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// 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 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 S = make_fvar<double, 2>(105); // Stock price.
double const sigma = 5; // Volatility.
double const tau = 30.0 / 365; // Time to expiration in years. (30 days).
double const r = 1.25 / 100; // 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);
std::cout << "black-scholes call price = " << call_price.derivative(0) << '\n'
<< "black-scholes put price = " << put_price.derivative(0) << '\n'
<< "call delta = " << call_price.derivative(1) << '\n'
<< "put delta = " << put_price.derivative(1) << '\n'
<< "call gamma = " << call_price.derivative(2) << '\n'
<< "put gamma = " << put_price.derivative(2) << '\n';
return 0;
}
/*
Output:
black-scholes call price = 56.5136
black-scholes put price = 51.4109
call delta = 0.773818
put delta = -0.226182
call gamma = 0.00199852
put gamma = 0.00199852
**/
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