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/* ************************************************************** * C++ Mathematical Expression Toolkit Library * * * * American Option Binomial Pricing Model * * Author: Arash Partow (1999-2024) * * URL: https://www.partow.net/programming/exprtk/index.html * * * * Copyright notice: * * Free use of the Mathematical Expression Toolkit Library is * * permitted under the guidelines and in accordance with the * * most current version of the MIT License. * * https://www.opensource.org/licenses/MIT * * SPDX-License-Identifier: MIT * * * ************************************************************** */ #include #include #include "exprtk.hpp" template void american_option_binomial_option_pricing_model() { typedef exprtk::symbol_table symbol_table_t; typedef exprtk::expression expression_t; typedef exprtk::parser parser_t; const std::string american_option_binomial_model_program = " var dt := t / n; " " var z := exp(r * dt); " " var u := exp(v * sqrt(dt)); " " var p_up := (z * u - 1) / (u^2 - 1); " " var p_down := 1 - p_up; " " var discount := 1 / z; " " " " var option_price[n + 1] := [0]; " " " " for (var i := 0; i <= n; i += 1) " " { " " var base_price := s * u^(2 * i - n); " " " " option_price[i] := " " switch " " { " " case callput_flag == 'call' : max(base_price - k, 0); " " case callput_flag == 'put' : max(k - base_price, 0); " " }; " " }; " " " " for (var j := n - 1; j >= 0; j -= 1) " " { " " for (var i := 0; i <= j; i += 1) " " { " " var base_price_at_node := s * u^(2 * i - j); " " " " var intrinsic_value := " " switch " " { " " case callput_flag == 'call' : base_price_at_node - k; " " case callput_flag == 'put' : k - base_price_at_node; " " }; " " " " var expected_value := discount * " " (p_up * option_price[i + 1] + p_down * option_price[i]); " " " " option_price[i] := max(expected_value, intrinsic_value); " " } " " }; " " " " option_price[0]; "; T s = T( 100.00); // Spot / Stock / Underlying / Base price T k = T( 110.00); // Strike price T v = T( 0.30); // Volatility T t = T( 2.22); // Years to maturity T r = T( 0.05); // Risk free rate T n = T(1000.00); // Number of time steps std::string callput_flag; symbol_table_t symbol_table(symbol_table_t::e_immutable); symbol_table.add_variable("s", s); symbol_table.add_variable("k", k); symbol_table.add_variable("t", t); symbol_table.add_variable("r", r); symbol_table.add_variable("v", v); symbol_table.add_constant("n", n); symbol_table.add_stringvar("callput_flag",callput_flag); expression_t expression; expression.register_symbol_table(symbol_table); parser_t parser; parser.compile(american_option_binomial_model_program,expression); callput_flag = "call"; const T binomial_call_option_price = expression.value(); callput_flag = "put"; const T binomial_put_option_price = expression.value(); printf("American BinomialPrice(call, %5.3f, %5.3f, %5.3f, %5.3f, %5.3f) = %22.18f\n", s, k, t, r, v, binomial_call_option_price); printf("American BinomialPrice(put , %5.3f, %5.3f, %5.3f, %5.3f, %5.3f) = %22.18f\n", s, k, t, r, v, binomial_put_option_price); // American option put-call 'parity': s - k < call - put < s - k * e^(-rt) const T callput_diff = (binomial_call_option_price - binomial_put_option_price); const T basestrike_diff = s - k; const T basepv_diff = s - k * std::exp(-r * t); const bool put_call_parity = (basestrike_diff < callput_diff) && (callput_diff < basepv_diff ) ; const T call_price_r0 = binomial_put_option_price + basestrike_diff; const T call_price_r1 = binomial_put_option_price + basepv_diff; const T put_price_r0 = binomial_call_option_price - basepv_diff; const T put_price_r1 = binomial_call_option_price - basestrike_diff; printf("Put-Call parity: %s\n", put_call_parity ? "True" : "False"); printf("Call price range: %7.4f < %7.4f < %7.4f\n", call_price_r0, binomial_call_option_price, call_price_r1); printf("Put price range: %7.4f < %7.4f < %7.4f\n", put_price_r0, binomial_put_option_price, put_price_r1); } int main() { american_option_binomial_option_pricing_model(); return 0; }