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470 lines (454 loc) · 24.8 KB
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#include "heston.h"
namespace Heston {
namespace MC {
namespace Direct {
std::vector<double>
call_price(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_price = 0.;
double sum_2_price = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S = S_0;
double V = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V = kappa * (theta - V) * dt + sigma * sqrt(std::abs(V)) * dW_v;
double d_S = r * S * dt + S * sqrt(std::abs(V)) * dW_s;
V += d_V;
S += d_S;
}
sum_price += D * std::max(S - K, 0.);
sum_2_price += std::pow(D * std::max(S - K, 0.), 2);
}
double m = sum_price / (double) M;
double v = ((1 / (double) M) * sum_2_price - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
std::vector<double>
call_delta(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M, double &delta_S) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_delta = 0.;
double sum_2_delta = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S = S_0;
double V = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V = kappa * (theta - V) * dt + sigma * sqrt(std::abs(V)) * dW_v;
double d_S = r * S * dt + S * sqrt(std::abs(V)) * dW_s;
V += d_V;
S += d_S;
}
double price_h = D * std::max(S + delta_S - K, 0.);
double price_m = D * std::max(S - K, 0.);
sum_delta += (price_h - price_m) / delta_S;
sum_2_delta += std::pow((price_h - price_m) / delta_S, 2);
}
double m = sum_delta / (double) M;
double v = ((1 / (double) M) * sum_2_delta - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
std::vector<double>
call_gamma(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M, double &delta_S) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_gamma = 0.;
double sum_2_gamma = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S = S_0;
double V = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V = kappa * (theta - V) * dt + sigma * sqrt(std::abs(V)) * dW_v;
double d_S = r * S * dt + S * sqrt(std::abs(V)) * dW_s;
V += d_V;
S += d_S;
}
double price_h = D * std::max(S + delta_S - K, 0.);
double price_m = D * std::max(S - K, 0.);
double price_l = D * std::max(S - delta_S - K, 0.);
sum_gamma += (price_h - 2 * price_m + price_l) / (delta_S * delta_S);
sum_2_gamma += std::pow((price_h - 2 * price_m + price_l) / (delta_S * delta_S), 2);
}
double m = sum_gamma / (double) M;
double v = ((1 / (double) M) * sum_2_gamma - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
std::vector<double>
call_vega(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M, double &delta_sigma) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_gamma = 0.;
double sum_2_gamma = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S_h = S_0;
double S_m = S_0;
double V_h = 0.04;
double V_m = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V_m = kappa * (theta - V_m) * dt + sigma * sqrt(std::abs(V_m)) * dW_v;
double d_V_h = d_V_m + delta_sigma / N;
double d_S_h = r * S_h * dt + S_h * sqrt(std::abs(V_h)) * dW_s;
double d_S_m = r * S_m * dt + S_m * sqrt(std::abs(V_m)) * dW_s;
V_h += d_V_h;
V_m += d_V_m;
S_h += d_S_h;
S_m += d_S_m;
}
double price_h = D * std::max(S_h - K, 0.);
double price_m = D * std::max(S_m - K, 0.);
sum_gamma += (price_h - price_m) / delta_sigma;
sum_2_gamma += std::pow((price_h - price_m) / delta_sigma, 2);
}
double m = sum_gamma / (double) M;
double v = ((1 / (double) M) * sum_2_gamma - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
}
namespace Antithetic {
std::vector<double>
call_price(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_price = 0.;
double sum_2_price = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S_1 = S_0;
double V_1 = 0.04;
double S_2 = S_0;
double V_2 = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V_1 = kappa * (theta - V_1) * dt + sigma * sqrt(std::abs(V_1)) * dW_v;
double d_S_1 = r * S_1 * dt + S_1 * sqrt(std::abs(V_1)) * dW_s;
double d_V_2 = kappa * (theta - V_2) * dt + sigma * sqrt(std::abs(V_2)) * (-dW_v);
double d_S_2 = r * S_2 * dt + S_2 * sqrt(std::abs(V_2)) * (-dW_s);
V_1 += d_V_1;
S_1 += d_S_1;
V_2 += d_V_2;
S_2 += d_S_2;
}
sum_price += D * (std::max(S_1 - K, 0.) + std::max(S_2 - K, 0.)) / 2;
sum_2_price += std::pow(D * (std::max(S_1 - K, 0.) + std::max(S_2 - K, 0.)) / 2, 2);
}
double m = sum_price / (double) M;
double v = ((1 / (double) M) * sum_2_price - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
std::vector<double>
call_delta(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M, double &delta_S) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_delta = 0.;
double sum_2_delta = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S_1 = S_0;
double V_1 = 0.04;
double S_2 = S_0;
double V_2 = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V_1 = kappa * (theta - V_1) * dt + sigma * sqrt(std::abs(V_1)) * dW_v;
double d_S_1 = r * S_1 * dt + S_1 * sqrt(std::abs(V_1)) * dW_s;
double d_V_2 = kappa * (theta - V_2) * dt + sigma * sqrt(std::abs(V_2)) * (-dW_v);
double d_S_2 = r * S_2 * dt + S_2 * sqrt(std::abs(V_2)) * (-dW_s);
V_1 += d_V_1;
S_1 += d_S_1;
V_2 += d_V_2;
S_2 += d_S_2;
}
double price_1_h = D * std::max(S_1 + delta_S - K, 0.);
double price_1_m = D * std::max(S_1 - K, 0.);
double price_2_h = D * std::max(S_2 + delta_S - K, 0.);
double price_2_m = D * std::max(S_2 - K, 0.);
sum_delta += ((price_1_h - price_1_m) / delta_S + (price_2_h - price_2_m) / delta_S) / 2;
sum_2_delta += std::pow(((price_1_h - price_1_m) / delta_S + (price_2_h - price_2_m) / delta_S) / 2, 2);
}
double m = sum_delta / (double) M;
double v = ((1 / (double) M) * sum_2_delta - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
std::vector<double>
call_gamma(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M, double &delta_S) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_gamma = 0.;
double sum_2_gamma = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S_1 = S_0;
double V_1 = 0.04;
double S_2 = S_0;
double V_2 = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V_1 = kappa * (theta - V_1) * dt + sigma * sqrt(std::abs(V_1)) * dW_v;
double d_S_1 = r * S_1 * dt + S_1 * sqrt(std::abs(V_1)) * dW_s;
double d_V_2 = kappa * (theta - V_2) * dt + sigma * sqrt(std::abs(V_2)) * (-dW_v);
double d_S_2 = r * S_2 * dt + S_2 * sqrt(std::abs(V_2)) * (-dW_s);
V_1 += d_V_1;
S_1 += d_S_1;
V_2 += d_V_2;
S_2 += d_S_2;
}
double price_1_h = D * std::max(S_1 + delta_S - K, 0.);
double price_1_m = D * std::max(S_1 - K, 0.);
double price_1_l = D * std::max(S_1 - delta_S - K, 0.);
double price_2_h = D * std::max(S_2 + delta_S - K, 0.);
double price_2_m = D * std::max(S_2 - K, 0.);
double price_2_l = D * std::max(S_2 - delta_S - K, 0.);
sum_gamma += ((price_1_h - 2 * price_1_m + price_1_l) / (delta_S * delta_S) + (price_2_h - 2 * price_2_m + price_2_l) / (delta_S * delta_S)) / 2;
sum_2_gamma += std::pow(((price_1_h - 2 * price_1_m + price_1_l) / (delta_S * delta_S) + (price_2_h - 2 * price_2_m + price_2_l) / (delta_S * delta_S)) / 2, 2);
}
double m = sum_gamma / (double) M;
double v = ((1 / (double) M) * sum_2_gamma - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
std::vector<double>
call_vega(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho,
double &T, int &M, double &delta_sigma) {
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
double sum_gamma = 0.;
double sum_2_gamma = 0.;
double D = exp(-r * T);
int N = 256;
double dt = T / (double) N;
for (int i = 0; i < M; ++i) {
double S_1_h = S_0;
double S_1_m = S_0;
double V_1_h = 0.04;
double V_1_m = 0.04;
double S_2_h = S_0;
double S_2_m = S_0;
double V_2_h = 0.04;
double V_2_m = 0.04;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double d_V_1_m = kappa * (theta - V_1_m) * dt + sigma * sqrt(std::abs(V_1_m)) * dW_v;
double d_V_1_h = d_V_1_m + delta_sigma / N;
double d_V_2_m = kappa * (theta - V_2_m) * dt + sigma * sqrt(std::abs(V_2_m)) * (-dW_v);
double d_V_2_h = d_V_2_m + delta_sigma / N;
double d_S_1_h = r * S_1_h * dt + S_1_h * sqrt(std::abs(V_1_h)) * dW_s;
double d_S_1_m = r * S_1_m * dt + S_1_m * sqrt(std::abs(V_1_m)) * dW_s;
double d_S_2_h = r * S_2_h * dt + S_2_h * sqrt(std::abs(V_2_h)) * (-dW_s);
double d_S_2_m = r * S_2_m * dt + S_2_m * sqrt(std::abs(V_2_m)) * (-dW_s);
V_1_h += d_V_1_h;
V_1_m += d_V_1_m;
S_1_h += d_S_1_h;
S_1_m += d_S_1_m;
V_2_h += d_V_2_h;
V_2_m += d_V_2_m;
S_2_h += d_S_2_h;
S_2_m += d_S_2_m;
}
double price_1_h = D * std::max(S_1_h - K, 0.);
double price_1_m = D * std::max(S_1_m - K, 0.);
double price_2_h = D * std::max(S_2_h - K, 0.);
double price_2_m = D * std::max(S_2_m - K, 0.);
sum_gamma += ((price_1_h - price_1_m) / delta_sigma + (price_2_h - price_2_m) / delta_sigma) / 2;
sum_2_gamma += std::pow(((price_1_h - price_1_m) / delta_sigma + (price_2_h - price_2_m) / delta_sigma) / 2, 2);
}
double m = sum_gamma / (double) M;
double v = ((1 / (double) M) * sum_2_gamma - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration};
return res;
}
}
namespace ControlVariate {
std::vector<double>
call_price(double &S_0, double &K, double &r, double &kappa, double &theta, double &sigma, double &rho, double &T, int &M){
if (2 * kappa * theta < sigma * sigma) {
std::cout << "Condition to ensure V_t > 0 not satisfied" << std::endl;
std::exit(0);
}
std::clock_t t = clock();
//pilot simulation
double corr = 0.;//correlation between Y and Z, just for information
double c = 0.;
int p = 1000;
int N = 256;
double V_0 = 0.04;
double dt = T / (double) N;
double D = exp(-r * T);
double E_Y = 0.;
double E_Y_2 = 0.;
std::vector<double> V_Y;
std::vector<double> V_Z;
double Var_Z = 0.;
//generate Y and Z samples
for (int i(0); i<p; ++i) {
double S_H = S_0;
double S_GBM = S_0;
double V = V_0;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double dW_GBM = dW_s;
double d_V = kappa * (theta - V) * dt + sigma * sqrt(std::abs(V)) * dW_v;
double d_S_H = r * S_H * dt + S_H * sqrt(std::abs(V)) * dW_s;
double d_S_GBM = r * S_GBM * dt + S_GBM * sqrt(V_0) * dW_s;
V += d_V;
S_H += d_S_H;
S_GBM += d_S_GBM;
}
V_Y.push_back(D * std::max(S_H - K, 0.));
V_Z.push_back(D * std::max(S_GBM - K, 0.));
E_Y += V_Y[i] / p;
E_Y_2 += std::pow(V_Y[i], 2) / p;
Var_Z += std::pow((V_Z[i] - S_0), 2) / p;
}
for (int i(0); i<p; ++i) {
c += - (V_Y[i] - E_Y) * (V_Z[i] - S_0) / (p*Var_Z);
corr += (V_Y[i] - E_Y) * (V_Z[i] - S_0) / (p*sqrt(Var_Z * (E_Y_2 - std::pow(E_Y, 2))));
}
double sum_price = 0.;
double sum_2_price = 0.;
for (int i = 0; i < M; ++i) {
double S_H = S_0;
double S_GBM = S_0;
double V = V_0;
for (int j = 0; j < N; ++j) {
double X_1 = (double) rand() / RAND_MAX;
double X_2 = (double) rand() / RAND_MAX;
double Z_1 = sqrt(std::abs(2 * log(X_1))) * sin(2 * M_PI * X_2);
double Z_2 = sqrt(std::abs(2 * log(X_1))) * cos(2 * M_PI * X_2);
double dW_v = sqrt(dt) * Z_1;
double dW_s = sqrt(dt) * (rho * Z_1 + sqrt(1 - rho * rho) * Z_2);
double dW_GBM = dW_s;
double d_V = kappa * (theta - V) * dt + sigma * sqrt(std::abs(V)) * dW_v;
double d_S_H = r * S_H * dt + S_H * sqrt(std::abs(V)) * dW_s;
double d_S_GBM = r * S_GBM * dt + S_GBM * sqrt(V_0) * dW_s;
V += d_V;
S_H += d_S_H;
S_GBM += d_S_GBM;
}
double Y = D*std::max(S_H - K, 0.);
double Z = D*std::max(S_GBM - K, 0.);
double E_Z = GBM::Analytic::call_price(S_0, K, r, sqrt(V_0), T);
double theta_c = Y + c*(Z - E_Z);
sum_price += theta_c;
sum_2_price += std::pow(theta_c, 2);
}
double m = sum_price / (double) M;
double v = ((1 / (double) M) * sum_2_price - m * m) / (double) M;
double duration = (std::clock() - t) / (double) CLOCKS_PER_SEC;
std::vector<double> res = {m, 1.96*sqrt(v / M), duration, corr};
return res;
}
}
}
}