1 // Copyright Matthew Pulver 2018 - 2019.
2 // Distributed under the Boost Software License, Version 1.0.
3 // (See accompanying file LICENSE_1_0.txt or copy at
4 // https://www.boost.org/LICENSE_1_0.txt)
5
6 #include <boost/math/differentiation/autodiff.hpp>
7 #include <iostream>
8 #include <stdexcept>
9
10 using namespace boost::math::constants;
11 using namespace boost::math::differentiation;
12
13 // Equations and function/variable names are from
14 // https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks
15
16 // Standard normal probability density function
17 template <typename X>
phi(X const & x)18 X phi(X const& x) {
19 return one_div_root_two_pi<X>() * exp(-0.5 * x * x);
20 }
21
22 // Standard normal cumulative distribution function
23 template <typename X>
Phi(X const & x)24 X Phi(X const& x) {
25 return 0.5 * erfc(-one_div_root_two<X>() * x);
26 }
27
28 enum class CP { call, put };
29
30 // Assume zero annual dividend yield (q=0).
31 template <typename Price, typename Sigma, typename Tau, typename Rate>
black_scholes_option_price(CP cp,double K,Price const & S,Sigma const & sigma,Tau const & tau,Rate const & r)32 promote<Price, Sigma, Tau, Rate> black_scholes_option_price(CP cp,
33 double K,
34 Price const& S,
35 Sigma const& sigma,
36 Tau const& tau,
37 Rate const& r) {
38 using namespace std;
39 auto const d1 = (log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
40 auto const d2 = (log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
41 switch (cp) {
42 case CP::call:
43 return S * Phi(d1) - exp(-r * tau) * K * Phi(d2);
44 case CP::put:
45 return exp(-r * tau) * K * Phi(-d2) - S * Phi(-d1);
46 default:
47 throw std::runtime_error("Invalid CP value.");
48 }
49 }
50
main()51 int main() {
52 double const K = 100.0; // Strike price.
53 auto const variables = make_ftuple<double, 3, 3, 1, 1>(105, 5, 30.0 / 365, 1.25 / 100);
54 auto const& S = std::get<0>(variables); // Stock price.
55 auto const& sigma = std::get<1>(variables); // Volatility.
56 auto const& tau = std::get<2>(variables); // Time to expiration in years. (30 days).
57 auto const& r = std::get<3>(variables); // Interest rate.
58 auto const call_price = black_scholes_option_price(CP::call, K, S, sigma, tau, r);
59 auto const put_price = black_scholes_option_price(CP::put, K, S, sigma, tau, r);
60
61 double const d1 = static_cast<double>((log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau)));
62 double const d2 = static_cast<double>((log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau)));
63 double const formula_call_delta = +Phi(+d1);
64 double const formula_put_delta = -Phi(-d1);
65 double const formula_vega = static_cast<double>(S * phi(d1) * sqrt(tau));
66 double const formula_call_theta =
67 static_cast<double>(-S * phi(d1) * sigma / (2 * sqrt(tau)) - r * K * exp(-r * tau) * Phi(+d2));
68 double const formula_put_theta =
69 static_cast<double>(-S * phi(d1) * sigma / (2 * sqrt(tau)) + r * K * exp(-r * tau) * Phi(-d2));
70 double const formula_call_rho = static_cast<double>(+K * tau * exp(-r * tau) * Phi(+d2));
71 double const formula_put_rho = static_cast<double>(-K * tau * exp(-r * tau) * Phi(-d2));
72 double const formula_gamma = static_cast<double>(phi(d1) / (S * sigma * sqrt(tau)));
73 double const formula_vanna = static_cast<double>(-phi(d1) * d2 / sigma);
74 double const formula_charm =
75 static_cast<double>(phi(d1) * (d2 * sigma * sqrt(tau) - 2 * r * tau) / (2 * tau * sigma * sqrt(tau)));
76 double const formula_vomma = static_cast<double>(S * phi(d1) * sqrt(tau) * d1 * d2 / sigma);
77 double const formula_veta = static_cast<double>(-S * phi(d1) * sqrt(tau) *
78 (r * d1 / (sigma * sqrt(tau)) - (1 + d1 * d2) / (2 * tau)));
79 double const formula_speed =
80 static_cast<double>(-phi(d1) * (d1 / (sigma * sqrt(tau)) + 1) / (S * S * sigma * sqrt(tau)));
81 double const formula_zomma = static_cast<double>(phi(d1) * (d1 * d2 - 1) / (S * sigma * sigma * sqrt(tau)));
82 double const formula_color =
83 static_cast<double>(-phi(d1) / (2 * S * tau * sigma * sqrt(tau)) *
84 (1 + (2 * r * tau - d2 * sigma * sqrt(tau)) * d1 / (sigma * sqrt(tau))));
85 double const formula_ultima =
86 -formula_vega * static_cast<double>((d1 * d2 * (1 - d1 * d2) + d1 * d1 + d2 * d2) / (sigma * sigma));
87
88 std::cout << std::setprecision(std::numeric_limits<double>::digits10)
89 << "autodiff black-scholes call price = " << call_price.derivative(0, 0, 0, 0) << '\n'
90 << "autodiff black-scholes put price = " << put_price.derivative(0, 0, 0, 0) << '\n'
91 << "\n## First-order Greeks\n"
92 << "autodiff call delta = " << call_price.derivative(1, 0, 0, 0) << '\n'
93 << " formula call delta = " << formula_call_delta << '\n'
94 << "autodiff call vega = " << call_price.derivative(0, 1, 0, 0) << '\n'
95 << " formula call vega = " << formula_vega << '\n'
96 << "autodiff call theta = " << -call_price.derivative(0, 0, 1, 0)
97 << '\n' // minus sign due to tau = T-time
98 << " formula call theta = " << formula_call_theta << '\n'
99 << "autodiff call rho = " << call_price.derivative(0, 0, 0, 1) << '\n'
100 << " formula call rho = " << formula_call_rho << '\n'
101 << '\n'
102 << "autodiff put delta = " << put_price.derivative(1, 0, 0, 0) << '\n'
103 << " formula put delta = " << formula_put_delta << '\n'
104 << "autodiff put vega = " << put_price.derivative(0, 1, 0, 0) << '\n'
105 << " formula put vega = " << formula_vega << '\n'
106 << "autodiff put theta = " << -put_price.derivative(0, 0, 1, 0) << '\n'
107 << " formula put theta = " << formula_put_theta << '\n'
108 << "autodiff put rho = " << put_price.derivative(0, 0, 0, 1) << '\n'
109 << " formula put rho = " << formula_put_rho << '\n'
110 << "\n## Second-order Greeks\n"
111 << "autodiff call gamma = " << call_price.derivative(2, 0, 0, 0) << '\n'
112 << "autodiff put gamma = " << put_price.derivative(2, 0, 0, 0) << '\n'
113 << " formula gamma = " << formula_gamma << '\n'
114 << "autodiff call vanna = " << call_price.derivative(1, 1, 0, 0) << '\n'
115 << "autodiff put vanna = " << put_price.derivative(1, 1, 0, 0) << '\n'
116 << " formula vanna = " << formula_vanna << '\n'
117 << "autodiff call charm = " << -call_price.derivative(1, 0, 1, 0) << '\n'
118 << "autodiff put charm = " << -put_price.derivative(1, 0, 1, 0) << '\n'
119 << " formula charm = " << formula_charm << '\n'
120 << "autodiff call vomma = " << call_price.derivative(0, 2, 0, 0) << '\n'
121 << "autodiff put vomma = " << put_price.derivative(0, 2, 0, 0) << '\n'
122 << " formula vomma = " << formula_vomma << '\n'
123 << "autodiff call veta = " << call_price.derivative(0, 1, 1, 0) << '\n'
124 << "autodiff put veta = " << put_price.derivative(0, 1, 1, 0) << '\n'
125 << " formula veta = " << formula_veta << '\n'
126 << "\n## Third-order Greeks\n"
127 << "autodiff call speed = " << call_price.derivative(3, 0, 0, 0) << '\n'
128 << "autodiff put speed = " << put_price.derivative(3, 0, 0, 0) << '\n'
129 << " formula speed = " << formula_speed << '\n'
130 << "autodiff call zomma = " << call_price.derivative(2, 1, 0, 0) << '\n'
131 << "autodiff put zomma = " << put_price.derivative(2, 1, 0, 0) << '\n'
132 << " formula zomma = " << formula_zomma << '\n'
133 << "autodiff call color = " << call_price.derivative(2, 0, 1, 0) << '\n'
134 << "autodiff put color = " << put_price.derivative(2, 0, 1, 0) << '\n'
135 << " formula color = " << formula_color << '\n'
136 << "autodiff call ultima = " << call_price.derivative(0, 3, 0, 0) << '\n'
137 << "autodiff put ultima = " << put_price.derivative(0, 3, 0, 0) << '\n'
138 << " formula ultima = " << formula_ultima << '\n';
139 return 0;
140 }
141 /*
142 Output:
143 autodiff black-scholes call price = 56.5136030677739
144 autodiff black-scholes put price = 51.4109161009333
145
146 ## First-order Greeks
147 autodiff call delta = 0.773818444921273
148 formula call delta = 0.773818444921274
149 autodiff call vega = 9.05493427705736
150 formula call vega = 9.05493427705736
151 autodiff call theta = -275.73013426444
152 formula call theta = -275.73013426444
153 autodiff call rho = 2.03320550539396
154 formula call rho = 2.03320550539396
155
156 autodiff put delta = -0.226181555078726
157 formula put delta = -0.226181555078726
158 autodiff put vega = 9.05493427705736
159 formula put vega = 9.05493427705736
160 autodiff put theta = -274.481417851526
161 formula put theta = -274.481417851526
162 autodiff put rho = -6.17753255212599
163 formula put rho = -6.17753255212599
164
165 ## Second-order Greeks
166 autodiff call gamma = 0.00199851912993254
167 autodiff put gamma = 0.00199851912993254
168 formula gamma = 0.00199851912993254
169 autodiff call vanna = 0.0410279463126531
170 autodiff put vanna = 0.0410279463126531
171 formula vanna = 0.0410279463126531
172 autodiff call charm = -1.2505564233679
173 autodiff put charm = -1.2505564233679
174 formula charm = -1.2505564233679
175 autodiff call vomma = -0.928114149313108
176 autodiff put vomma = -0.928114149313108
177 formula vomma = -0.928114149313107
178 autodiff call veta = 26.7947073115641
179 autodiff put veta = 26.7947073115641
180 formula veta = 26.7947073115641
181
182 ## Third-order Greeks
183 autodiff call speed = -2.90117322380992e-05
184 autodiff put speed = -2.90117322380992e-05
185 formula speed = -2.90117322380992e-05
186 autodiff call zomma = -0.000604548369901419
187 autodiff put zomma = -0.000604548369901419
188 formula zomma = -0.000604548369901419
189 autodiff call color = -0.0184014426606065
190 autodiff put color = -0.0184014426606065
191 formula color = -0.0184014426606065
192 autodiff call ultima = -0.0922426864775683
193 autodiff put ultima = -0.0922426864775683
194 formula ultima = -0.0922426864775685
195 **/
196