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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2006 Ferdinando Ametrano
Copyright (C) 2006 Mario Pucci
Copyright (C) 2006 StatPro Italia srl
Copyright (C) 2015 Peter Caspers
Copyright (C) 2019 Klaus Spanderen
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
#include <ql/termstructures/volatility/sabr.hpp>
#include <ql/utilities/dataformatters.hpp>
#include <ql/math/comparison.hpp>
#include <ql/math/functional.hpp>
#include <ql/errors.hpp>
#include <ql/termstructures/volatility/volatilitytype.hpp>
#if BOOST_VERSION >= 107800
#include <boost/math/special_functions/sign.hpp>
#include <boost/math/tools/cubic_roots.hpp>
#endif
namespace QuantLib {
Real unsafeSabrLogNormalVolatility(
Rate strike,
Rate forward,
Time expiryTime,
Real alpha,
Real beta,
Real nu,
Real rho) {
const Real oneMinusBeta = 1.0-beta;
const Real A = std::pow(forward*strike, oneMinusBeta);
const Real sqrtA= std::sqrt(A);
Real logM;
if (!close(forward, strike))
logM = std::log(forward/strike);
else {
const Real epsilon = (forward-strike)/strike;
logM = epsilon - .5 * epsilon * epsilon ;
}
const Real z = (nu/alpha)*sqrtA*logM;
const Real B = 1.0-2.0*rho*z+z*z;
const Real C = oneMinusBeta*oneMinusBeta*logM*logM;
const Real tmp = (std::sqrt(B)+z-rho)/(1.0-rho);
const Real xx = std::log(tmp);
const Real D = sqrtA*(1.0+C/24.0+C*C/1920.0);
const Real d = 1.0 + expiryTime *
(oneMinusBeta*oneMinusBeta*alpha*alpha/(24.0*A)
+ 0.25*rho*beta*nu*alpha/sqrtA
+(2.0-3.0*rho*rho)*(nu*nu/24.0));
Real multiplier;
// computations become precise enough if the square of z worth
// slightly more than the precision machine (hence the m)
static const Real m = 10;
if (std::fabs(z*z)>QL_EPSILON * m)
multiplier = z/xx;
else {
multiplier = 1.0 - 0.5*rho*z - (3.0*rho*rho-2.0)*z*z/12.0;
}
return (alpha/D)*multiplier*d;
}
Real unsafeShiftedSabrVolatility(Rate strike,
Rate forward,
Time expiryTime,
Real alpha,
Real beta,
Real nu,
Real rho,
Real shift,
VolatilityType volatilityType) {
if (volatilityType == VolatilityType::Normal) {
return unsafeSabrNormalVolatility(strike + shift, forward + shift, expiryTime, alpha, beta, nu, rho);
} else {
return unsafeSabrLogNormalVolatility(strike + shift, forward + shift, expiryTime, alpha, beta, nu, rho);
}
}
Real unsafeSabrNormalVolatility(
Rate strike, Rate forward, Time expiryTime, Real alpha, Real beta, Real nu, Real rho) {
const Real oneMinusBeta = 1.0 - beta;
const Real minusBeta = -1.0 * beta;
const Real A = std::pow(forward * strike, oneMinusBeta);
const Real sqrtA = std::sqrt(A);
Real logM;
if (!close(forward, strike))
logM = std::log(forward / strike);
else {
const Real epsilon = (forward - strike) / strike;
logM = epsilon - .5 * epsilon * epsilon;
}
const Real z = (nu / alpha) * sqrtA * logM;
const Real B = 1.0 - 2.0 * rho * z + z * z;
const Real C = oneMinusBeta * oneMinusBeta * logM * logM;
const Real D = logM * logM;
const Real tmp = (std::sqrt(B) + z - rho) / (1.0 - rho);
const Real xx = std::log(tmp);
const Real E_1 = (1.0 + D / 24.0 + D * D / 1920.0);
const Real E_2 = (1.0 + C / 24.0 + C * C / 1920.0);
const Real E = E_1 / E_2;
const Real d = 1.0 + expiryTime * (minusBeta * (2 - beta) * alpha * alpha / (24.0 * A) +
0.25 * rho * beta * nu * alpha / sqrtA +
(2.0 - 3.0 * rho * rho) * (nu * nu / 24.0));
Real multiplier;
// computations become precise enough if the square of z worth
// slightly more than the precision machine (hence the m)
static const Real m = 10;
if (std::fabs(z * z) > QL_EPSILON * m)
multiplier = z / xx;
else {
multiplier = 1.0 - 0.5 * rho * z - (3.0 * rho * rho - 2.0) * z * z / 12.0;
}
const Real F = alpha * std::pow(forward * strike, beta / 2.0);
return F * E * multiplier * d;
}
Real unsafeSabrVolatility(Rate strike,
Rate forward,
Time expiryTime,
Real alpha,
Real beta,
Real nu,
Real rho,
VolatilityType volatilityType) {
if (volatilityType == VolatilityType::Normal) {
return unsafeSabrNormalVolatility(strike, forward, expiryTime, alpha, beta, nu, rho);
} else {
return unsafeSabrLogNormalVolatility(strike, forward, expiryTime, alpha, beta, nu, rho);
}
}
void validateSabrParameters(Real alpha,
Real beta,
Real nu,
Real rho) {
QL_REQUIRE(alpha>0.0, "alpha must be positive: "
<< alpha << " not allowed");
QL_REQUIRE(beta>=0.0 && beta<=1.0, "beta must be in (0.0, 1.0): "
<< beta << " not allowed");
QL_REQUIRE(nu>=0.0, "nu must be non negative: "
<< nu << " not allowed");
QL_REQUIRE(rho*rho<1.0, "rho square must be less than one: "
<< rho << " not allowed");
}
Real sabrVolatility(Rate strike,
Rate forward,
Time expiryTime,
Real alpha,
Real beta,
Real nu,
Real rho,
VolatilityType volatilityType) {
QL_REQUIRE(strike>0.0, "strike must be positive: "
<< io::rate(strike) << " not allowed");
QL_REQUIRE(forward>0.0, "at the money forward rate must be "
"positive: " << io::rate(forward) << " not allowed");
QL_REQUIRE(expiryTime>=0.0, "expiry time must be non-negative: "
<< expiryTime << " not allowed");
validateSabrParameters(alpha, beta, nu, rho);
return unsafeSabrVolatility(strike, forward, expiryTime, alpha, beta, nu, rho,
volatilityType);
}
Real shiftedSabrVolatility(Rate strike,
Rate forward,
Time expiryTime,
Real alpha,
Real beta,
Real nu,
Real rho,
Real shift,
VolatilityType volatilityType) {
QL_REQUIRE(strike + shift > 0.0, "strike+shift must be positive: "
<< io::rate(strike) << "+" << io::rate(shift) << " not allowed");
QL_REQUIRE(forward + shift > 0.0, "at the money forward rate + shift must be "
"positive: " << io::rate(forward) << " " << io::rate(shift) << " not allowed");
QL_REQUIRE(expiryTime>=0.0, "expiry time must be non-negative: "
<< expiryTime << " not allowed");
validateSabrParameters(alpha, beta, nu, rho);
return unsafeShiftedSabrVolatility(strike, forward, expiryTime,
alpha, beta, nu, rho,shift, volatilityType);
}
namespace {
struct SabrFlochKennedyVolatility {
Real F, alpha, beta, nu, rho, t;
Real y(Real k) const {
return -1.0/(1.0-beta)*(std::pow(F,1-beta)-std::pow(k,1-beta));
}
Real Dint(Real k) const {
return 1/nu*std::log( ( std::sqrt(1+2*rho*nu/alpha*y(k)
+ squared(nu/alpha*y(k)) )
- rho - nu/alpha*y(k) ) / (1-rho) );
}
Real D(Real k) const {
return std::sqrt(alpha*alpha+2*alpha*rho*nu*y(k)
+ squared(nu*y(k)))*std::pow(k,beta);
}
Real omega0(Real k) const {
return std::log(F/k)/Dint(k);
}
Real operator()(Real k) const {
const Real m = F/k;
if (m > 1.0025 || m < 0.9975) {
return omega0(k)*(1+0.25*rho*nu*alpha*
(std::pow(k,beta)-std::pow(F,beta))/(k-F)*t)
-omega0(k)/squared(Dint(k))*(std::log(
omega0(k)) + 0.5*std::log((F*k/(D(F)*D(k))) ))*t;
}
else {
return taylorExpansion(k);
}
}
Real taylorExpansion(Real k) const {
const Real F2 = F*F;
const Real alpha2 = alpha*alpha;
const Real rho2 = rho*rho;
return
(alpha*std::pow(F,-3 + beta)*(alpha2*squared(-1 + beta)*std::pow(F,2*beta)*t + 6*alpha*beta*nu*std::pow(F,1 + beta)*rho*t +
F2*(24 + nu*nu*(2 - 3*rho2)*t)))/24.0 +
(3*alpha2*alpha*std::pow(-1 + beta,3)*std::pow(F,3*beta)*t +
3*alpha2*(-1 + beta)*(-1 + 5*beta)*nu*std::pow(F,1 + 2*beta)*rho*t + nu*F2*F*rho*(24 + nu*nu*(-4 + 3*rho2)*t) +
alpha*std::pow(F,2 + beta)*(24*(-1 + beta) + nu*nu*(2*(-1 + beta) + 3*(1 + beta)*rho2)*t))/(48.*F2*F2) * (k-F) +
(std::pow(F,-5 - beta)*(alpha2*alpha2*std::pow(-1 + beta,3)*(-209 + 119*beta)*std::pow(F,4*beta)*t + 30*alpha2*alpha*(-1 + beta)*(9 + beta*(-37 + 18*beta))*nu*std::pow(F,1 + 3*beta)*rho*t -
30*alpha*nu*std::pow(F,3 + beta)*rho*(24 + nu*nu*(-4*(1 + beta) + 3*(1 + 2*beta)*rho2)*t) +
10*alpha2*std::pow(F,2 + 2*beta)*(24*(-4 + beta)*(-1 + beta) + nu*nu*(2*(-1 + beta)*(-7 + 4*beta) + 3*(-4 + beta*(-7 + 5*beta))*rho2)*t) +
nu*nu*F2*F2*(480 - 720*rho2 + nu*nu*(-64 + 75*rho2*(4 - 3*rho2))*t)))/(2880*alpha) * (k-F)*(k-F);
}
};
}
Real sabrFlochKennedyVolatility(Rate strike,
Rate forward,
Time expiryTime,
Real alpha,
Real beta,
Real nu,
Real rho) {
const SabrFlochKennedyVolatility v =
{forward, alpha, beta, nu, rho, expiryTime};
return v(strike);
}
#if BOOST_VERSION >= 107800
namespace {
Real smallest_positive_root(Real c1, Real c2, Real c3, Real c4) {
auto [r1, r2, r3] = boost::math::tools::cubic_roots(c1, c2, c3, c4);
if (std::isnan(r3)) {
// single root (or two equal ones), check that it's positive
QL_REQUIRE(r1 > 0.0, "no positive root");
return r1;
} else {
// three roots in non-decreasing order, return the first positive one
QL_REQUIRE(r3 > 0.0, "no positive root");
return r1 > 0.0 ? r1 : (r2 > 0.0 ? r2 : r3);
}
}
}
std::array<Real, 4> sabrGuess(Real k_m, Volatility vol_m,
Real k_0, Volatility vol_0,
Real k_p, Volatility vol_p,
Rate forward,
Time expiryTime,
Real beta,
Real shift,
VolatilityType volatilityType) {
// same variable names as in the equations for ease of reference:
Real f = forward, b = shift, T = expiryTime;
// change to log-moneyness
Real z_m = std::log((k_m + b) / (f + b));
Real z_0 = std::log((k_0 + b) / (f + b));
Real z_p = std::log((k_p + b) / (f + b));
// calculate atm, skew, curvature
Real w_m = 1 / ((z_m - z_0) * (z_m - z_p)); // eq. (42) in the paper
Real w_0 = 1 / ((z_0 - z_m) * (z_0 - z_p)); // eq. (43)
Real w_p = 1 / ((z_p - z_m) * (z_p - z_0)); // eq. (44)
Real sigma_0 = z_0 * z_p * w_m * vol_m + z_m * z_p * w_0 * vol_0 + z_m * z_0 * w_p * vol_p; // (39)
Real sigma_1 = - (z_0 + z_p) * w_m * vol_m - (z_m + z_p) * w_0 * vol_0 - (z_m + z_0) * w_p * vol_p; // (40)
Real sigma_2 = 2 * w_m * vol_m + 2 * w_0 * vol_0 + 2 * w_p * vol_p; // (41)
switch (volatilityType) {
case ShiftedLognormal: {
// equations (32)
Real alpha = sigma_0 * std::pow(f + b, 1.0-beta); // NOLINT(clang-analyzer-deadcode.DeadStores)
Real nu2 =
3 * sigma_0 * sigma_2
- 0.5 * squared(1-beta) * sigma_0 * sigma_0
+ 1.5 * squared(2*sigma_1 + (1-beta)*sigma_0);
Real nu, rho;
if (nu2 > 0.0) {
nu = std::sqrt(nu2);
rho = (1/nu) * (2*sigma_1 + (1-beta)*sigma_0);
} else {
rho = boost::math::sign(2*sigma_1 + (1-beta)*sigma_0);
nu = (1/rho) * (2*sigma_1 + (1-beta)*sigma_0);
}
// coefficients of the polynomial in equation (33)
Real c1 = squared(1 - beta) * T / (24 * std::pow(f + b, 2 - 2 * beta));
Real c2 = rho * beta * nu * T / (4 * std::pow(f + b, 1 - beta));
Real c3 = 1 + ((2 - 3 * rho*rho) / 24) * nu*nu * T;
Real c4 = - sigma_0 * std::pow(f + b, 1-beta);
try {
alpha = smallest_positive_root(c1, c2, c3, c4);
} catch (Error&) {}
return { alpha, beta, nu, rho };
}
case Normal: {
// equations (37)
Real alpha = sigma_0 * std::pow(f + b, -beta); // NOLINT(clang-analyzer-deadcode.DeadStores)
Real nu2 = squared(1 / (f + b)) * (
3 * sigma_0 * sigma_2
- 0.5 * (beta*beta + beta) * (sigma_0*sigma_0)
- 3 * sigma_0 * (sigma_1 - 0.5 * beta * sigma_0)
+ 1.5 * squared(2 * sigma_1 - beta * sigma_0)
);
Real nu, rho;
if (nu2 > 0.0) {
nu = std::sqrt(nu2);
rho = (1 / (nu * (f + b))) * (2 * sigma_1 - beta * sigma_0);
} else {
rho = boost::math::sign((1 / (f + b)) * (2 * sigma_1 - beta * sigma_0));
nu = (1 / (rho * (f + b))) * (2 * sigma_1 - beta * sigma_0);
}
// coefficients of the polynomial in equation (38)
Real c1 = (beta * beta - 2 * beta) * T / (24 * std::pow(f + b, 2 - 2 * beta));
Real c2 = rho * beta * nu * T / (4 * std::pow(f + b, 1 - beta));
Real c3 = 1 + ((2 - 3 * rho*rho) / 24) * nu*nu * T;
Real c4 = - sigma_0 * std::pow(f + b, -beta);
try {
alpha = smallest_positive_root(c1, c2, c3, c4);
} catch (Error&) {}
return { alpha, beta, nu, rho };
}
default:
QL_FAIL("unknown volatility type: " << Integer(volatilityType));
}
}
#else
std::array<Real, 4> sabrGuess(Real, Volatility,
Real, Volatility,
Real, Volatility,
Rate,
Time,
Real,
Real,
VolatilityType) {
QL_FAIL("Boost 1.78 or later is required for the implementation of this functionality");
}
#endif
}
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