < nS - 1; ++i)
{
for (size_t j = 0; j < nz - 1; ++j)
{
int_p = int_p + (1.0 / 4.0) * (initial_p(i, j) + initial_p(i + 1, j) + initial_p(i, j + 1), initial_p(i + 1, j + 1)) * dz * dS;
}
}
int_diff = int_p - 1;
if (int_diff < 0)dt2 = dt;
else dt1 = dt;
dt = (dt1 + dt2) / 2.0;
t = t + 1;
if (t > 15)throw"The partition is too sparse";
}
for (size_t i = 0; i < initial_p.size1(); ++i)
for (size_t j = 0; j < initial_p.size2(); ++j)
if (initial_p(i, j) < 0)initial_p(i, j) = 0;
}
void identify_neighbors(
ublas::matrix<double> A,
ublas::matrix<double> nan_list,
ublas::matrix<double> talks_to,
ublas::matrix<double>& neighbor_list
)
{
size_t n1 = A.size1();
size_t n2 = A.size2();
size_t n3 = nan_list.size1();
if (n3 != 0)
{
size_t nan_count = nan_list.size1();
size_t talk_count = talks_to.size1();
ublas::matrix<double> nn(nan_count * talk_count, 2, 0);
//可能有误
for (size_t i = 0; i < talk_count; ++i)
{
for (size_t k = 0; k < nan_count; ++k)
{
nn(i * nan_count + k, 0) = nan_list(k, 1) + talks_to(i, 0);
nn(i * nan_count + k, 1) = nan_list(k, 2) + talks_to(i, 1);
}
}
//drop those nodes which fall outside the bounds of the original array
double t1 = 0;
for (size_t i = 0; i < nn.size1(); i++)
{
if (nn(i, 0) < 1 || nn(i, 0) > n1 || nn(i, 1) < 1 || nn(i, 1) > n2)t1 = t1 + 1;
}
ublas::matrix<double> L(nn.size1() - t1, 2, 0);
double count = 0;
for (size_t i = 0; i < nn.size1(); i++) //删掉没有在范围内的点,得到更新后的nn,也就是L
{
if (nn(i, 0) < 1 || nn(i, 0) > n1 || nn(i, 1) < 1 || nn(i, 1) > n2) {}
else
{
L(count, 0) = nn(i, 0);
L(count, 1) = nn(i, 1);
count = count + 1;
}
}
ublas::matrix<double> neighbor_list0(L.size1(), 3);
for (size_t i = 0; i < L.size1(); i++)
{
for (size_t j = 0; j < 3; j++)
{
if (j > 0) {
neighbor_list0(i, j) = L(i, j - 1);
}
else {
neighbor_list0(i, j) = (L(i, 1) - 1) *L.size1() + L(i, 0);
}
}
}
//把ublas vector 转成 std vector做了，因为前者好像不能用unique这些函数
// 两者之间函数可以封成函数，但目前没有做
//把neighbor_list转换
vector<vector<int> > v1(neighbor_list0.size1()); //行
for (int i = 0; i < v1.size(); i++) {
v1[i].resize(neighbor_list0.size2()); //列
for (int j = 0; j < v1[i].size(); j++) {
v1[i][j] = neighbor_list0(i, j); //赋值
}
}
sort(v1.begin(), v1.end());
v1.erase(unique(v1.begin(), v1.end()), v1.end()); // unique那一步
//把nan_list转换
vector<vector<int> > v2(nan_list.size1()); //行
for (int i = 0; i < v2.size(); i++) {
v2[i].resize(nan_list.size2()); //列
for (int j = 0; j < v2[i].size(); j++) {
v2[i][j] = nan_list(i, j); //赋值
}
}
sort(v2.begin(), v2.end());
v2.erase(unique(v2.begin(), v2.end()), v2.end());
//比较二者差异
vector<vector<int> >diff;
std::set_difference(v1.begin(), v1.end(), v2.begin(), v2.end(), std::inserter(diff, diff.begin()));
//把得到的结果再转回到ublas vector
ublas::matrix<double> trans(diff.size(), diff[0].size());
for (size_t i = 0; i < diff.size(); i++)
{
for (int j = 0; j < diff[i].size(); j++) {
trans(i, j) = diff[i][j]; //赋值
}
}
ublas::matrix<double> neighbor_list = trans;
}
else {
ublas::vector<double> neighbor_list(0);
}
}
void inpaint_nans(
ublas::matrix<double> A,
int k,
ublas::matrix<double>& B
)
{
size_t n1 = A.size1();
size_t n2 = A.size2();
vector<vector<double> > nan_list;
vector<vector<double> > known_list;
double k1 = 0;
double k2 = 0;
for (size_t j = 0; j < n2; j++)
{
for (size_t i = 0; i < n1; i++)
{
//if (std::isnan(A(i, j)))
if (std::isnan(A(i, j)) || std::isinf(A(i, j)))
{
nan_list.push_back(vector<double>(3));
nan_list[k1][0] = j * n1 + i;
nan_list[k1][1] = i;
nan_list[k1][2] = j;
k1 = k1 + 1;
}
else
{
known_list.push_back(vector<double>(2));
known_list[k2][0] = j * n1 + i;
known_list[k2][1] = A(i, j);
k2 = k2 + 1;
}
}
}
B.resize(A.size1(), A.size2());
B=A;
if(nan_list.size()==0){}
else{
//把 标准vector转化成 ublas
ublas::matrix<double> nan_list1(nan_list.size(), 3);
for (int i = 0; i < nan_list.size(); i++)
{
for (int j = 0; j < 3; j++) {
nan_list1(i, j) = nan_list[i][j]; //赋值
}
}
ublas::matrix<double> talks_to(12, 2);
talks_to <<= -2, 0,
-1, -1,
-1, 0,
-1, 1,
0, -2,
0, -1,
0, 1,
0, 2,
1, -1,
1, 0,
1, 1,
2, 0;
ublas::matrix<double> neighbor_list;
identify_neighbors(A, nan_list1, talks_to, neighbor_list);
//可用subrange优化
ublas::matrix<double> all_list(nan_list1.size1() + neighbor_list.size1(), 3);
for (size_t i = 0; i < all_list.size1(); i++)
{
for (size_t j = 0; j < 3; j++)
{
if (i < nan_list1.size1())
{
all_list(i, j) = nan_list1(i, j);
}
else
{
all_list(i, j) = neighbor_list(i - nan_list1.size1(), j);
}
}
}
vector<double> L1, L2, L3, L4;
for (size_t i = 0; i < all_list.size1(); i++)
{
if (all_list(i, 1) >= 3 && all_list(i, 1) <= n1-2 && all_list(i, 2) >= 3 && all_list(i, 2) <= n2 - 2)
{
L1.push_back(i);
}
if ((((all_list(i, 1) == 2) || (all_list(i, 1) == (n1 - 1))) && (all_list(i, 2) >= 2) && (all_list(i, 2) <= (n2 - 1))) || (((all_list(i, 2) == 2) || (all_list(i, 2) == (n2 - 1))) && (all_list(i, 1) >= 2) && (all_list(i, 1) <= (n1 - 1))))
{
L2.push_back(i);
}
if (((all_list(i, 1) == 1) || (all_list(i, 1) == n1)) && (all_list(i, 2) >= 2) && (all_list(i, 2) <= (n2 - 1)))
{
L3.push_back(i);
}
if (((all_list(i, 2) == 1) || (all_list(i, 2) == n2)) && (all_list(i, 1) >= 2) && (all_list(i, 1) <= (n1 - 1)))
{
L4.push_back(i);
}
}
ublas::compressed_matrix<double> fda(n1 * n2, n1 * n2);
if (L1.size() > 0)
{
ublas::vector<double> T1(13);
T1 <<= -2 * n1, -n1 - 1, -n1, -n1 + 1, -2, -1, 0, 1, 2, n1 - 1, n1, n1 + 1, 2 * n1;
ublas::vector<double> T2(13);
T2 <<= 1, 2, -8, 2, 1, -8, 20, -8, 1, 2, -8, 2, 1;
for (size_t i = 0; i < L1.size(); i++)
{
for (size_t j = 0; j < 13; j++)
{
fda(all_list(L1[i]), all_list(L1[i]) + T1(j)) = T2(j);
}
}
}
else
{
ublas::compressed_matrix<double> fda(n1 * n2, n1 * n2, all_list.size1() * 5);
}
if (L2.size() > 0)
{
ublas::vector<double> T1(5);
T1 <<= -n1, -1, 0, 1, n1;
ublas::vector<double> T2(5);
T2 <<= 1, 1, -4, 1, 1;
for (size_t i = 0; i < L2.size(); i++)
{
for (size_t j = 0; j < 5; j++)
{
fda(all_list(L2[i]), all_list(L2[i]) + T1(j)) += T2(j);
}
}
}
if (L3.size() > 0)
{
ublas::vector<double> T1(3);
T1 <<= -n1, -0, n1;
ublas::vector<double> T2(3);
T2 <<= 1, -2, 1;
for (size_t i = 0; i < L3.size(); i++)
{
for (size_t j = 0; i < 3; i++)
{
fda(all_list(L3[i]), all_list(L3[i]) + T1(j)) += T2(j);
}
}
}
if (L4.size() > 0)
{
ublas::vector<double> T1(3);
T1 <<= -1, -0, 1;
ublas::vector<double> T2(3);
T2 <<= 1, -2, 1;
for (size_t i = 0; i < L4.size(); i++)
{
for (size_t j = 0; i < 3; i++)
{
fda(all_list(L4[i]), all_list(L4[i]) + T1(j)) += T2(j);
}
}
}
ublas::matrix<double> fdaknown(fda.size1(), known_list.size());
for (size_t i = 0; i < fdaknown.size1(); i++)
{
for (size_t j = 0; j < fdaknown.size2(); j++) {
fdaknown(i, j) = fda(i, known_list[j][0]);
}
}
ublas::vector<double> Aknown(known_list.size());
for (size_t i = 0; i < Aknown.size(); i++)Aknown(i) = known_list[i][1];
ublas::vector<double> rhs = -prod(fdaknown, Aknown);
// //找到非零行
vector<double> k0;
for (size_t i = 0; i < n1 * n2; i++)
{
for (size_t j = 0; j < nan_list1.size1(); j++)
{
if (fda(i, nan_list1(j, 0)) == 0)
{
k0.push_back(i);
break;
}
}
}
// cout << k0.size() << std::endl;
//cout << nan_list.size() << std::endl;
Eigen::MatrixXd A0(k0.size(),nan_list1.size1());
for (size_t i = 0; i < k0.size(); i++)
{
for (size_t j = 0; j < nan_list1.size1(); j++)
{
A0(i,j)=1/i+1/j;
(k0[i],nan_list1(j,0));
}
}
Eigen::VectorXd rhs0(k0.size());
for (size_t i = 0; i < k0.size(); i++)
{
rhs0(i)=i;
//rhs(k0[i]);
}
Eigen::VectorXd b2(nan_list.size());
b2=pseudoInverse(A0)*rhs0;
for (size_t i = 0; i < nan_list1.size1(); i++)
{
B(nan_list1(i,0))=b2(i);
}
//cout << A << std::endl;
}
}
void OptionPrice(
ublas::vector<double>x,
ublas::vector<double>z,
ublas::vector<double>L,
ublas::vector<double>V_initial,
ublas::matrix<double>psi,
double T1,
double T2,
size_t nt,
double r,
double q,
double y0,
int call_put,//call:1, put:-1
ublas::vector<double>& V //return
)
{
// if (call_put != 1 || call_put != -1) throw"error of option type, call_put must set to 1(call), or -1(put).";
//M为稀疏矩阵
size_t nx = x.size();
size_t nz = z.size();
double x_max = x[nx - 1], x_min = x[0];
double z_max = z[nz - 1], z_min = z[0];
double dx = (x_max - x_min) / (nx - 1);
double dt = (T2 - T1) / (nt - 1);
double dz = (z_max - z_min) / (nz - 1);
ublas::vector<double>z1(z.size());
fnexp(z,z1);
ublas::vector<double>z2(z.size(),1);
ublas::vector<double>C(z.size());
ublas::vector<double>kb1=prod(psi,z1);
ublas::vector<double>kb2=prod(psi,z2);
C = y0 * element_div( kb1* dz, kb2* dz);
//这里本不应注释
// ublas::matrix<double> C1(C.size(),1);
// for (size_t i = 0; i < C1.size1(); i++)C1(i,0)=C(i);
// //std::cout<< C1<<std::endl;
// inpaint_nans(C1, 3, C1);
// ublas::vector<double> C2(C1.size1());
// for (size_t i = 0; i < C2.size(); i++)C2(i)=C1(i,0);
// C=C2;
for (size_t i = 0; i < C.size(); i++)
{
if (C(i)< 0)C(i) = 0;
}
ublas::vector<double> A(nx - 2), A_plus(nx - 2), A_minus(nx - 2);
for (size_t i = 0; i < A.size(); ++i)
{
A(i) = 1 + L(i + 1) * L(i + 1) * C(i + 1) * dt / dx / dx + dt * q;
A_plus(i) = (r - q + 0.5 * L(i + 1) * L(i + 1) * C(i + 1)) * dt / 2 / dx - 0.5 * L(i + 1) * L(i + 1) * C(i + 1) * dt / dx / dx;
A_minus(i) = -(r - q + 0.5 * L(i + 1) * L(i + 1) * C(i + 1)) * dt / 2 / dx - 0.5 * L(i + 1) * L(i + 1) * C(i + 1) * dt / dx / dx;
}
//以三对角矩阵(banded_matrix)存储M
//ublas::banded_matrix<double> M(nx, nx, 1, 1);//size1,size2,lower,upper; Allocates an uninitialized banded_matrix that holds (lower + 1 + upper) diagonals
ublas::matrix<double> M(nx, nx,0);
M(0, 0) = 1.0; M(0, 1) = -1.0;
M(nx - 1, nx - 1) = 1.0; M(nx - 1, nx - 2) = -1.0;
for (size_t i = 1; i < nx - 1; ++i)
{
for (size_t j = i - 1; j <= i + 1; ++j)
{
if (j == i - 1) M(i, j) = A_minus(i - 1); //upper, (i,j)=(1,0)处存储的是A_minus(0)
else if (j == i)M(i, j) = A(i - 1); //mid, (i,j)=(1,1)处存储的是A(0)
else if (j == i + 1) M(i, j) = A_plus(i - 1); //lower, (i,j)=(1,2)处存储的是A_plus(0)
}
}
V = V_initial;
for (size_t t = 1; t < nt; ++t)
{
//ublas::vector<double> f = V;
if (call_put == 1)
{
V(0) = exp(x(0)) * exp(-r * (T1 + t * dt)) * dx + 1.0 / 2.0 * exp(x(0)) * exp(-r * (T1 + t * dt)) * pow(dx, 2) + 1 / 6 * exp(x(0)) * exp(-r * (T1 + t * dt)) * pow(dx, 3) +
1.0 / 24.0 * exp(x(0)) * exp(-r * (T1 + t * dt)) * pow(dx, 4) + 1.0 / 120.0 * exp(x(0)) * exp(-r * (T1 + t * dt)) * pow(dx, 5);
V(V.size() - 1) = 0.0;
}
else if (call_put == -1)
{
V(0) = 0.0;
V(V.size() - 1) = exp(x(nx - 1)) * exp(-r * (T1 + t * dt)) * dx + 1.0 / 2.0 * exp(x(nx - 1)) * exp(-r * (T1 + t * dt)) * pow(dx, 2) + 1 / 6 * exp(x(nx - 1)) * exp(-r * (T1 + t * dt)) * pow(dx, 3) +
1.0 / 24.0 * exp(x(nx - 1)) * exp(-r * (T1 + t * dt)) * pow(dx, 4) + 1.0 / 120.0 * exp(x(nx - 1)) * exp(-r * (T1 + t * dt)) * pow(dx, 5);
}
//Matlab: V = M\f
//===============LU分解求解线性方程组======================
//link1: https://stackoverflow.com/questions/1225411/boosts-linear-algebra-solution-for-y-ax(关于ublas中如何使用LU分解)
//link2：https://stackoverflow.com/questions/26404106/what-does-lu-factorize-return(关于LU分解)
ublas::permutation_matrix<size_t> pm(M.size1());//列主元的高斯消元法
int res = lu_factorize(M, pm);//LU分解并将L(对角线为1)保存在M左下角；U(对角线不为1)保存在M右上角
if (res != 0)throw"error in LV decomposition.";
else {
lu_substitute(M, pm, V);//求解线性方程组,MV=f，并将结果保存在f中
}
// //===============LU分解求解线性方程组======================
for (size_t i = 0; i < V.size(); i++)
{
if(V(i)<0)V(i)=0;
}
// // for (auto iter = V.begin(); iter < V.end(); ++iter)
// // if (*iter < 0)*iter = 0.0;
}
}
void finite_difference_mat_b2(
ublas::vector<double> x,
ublas::vector<double> z,
hestonParas hp,
ublas::matrix<double>& T_x1,
ublas::matrix<double>& T_x2,
ublas::matrix<double>& T_z1,
ublas::matrix<double>& T_z2,
ublas::matrix<double>& T_zb1,
ublas::matrix<double>& T_zb2
)
{
size_t nx1 = x.size();
size_t nz1 = z.size();
ublas::vector<double> dx = subrange(x, 1, nx1) - subrange(x, 0, nx1 - 1);
ublas::vector<double> dx_i_minus = subrange(dx, 0, dx.size() - 1);
ublas::vector<double> dx_i = subrange(dx, 1, dx.size());
ublas::vector<double> c_x_minus = -element_div(dx_i, element_prod(dx_i_minus, dx_i_minus + dx_i));
ublas::vector<double> c_x = element_div(dx_i - dx_i_minus, element_prod(dx_i_minus, dx_i));
ublas::vector<double> c_x_plus = element_div(dx_i_minus, element_prod(dx_i, dx_i_minus + dx_i));
c_x(0) = c_x(0) + c_x_minus(0);
c_x(c_x.size() - 1) = c_x(c_x.size() - 1) + c_x_plus(c_x_plus.size() - 1);
ublas::vector<double> one1(dx_i_minus.size(), 2);
ublas::vector<double> x_x_minus = element_div(one1, element_prod(dx_i_minus, dx_i_minus + dx_i));
ublas::vector<double> x_x = -element_div(one1, element_prod(dx_i_minus, dx_i));
ublas::vector<double> x_x_plus = element_div(one1, element_prod(dx_i, dx_i_minus + dx_i));
x_x(0) = x_x(0) + x_x_minus(0);
x_x(x_x.size() - 1) = x_x(x_x.size() - 1) + x_x_plus(x_x_plus.size() - 1);
//这段还未检查
T_x1.resize(nx1 - 2, nx1 - 2);
T_x2.resize(nx1 - 2, nx1 - 2);
for (size_t i = 1; i < nx1 - 3; ++i)
{
for (size_t j = i - 1; j <= i + 1; ++j)
{
if (j == i - 1) T_x1(i, j) = c_x_minus(i - 1), T_x2(i, j) = x_x_minus(i - 1);
else if (j == i)T_x1(i, j) = c_x(i - 1), T_x2(i, j) = x_x(i - 1);
else if (j == i + 1) T_x1(i, j) = c_x_plus(i - 1), T_x2(i, j) = x_x_plus(i - 1);
}
}
ublas::vector<double> dz = subrange(z, 1, nz1) - subrange(z, 0, nz1 - 1);
ublas::vector<double> dz_i_minus = subrange(dz, 0, dz.size() - 1);
ublas::vector<double> dz_i = subrange(dz, 1, dz.size());
ublas::vector<double> c_z_minus = -element_div(dz_i, element_prod(dz_i_minus, dz_i_minus + dz_i));
ublas::vector<double> c_z = element_div(dz_i - dz_i_minus, element_prod(dz_i_minus, dz_i));
ublas::vector<double> c_z_plus = element_div(dz_i_minus, element_prod(dz_i, dz_i_minus + dz_i));
// here we give a boundary condition for pe^z
ublas::vector<double> c_z_b = c_z;
c_z_b(0) = c_z_b(0) + (1 + dz_i_minus(0) - dz_i_minus(0) * 2 * hp.kappa * hp.theta / pow(hp.VoV, 2)) * c_z_minus(0);
//
// c_z(0) = c_z(0) + c_z_minus(0);
// c_z(c_z.size() - 1) = c_z(c_z.size() - 1) + c_z_plus(c_z_plus.size() - 1);
c_z=c_z_b;
ublas::vector<double> one0(dz_i_minus.size(), 2);
ublas::vector<double> x_z_minus = element_div(one0, element_prod(dz_i_minus, dz_i_minus + dz_i));
ublas::vector<double> x_z = -element_div(one0, element_prod(dz_i_minus, dz_i));
ublas::vector<double> x_z_plus = element_div(one0, element_prod(dz_i, dz_i_minus + dz_i));
// here we give a boundary condition for pe^-z from zero flux condition
ublas::vector<double> x_z_b = x_z;
x_z_b(0) = x_z_b(0) + (1 + dz_i_minus(0) - dz_i_minus(0) * 2 * hp.kappa * hp.theta / pow(hp.VoV, 2)) * x_z_minus(0);
//
// x_z(0) = x_z(0) + x_z_minus(0);
// x_z(x_z.size() - 1) = x_z(x_z.size() - 1) + x_z_plus(x_z_plus.size() - 1);
x_z=x_z_b;
// //这里还没有检查？
T_z1.resize(nz1 - 2, nz1 - 2);
T_z2.resize(nz1 - 2, nz1 - 2);
T_zb1.resize(nz1 - 2, nz1 - 2);
T_zb2.resize(nz1 - 2, nz1 - 2);
for (size_t i = 1; i < nz1 - 3; ++i)
{
for (size_t j = i - 1; j <= i + 1; ++j)
{
if (j == i - 1){
T_z1(i, j) = c_z_minus(i - 1);
T_z2(i, j) = x_z_minus(i - 1);
T_zb1(i, j) = c_z_minus(i - 1);
T_zb2(i, j) = x_z_minus(i - 1);
}
else if (j == i){
T_z1(i, j) = c_z(i - 1);
T_z2(i, j) = x_z(i - 1);
T_zb1(i, j) = c_z_b(i - 1);
T_zb2(i, j) = x_z_b(i - 1);
}
else if (j == i + 1){
T_z1(i, j) = c_z_plus(i - 1);
T_z2(i, j) = x_z_plus(i - 1);
T_zb1(i, j) = c_z_plus(i - 1);
T_zb2(i, j) = x_z_plus(i - 1);
}
}
}
}
void adi_p_b(ublas::vector<double> x,
ublas::vector<double> z,
ublas::vector<double> sigma,
ublas::matrix<double> initial_p,
double t_start,
double t_end,
double dt,
hestonParas& hp,
double r,
double q,
double alpha,
ublas::matrix<double>& psi
)
{
size_t nt = floor((t_end - t_start) / dt + 1);
ublas::vector<double> t(nt);
for (size_t i = 0; i < nt; ++i)t(i) = t_start + i * dt;
size_t nx2 = x.size();
size_t nz2 = z.size();
double dx = (x(nx2 - 1) - x(0)) / (nx2 - 1);
double dz = (z(nz2 - 1) - z(0)) / (nz2 - 1);
ublas::matrix<double> T_S1, T_S2, T_z1, T_z2, T_zb1, T_zb2;
finite_difference_mat_b2(x, z, hp, T_S1, T_S2, T_z1, T_z2, T_zb1, T_zb2);
ublas::matrix<double> p = subrange(initial_p, 1, nx2 - 1, 1, nz2 - 1);//这里或许可以改进用切片
ublas::vector<double> x_inner(nx2 - 2);//可以改进用切片
ublas::vector<double> z_inner(nz2 - 2);
ublas::vector<double> sigma_inner(sigma.size() - 2);
for (size_t i = 0; i < x_inner.size(); ++i)x_inner(i) = x(i + 1);
for (size_t i = 0; i < z_inner.size(); ++i)z_inner(i) = z(i + 1);
for (size_t i = 0; i < sigma_inner.size(); ++i)sigma_inner(i) = sigma(i + 1);
psi.resize(nx2, nz2);
for (size_t i = 0; i <nx2; i++)
{
for (size_t j = 0; j < nz2; j++)
{
psi(i,j)=0;
}
}
for (size_t i = 0; i < nt - 1; ++i)//共nt-1次循环
{
//std::cout << 5 << std::endl;
ublas::matrix<double> sv;
stackvector(sigma_inner, 1, sv);
ublas::matrix<double> f0p1=element_prod(sv, T_S1);
ublas::matrix<double> f0p2=prod(f0p1,p);
ublas::matrix<double> F0p = hp.VoV * hp.rho * prod(f0p2, T_z1);
// ublas::matrix<double> F0p = hp.VoV * hp.rho *p;
ublas::vector<double> temp(z_inner.size());
for (size_t j = 0; j < z_inner.size(); ++j)temp(j) = exp(-z_inner(j)) / hp.Y0;
stackvector(temp, 2, sv);
ublas::matrix<double> F1 = -(hp.kappa * hp.theta - 0.5 * pow(hp.VoV,2)) * element_prod(T_zb1, sv) + hp.kappa * T_z1 + 0.5 * pow(hp.VoV,2)* element_prod(T_zb2, sv);
//这里如果添加上后两项会变得不ok
temp.resize(sigma_inner.size());
for (size_t j = 0; j < sigma_inner.size(); ++j)temp(j) = sigma_inner(j) * sigma_inner(j);
stackvector(temp, 1, sv);
ublas::matrix<double> temp1(z_inner.size(),z_inner.size());
for (size_t i = 0; i < temp1.size1(); i++)
{
temp1(i,i) = exp(z_inner(i)) * hp.Y0;
}
ublas::matrix<double> f2p1=element_prod(sv, T_S1);
ublas::matrix<double> f2p2=prod(f2p1, p);
ublas::matrix<double> f2p3=element_prod(sv, T_S2);
ublas::matrix<double> f2p4=prod(f2p3, p);
ublas::matrix<double> F2p = (r - q) * prod(T_S1, p) + 0.5 * prod(f2p2, temp1) + 0.5 * prod(f2p4, temp1);
ublas::matrix<double> A = p + dt * (F0p + prod(p, F1) + F2p);
ublas::matrix<double> B1 = (A - alpha * dt * prod(p, F1));
ublas::matrix<double> B2(nz2 - 2, nz2 - 2);
for (size_t i = 0; i < nz2 - 2; i++)B2(i, i) = 1;
ublas::matrix<double> B3 = B2 - alpha * dt * F1;
ublas::permutation_matrix<size_t> pm2(B3.size1());
int res3 = lu_factorize(B3, pm2);
if (res3 != 0)throw"error in LV decomposition.";
ublas::matrix<double> B3inv = ublas::identity_matrix<double>(B3.size1());
lu_substitute(B3, pm2, B3inv);
ublas::matrix<double> B = prod(B1, B3inv);
// // std::cout << B.size1() <<std::endl;
//ublas::matrix<double> B=A;
ublas::compressed_matrix<double>I1(nx2 - 2, nx2 - 2);
for (size_t i = 0; i < nx2 - 2; i++)I1(i, i) = 1;
ublas::compressed_matrix<double>K1((nx2 - 2) * (nz2 - 2), (nx2 - 2) * (nz2 - 2), 0);
ublas::compressed_matrix<double>K2((nx2 - 2) * (nz2 - 2), (nx2 - 2) * (nz2 - 2), 0);
ublas::compressed_matrix<double>K4((nx2 - 2) * (nz2 - 2), (nx2 - 2) * (nz2 - 2), 0);
ublas::compressed_matrix<double>K3((nx2 - 2) * (nz2 - 2), (nx2 - 2) * (nz2 - 2), 0);
for (size_t i = 0; i < nz2 - 2; i++)
{
subrange(K1, i * (nx2 - 2), (i + 1) * (nx2 - 2) , i * (nx2 - 2), (i + 1) * (nx2 - 2) ) = T_S1;
subrange(K2, i * (nx2 - 2), (i + 1) * (nx2 - 2) , i * (nx2 - 2), (i + 1) * (nx2 - 2)) = f2p1;
subrange(K4, i * (nx2 - 2), (i + 1) * (nx2 - 2) , i * (nx2 - 2), (i + 1) * (nx2 - 2)) = f2p3;
subrange(K3, i * (nx2 - 2), (i + 1) * (nx2 - 2) , i * (nx2 - 2), (i + 1) * (nx2 - 2)) = exp(z_inner(i)) * hp.Y0 * I1;
}
ublas::compressed_matrix<double>I2((nx2 - 2)*(nz2-2), (nx2 - 2)*(nz2-2),0);
for (size_t i = 0; i < (nx2 - 2)*(nz2-2); i++)I2(i, i) = 1;
ublas::compressed_matrix<double> Cnew((nx2 - 2)*(nz2 - 2), (nx2 - 2)*(nz2 - 2),0);
Cnew= I2-alpha*dt*(-(r-q)*K1+0.5*prod(K3,K2)+0.5*prod(K3,K4));
//Cnew=I2;
ublas::vector<double> resh((nx2-2)*(nz2-2));
for (size_t i = 0; i < nx2-2; i++)
{
for (size_t j = 0; j < nz2-2; j++)
{
resh(j*(nx2-2)+i)=B(i,j)-alpha*dt*F2p(i,j);
}
}
ublas::permutation_matrix<size_t> pm1(Cnew.size1());
int res2 = lu_factorize(Cnew, pm1);
lu_substitute(Cnew, pm1, resh);
ublas::matrix<double> p(nx2 - 2, nz2 - 2);
for (size_t i = 0; i < nx2-2; i++)
{
for (size_t j = 0; j < nz2-2; j++)
{
if(resh(j*(nx2-2)+i)<0)p(i,j)=0;
//else p(i,j)=resh(j*(nx2-2)+i);
p(i,j)=1;
}
}
}
subrange(psi, 1, nx2 - 1, 1, nz2 - 1) = p;
}
void meshgrid(ublas::vector<double>& x, ublas::vector<double>& y, ublas::matrix<double>& X, ublas::matrix<double>& Y)
{
/*
* X : x作为每一行，共y.size()行
* Y ：y作为每一列, 共x.size()列
*
* matlab:
[A,B]=Meshgrid(a,b)
生成size(b)Xsize(a)大小的矩阵A和B。它相当于a从一行重复增加到size(b)行，把b转置成一列再重复增加到size(a)列
*/
size_t nx = x.size();
size_t ny = y.size();
X.resize(ny, nx);
Y.resize(ny, nx);
for (size_t i = 0; i < ny; ++i)
for (size_t j = 0; j < nx; ++j)
{
X(i, j) = x[j];
Y(i, j) = y[i];
}
}
static double Mtools_T[] = {
9.60497373987051638749E0, 9.00260197203842689217E1, 2.23200534594684319226E3,
7.00332514112805075473E3, 5.55923013010394962768E4 };
static double Mtools_U[] = {/* 1.00000000000000000000E0,*/
3.35617141647503099647E1, 5.21357949780152679795E2, 4.59432382970980127987E3,
2.26290000613890934246E4, 4.92673942608635921086E4 };
static double Mtools_nep[] = {
2.46196981473530512524E-10, 5.64189564831068821977E-1, 7.46321056442269912687E0,
4.86371970985681366614E1, 1.96520832956077098242E2, 5.26445194995477358631E2,
9.34528527171957607540E2, 1.02755188689515710272E3, 5.57535335369399327526E2 };
static double Mtools_neq[] = {/* 1.00000000000000000000E0,*/
1.32281951154744992508E1, 8.67072140885989742329E1, 3.54937778887819891062E2,
9.75708501743205489753E2, 1.82390916687909736289E3, 2.24633760818710981792E3,
1.65666309194161350182E3, 5.57535340817727675546E2 };
static double Mtools_ner[] = {
5.64189583547755073984E-1, 1.27536670759978104416E0, 5.01905042251180477414E0,
6.16021097993053585195E0, 7.40974269950448939160E0, 2.97886665372100240670E0 };
static double Mtools_nes[] = {/* 1.00000000000000000000E0,*/
2.26052863220117276590E0, 9.39603524938001434673E0, 1.20489539808096656605E1,
1.70814450747565897222E1, 9.60896809063285878198E0, 3.36907645100081516050E0 };
double normal(double a)
{
double x;
double y;
double z;
x = a * DBA_SQRT2OVER2;
z = (x > 0 ? x : -x);
if (z < DBA_SQRT2OVER2)
y = 0.5 + 0.5 * normal_error(x);
else {
y = 0.5 * normal_error_comp(z);
if (x > 0.0)
y = 1.0 - y;
}
return(y);
}
inline
double polynomial_1(double x, double coef[], int N)
{
double* p = coef;
double ans = x + *p++;
int i = N - 1;
do ans = ans * x + *p++;
while (--i);
return(ans);
}
inline
double polynomial(double x, double coef[], int N)
{
double* p = coef;
double ans = *p++;
do ans = ans * x + *p++;
while (--N);
return(ans);
}
double normal_error(double x)
{
double y;
double z;
z = x * x;
if (z > 1.0)
return(1.0 - normal_error_comp(x));
y = x * polynomial(z, Mtools_T, 4) / polynomial_1(z, Mtools_U, 5);
return(y);
}
double normal_error_comp(double a)
{
double p;
double q;
double x;
double y;
double z;
x = (a > 0.0 ? a : -a);
if (x < 1.0)
return(1.0 - normal_error(a));
z = -a * a;
if (z < -DBA_MAXLOG) {
under:
dba_error("normal_error_comp", DBA_ERROR_UNDERFLOW);
return(0.0);
}
z = exp(z);
if (x < 8.0) {
p = polynomial(x, Mtools_nep, 8);
q = polynomial_1(x, Mtools_neq, 8);
}
else {
p = polynomial(x, Mtools_ner, 5);
q = polynomial_1(x, Mtools_nes, 6);
}
y = (z * p) / q;
if (a < 0.0)
y = 2.0 - y;
if (y == 0.0)
goto under;
return(y);
}