196 lines
10 KiB
C++
196 lines
10 KiB
C++
// Copyright Matthew Pulver 2018 - 2019.
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// Distributed under the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt or copy at
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// https://www.boost.org/LICENSE_1_0.txt)
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#include <boost/math/differentiation/autodiff.hpp>
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#include <iostream>
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#include <stdexcept>
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using namespace boost::math::constants;
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using namespace boost::math::differentiation;
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// Equations and function/variable names are from
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// https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks
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// Standard normal probability density function
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template <typename X>
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X phi(X const& x) {
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return one_div_root_two_pi<X>() * exp(-0.5 * x * x);
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}
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// Standard normal cumulative distribution function
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template <typename X>
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X Phi(X const& x) {
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return 0.5 * erfc(-one_div_root_two<X>() * x);
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}
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enum class CP { call, put };
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// Assume zero annual dividend yield (q=0).
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template <typename Price, typename Sigma, typename Tau, typename Rate>
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promote<Price, Sigma, Tau, Rate> black_scholes_option_price(CP cp,
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double K,
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Price const& S,
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Sigma const& sigma,
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Tau const& tau,
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Rate const& r) {
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using namespace std;
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auto const d1 = (log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
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auto const d2 = (log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
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switch (cp) {
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case CP::call:
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return S * Phi(d1) - exp(-r * tau) * K * Phi(d2);
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case CP::put:
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return exp(-r * tau) * K * Phi(-d2) - S * Phi(-d1);
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default:
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throw std::runtime_error("Invalid CP value.");
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}
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}
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int main() {
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double const K = 100.0; // Strike price.
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auto const variables = make_ftuple<double, 3, 3, 1, 1>(105, 5, 30.0 / 365, 1.25 / 100);
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auto const& S = std::get<0>(variables); // Stock price.
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auto const& sigma = std::get<1>(variables); // Volatility.
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auto const& tau = std::get<2>(variables); // Time to expiration in years. (30 days).
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auto const& r = std::get<3>(variables); // Interest rate.
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auto const call_price = black_scholes_option_price(CP::call, K, S, sigma, tau, r);
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auto const put_price = black_scholes_option_price(CP::put, K, S, sigma, tau, r);
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double const d1 = static_cast<double>((log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau)));
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double const d2 = static_cast<double>((log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau)));
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double const formula_call_delta = +Phi(+d1);
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double const formula_put_delta = -Phi(-d1);
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double const formula_vega = static_cast<double>(S * phi(d1) * sqrt(tau));
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double const formula_call_theta =
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static_cast<double>(-S * phi(d1) * sigma / (2 * sqrt(tau)) - r * K * exp(-r * tau) * Phi(+d2));
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double const formula_put_theta =
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static_cast<double>(-S * phi(d1) * sigma / (2 * sqrt(tau)) + r * K * exp(-r * tau) * Phi(-d2));
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double const formula_call_rho = static_cast<double>(+K * tau * exp(-r * tau) * Phi(+d2));
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double const formula_put_rho = static_cast<double>(-K * tau * exp(-r * tau) * Phi(-d2));
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double const formula_gamma = static_cast<double>(phi(d1) / (S * sigma * sqrt(tau)));
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double const formula_vanna = static_cast<double>(-phi(d1) * d2 / sigma);
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double const formula_charm =
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static_cast<double>(phi(d1) * (d2 * sigma * sqrt(tau) - 2 * r * tau) / (2 * tau * sigma * sqrt(tau)));
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double const formula_vomma = static_cast<double>(S * phi(d1) * sqrt(tau) * d1 * d2 / sigma);
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double const formula_veta = static_cast<double>(-S * phi(d1) * sqrt(tau) *
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(r * d1 / (sigma * sqrt(tau)) - (1 + d1 * d2) / (2 * tau)));
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double const formula_speed =
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static_cast<double>(-phi(d1) * (d1 / (sigma * sqrt(tau)) + 1) / (S * S * sigma * sqrt(tau)));
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double const formula_zomma = static_cast<double>(phi(d1) * (d1 * d2 - 1) / (S * sigma * sigma * sqrt(tau)));
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double const formula_color =
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static_cast<double>(-phi(d1) / (2 * S * tau * sigma * sqrt(tau)) *
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(1 + (2 * r * tau - d2 * sigma * sqrt(tau)) * d1 / (sigma * sqrt(tau))));
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double const formula_ultima =
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-formula_vega * static_cast<double>((d1 * d2 * (1 - d1 * d2) + d1 * d1 + d2 * d2) / (sigma * sigma));
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std::cout << std::setprecision(std::numeric_limits<double>::digits10)
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<< "autodiff black-scholes call price = " << call_price.derivative(0, 0, 0, 0) << '\n'
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<< "autodiff black-scholes put price = " << put_price.derivative(0, 0, 0, 0) << '\n'
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<< "\n## First-order Greeks\n"
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<< "autodiff call delta = " << call_price.derivative(1, 0, 0, 0) << '\n'
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<< " formula call delta = " << formula_call_delta << '\n'
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<< "autodiff call vega = " << call_price.derivative(0, 1, 0, 0) << '\n'
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<< " formula call vega = " << formula_vega << '\n'
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<< "autodiff call theta = " << -call_price.derivative(0, 0, 1, 0)
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<< '\n' // minus sign due to tau = T-time
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<< " formula call theta = " << formula_call_theta << '\n'
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<< "autodiff call rho = " << call_price.derivative(0, 0, 0, 1) << '\n'
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<< " formula call rho = " << formula_call_rho << '\n'
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<< '\n'
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<< "autodiff put delta = " << put_price.derivative(1, 0, 0, 0) << '\n'
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<< " formula put delta = " << formula_put_delta << '\n'
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<< "autodiff put vega = " << put_price.derivative(0, 1, 0, 0) << '\n'
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<< " formula put vega = " << formula_vega << '\n'
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<< "autodiff put theta = " << -put_price.derivative(0, 0, 1, 0) << '\n'
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<< " formula put theta = " << formula_put_theta << '\n'
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<< "autodiff put rho = " << put_price.derivative(0, 0, 0, 1) << '\n'
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<< " formula put rho = " << formula_put_rho << '\n'
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<< "\n## Second-order Greeks\n"
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<< "autodiff call gamma = " << call_price.derivative(2, 0, 0, 0) << '\n'
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<< "autodiff put gamma = " << put_price.derivative(2, 0, 0, 0) << '\n'
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<< " formula gamma = " << formula_gamma << '\n'
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<< "autodiff call vanna = " << call_price.derivative(1, 1, 0, 0) << '\n'
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<< "autodiff put vanna = " << put_price.derivative(1, 1, 0, 0) << '\n'
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<< " formula vanna = " << formula_vanna << '\n'
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<< "autodiff call charm = " << -call_price.derivative(1, 0, 1, 0) << '\n'
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<< "autodiff put charm = " << -put_price.derivative(1, 0, 1, 0) << '\n'
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<< " formula charm = " << formula_charm << '\n'
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<< "autodiff call vomma = " << call_price.derivative(0, 2, 0, 0) << '\n'
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<< "autodiff put vomma = " << put_price.derivative(0, 2, 0, 0) << '\n'
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<< " formula vomma = " << formula_vomma << '\n'
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<< "autodiff call veta = " << call_price.derivative(0, 1, 1, 0) << '\n'
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<< "autodiff put veta = " << put_price.derivative(0, 1, 1, 0) << '\n'
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<< " formula veta = " << formula_veta << '\n'
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<< "\n## Third-order Greeks\n"
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<< "autodiff call speed = " << call_price.derivative(3, 0, 0, 0) << '\n'
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<< "autodiff put speed = " << put_price.derivative(3, 0, 0, 0) << '\n'
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<< " formula speed = " << formula_speed << '\n'
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<< "autodiff call zomma = " << call_price.derivative(2, 1, 0, 0) << '\n'
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<< "autodiff put zomma = " << put_price.derivative(2, 1, 0, 0) << '\n'
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<< " formula zomma = " << formula_zomma << '\n'
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<< "autodiff call color = " << call_price.derivative(2, 0, 1, 0) << '\n'
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<< "autodiff put color = " << put_price.derivative(2, 0, 1, 0) << '\n'
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<< " formula color = " << formula_color << '\n'
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<< "autodiff call ultima = " << call_price.derivative(0, 3, 0, 0) << '\n'
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<< "autodiff put ultima = " << put_price.derivative(0, 3, 0, 0) << '\n'
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<< " formula ultima = " << formula_ultima << '\n';
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return 0;
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}
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/*
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Output:
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autodiff black-scholes call price = 56.5136030677739
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autodiff black-scholes put price = 51.4109161009333
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## First-order Greeks
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autodiff call delta = 0.773818444921273
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formula call delta = 0.773818444921274
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autodiff call vega = 9.05493427705736
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formula call vega = 9.05493427705736
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autodiff call theta = -275.73013426444
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formula call theta = -275.73013426444
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autodiff call rho = 2.03320550539396
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formula call rho = 2.03320550539396
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autodiff put delta = -0.226181555078726
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formula put delta = -0.226181555078726
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autodiff put vega = 9.05493427705736
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formula put vega = 9.05493427705736
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autodiff put theta = -274.481417851526
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formula put theta = -274.481417851526
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autodiff put rho = -6.17753255212599
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formula put rho = -6.17753255212599
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## Second-order Greeks
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autodiff call gamma = 0.00199851912993254
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autodiff put gamma = 0.00199851912993254
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formula gamma = 0.00199851912993254
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autodiff call vanna = 0.0410279463126531
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autodiff put vanna = 0.0410279463126531
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formula vanna = 0.0410279463126531
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autodiff call charm = -1.2505564233679
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autodiff put charm = -1.2505564233679
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formula charm = -1.2505564233679
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autodiff call vomma = -0.928114149313108
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autodiff put vomma = -0.928114149313108
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formula vomma = -0.928114149313107
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autodiff call veta = 26.7947073115641
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autodiff put veta = 26.7947073115641
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formula veta = 26.7947073115641
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## Third-order Greeks
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autodiff call speed = -2.90117322380992e-05
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autodiff put speed = -2.90117322380992e-05
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formula speed = -2.90117322380992e-05
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autodiff call zomma = -0.000604548369901419
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autodiff put zomma = -0.000604548369901419
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formula zomma = -0.000604548369901419
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autodiff call color = -0.0184014426606065
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autodiff put color = -0.0184014426606065
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formula color = -0.0184014426606065
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autodiff call ultima = -0.0922426864775683
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autodiff put ultima = -0.0922426864775683
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formula ultima = -0.0922426864775685
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**/
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