EQ2EMu/EQ2/source/depends/boost_1_72_0/libs/math/example/autodiff_black_scholes.cpp

//           Copyright Matthew Pulver 2018 - 2019.
// Distributed under the Boost Software License, Version 1.0.
//      (See accompanying file LICENSE_1_0.txt or copy at
//           https://www.boost.org/LICENSE_1_0.txt)

#include <boost/math/differentiation/autodiff.hpp>
#include <iostream>
#include <stdexcept>

using namespace boost::math::constants;
using namespace boost::math::differentiation;

// Equations and function/variable names are from
// https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks

// Standard normal probability density function
template <typename X>
X phi(X const& x) {
  return one_div_root_two_pi<X>() * exp(-0.5 * x * x);
}

// Standard normal cumulative distribution function
template <typename X>
X Phi(X const& x) {
  return 0.5 * erfc(-one_div_root_two<X>() * x);
}

enum class CP { call, put };

// Assume zero annual dividend yield (q=0).
template <typename Price, typename Sigma, typename Tau, typename Rate>
promote<Price, Sigma, Tau, Rate> black_scholes_option_price(CP cp,
                                                            double K,
                                                            Price const& S,
                                                            Sigma const& sigma,
                                                            Tau const& tau,
                                                            Rate const& r) {
  using namespace std;
  auto const d1 = (log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
  auto const d2 = (log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
  switch (cp) {
    case CP::call:
      return S * Phi(d1) - exp(-r * tau) * K * Phi(d2);
    case CP::put:
      return exp(-r * tau) * K * Phi(-d2) - S * Phi(-d1);
    default:
      throw std::runtime_error("Invalid CP value.");
  }
}

int main() {
  double const K = 100.0;  // Strike price.
  auto const variables = make_ftuple<double, 3, 3, 1, 1>(105, 5, 30.0 / 365, 1.25 / 100);
  auto const& S = std::get<0>(variables);      // Stock price.
  auto const& sigma = std::get<1>(variables);  // Volatility.
  auto const& tau = std::get<2>(variables);    // Time to expiration in years. (30 days).
  auto const& r = std::get<3>(variables);      // Interest rate.
  auto const call_price = black_scholes_option_price(CP::call, K, S, sigma, tau, r);
  auto const put_price = black_scholes_option_price(CP::put, K, S, sigma, tau, r);

  double const d1 = static_cast<double>((log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau)));
  double const d2 = static_cast<double>((log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau)));
  double const formula_call_delta = +Phi(+d1);
  double const formula_put_delta = -Phi(-d1);
  double const formula_vega = static_cast<double>(S * phi(d1) * sqrt(tau));
  double const formula_call_theta =
      static_cast<double>(-S * phi(d1) * sigma / (2 * sqrt(tau)) - r * K * exp(-r * tau) * Phi(+d2));
  double const formula_put_theta =
      static_cast<double>(-S * phi(d1) * sigma / (2 * sqrt(tau)) + r * K * exp(-r * tau) * Phi(-d2));
  double const formula_call_rho = static_cast<double>(+K * tau * exp(-r * tau) * Phi(+d2));
  double const formula_put_rho = static_cast<double>(-K * tau * exp(-r * tau) * Phi(-d2));
  double const formula_gamma = static_cast<double>(phi(d1) / (S * sigma * sqrt(tau)));
  double const formula_vanna = static_cast<double>(-phi(d1) * d2 / sigma);
  double const formula_charm =
      static_cast<double>(phi(d1) * (d2 * sigma * sqrt(tau) - 2 * r * tau) / (2 * tau * sigma * sqrt(tau)));
  double const formula_vomma = static_cast<double>(S * phi(d1) * sqrt(tau) * d1 * d2 / sigma);
  double const formula_veta = static_cast<double>(-S * phi(d1) * sqrt(tau) *
                                                  (r * d1 / (sigma * sqrt(tau)) - (1 + d1 * d2) / (2 * tau)));
  double const formula_speed =
      static_cast<double>(-phi(d1) * (d1 / (sigma * sqrt(tau)) + 1) / (S * S * sigma * sqrt(tau)));
  double const formula_zomma = static_cast<double>(phi(d1) * (d1 * d2 - 1) / (S * sigma * sigma * sqrt(tau)));
  double const formula_color =
      static_cast<double>(-phi(d1) / (2 * S * tau * sigma * sqrt(tau)) *
                          (1 + (2 * r * tau - d2 * sigma * sqrt(tau)) * d1 / (sigma * sqrt(tau))));
  double const formula_ultima =
      -formula_vega * static_cast<double>((d1 * d2 * (1 - d1 * d2) + d1 * d1 + d2 * d2) / (sigma * sigma));

  std::cout << std::setprecision(std::numeric_limits<double>::digits10)
            << "autodiff black-scholes call price = " << call_price.derivative(0, 0, 0, 0) << '\n'
            << "autodiff black-scholes put  price = " << put_price.derivative(0, 0, 0, 0) << '\n'
            << "\n## First-order Greeks\n"
            << "autodiff call delta = " << call_price.derivative(1, 0, 0, 0) << '\n'
            << " formula call delta = " << formula_call_delta << '\n'
            << "autodiff call vega  = " << call_price.derivative(0, 1, 0, 0) << '\n'
            << " formula call vega  = " << formula_vega << '\n'
            << "autodiff call theta = " << -call_price.derivative(0, 0, 1, 0)
            << '\n'  // minus sign due to tau = T-time
            << " formula call theta = " << formula_call_theta << '\n'
            << "autodiff call rho   = " << call_price.derivative(0, 0, 0, 1) << '\n'
            << " formula call rho   = " << formula_call_rho << '\n'
            << '\n'
            << "autodiff put delta = " << put_price.derivative(1, 0, 0, 0) << '\n'
            << " formula put delta = " << formula_put_delta << '\n'
            << "autodiff put vega  = " << put_price.derivative(0, 1, 0, 0) << '\n'
            << " formula put vega  = " << formula_vega << '\n'
            << "autodiff put theta = " << -put_price.derivative(0, 0, 1, 0) << '\n'
            << " formula put theta = " << formula_put_theta << '\n'
            << "autodiff put rho   = " << put_price.derivative(0, 0, 0, 1) << '\n'
            << " formula put rho   = " << formula_put_rho << '\n'
            << "\n## Second-order Greeks\n"
            << "autodiff call gamma = " << call_price.derivative(2, 0, 0, 0) << '\n'
            << "autodiff put  gamma = " << put_price.derivative(2, 0, 0, 0) << '\n'
            << "      formula gamma = " << formula_gamma << '\n'
            << "autodiff call vanna = " << call_price.derivative(1, 1, 0, 0) << '\n'
            << "autodiff put  vanna = " << put_price.derivative(1, 1, 0, 0) << '\n'
            << "      formula vanna = " << formula_vanna << '\n'
            << "autodiff call charm = " << -call_price.derivative(1, 0, 1, 0) << '\n'
            << "autodiff put  charm = " << -put_price.derivative(1, 0, 1, 0) << '\n'
            << "      formula charm = " << formula_charm << '\n'
            << "autodiff call vomma = " << call_price.derivative(0, 2, 0, 0) << '\n'
            << "autodiff put  vomma = " << put_price.derivative(0, 2, 0, 0) << '\n'
            << "      formula vomma = " << formula_vomma << '\n'
            << "autodiff call veta = " << call_price.derivative(0, 1, 1, 0) << '\n'
            << "autodiff put  veta = " << put_price.derivative(0, 1, 1, 0) << '\n'
            << "      formula veta = " << formula_veta << '\n'
            << "\n## Third-order Greeks\n"
            << "autodiff call speed = " << call_price.derivative(3, 0, 0, 0) << '\n'
            << "autodiff put  speed = " << put_price.derivative(3, 0, 0, 0) << '\n'
            << "      formula speed = " << formula_speed << '\n'
            << "autodiff call zomma = " << call_price.derivative(2, 1, 0, 0) << '\n'
            << "autodiff put  zomma = " << put_price.derivative(2, 1, 0, 0) << '\n'
            << "      formula zomma = " << formula_zomma << '\n'
            << "autodiff call color = " << call_price.derivative(2, 0, 1, 0) << '\n'
            << "autodiff put  color = " << put_price.derivative(2, 0, 1, 0) << '\n'
            << "      formula color = " << formula_color << '\n'
            << "autodiff call ultima = " << call_price.derivative(0, 3, 0, 0) << '\n'
            << "autodiff put  ultima = " << put_price.derivative(0, 3, 0, 0) << '\n'
            << "      formula ultima = " << formula_ultima << '\n';
  return 0;
}
/*
Output:
autodiff black-scholes call price = 56.5136030677739
autodiff black-scholes put  price = 51.4109161009333

## First-order Greeks
autodiff call delta = 0.773818444921273
 formula call delta = 0.773818444921274
autodiff call vega  = 9.05493427705736
 formula call vega  = 9.05493427705736
autodiff call theta = -275.73013426444
 formula call theta = -275.73013426444
autodiff call rho   = 2.03320550539396
 formula call rho   = 2.03320550539396

autodiff put delta = -0.226181555078726
 formula put delta = -0.226181555078726
autodiff put vega  = 9.05493427705736
 formula put vega  = 9.05493427705736
autodiff put theta = -274.481417851526
 formula put theta = -274.481417851526
autodiff put rho   = -6.17753255212599
 formula put rho   = -6.17753255212599

## Second-order Greeks
autodiff call gamma = 0.00199851912993254
autodiff put  gamma = 0.00199851912993254
      formula gamma = 0.00199851912993254
autodiff call vanna = 0.0410279463126531
autodiff put  vanna = 0.0410279463126531
      formula vanna = 0.0410279463126531
autodiff call charm = -1.2505564233679
autodiff put  charm = -1.2505564233679
      formula charm = -1.2505564233679
autodiff call vomma = -0.928114149313108
autodiff put  vomma = -0.928114149313108
      formula vomma = -0.928114149313107
autodiff call veta = 26.7947073115641
autodiff put  veta = 26.7947073115641
      formula veta = 26.7947073115641

## Third-order Greeks
autodiff call speed = -2.90117322380992e-05
autodiff put  speed = -2.90117322380992e-05
      formula speed = -2.90117322380992e-05
autodiff call zomma = -0.000604548369901419
autodiff put  zomma = -0.000604548369901419
      formula zomma = -0.000604548369901419
autodiff call color = -0.0184014426606065
autodiff put  color = -0.0184014426606065
      formula color = -0.0184014426606065
autodiff call ultima = -0.0922426864775683
autodiff put  ultima = -0.0922426864775683
      formula ultima = -0.0922426864775685
**/