LetsBeRational (empty) → 1.0.0.0
raw patch · 13 files changed
+1632/−0 lines, 13 filesdep +base
Dependencies added: base
Files
- CHANGELOG.md +11/−0
- LICENSE +21/−0
- LetsBeRational.cabal +67/−0
- README.md +10/−0
- external/include/importexport.h +36/−0
- external/include/lets_be_rational.h +36/−0
- external/include/normaldistribution.h +34/−0
- external/include/rationalcubic.h +34/−0
- external/src/erf_cody.cpp +455/−0
- external/src/lets_be_rational.cpp +639/−0
- external/src/normaldistribution.cpp +147/−0
- external/src/rationalcubic.cpp +115/−0
- src/LetsBeRational.hs +27/−0
+ CHANGELOG.md view
@@ -0,0 +1,11 @@+# Changelog++`LetsBeRational` uses [PVP Versioning][1].+The changelog is available [on GitHub][2].++## 0.0.0.0++* Initially created.++[1]: https://pvp.haskell.org+[2]: https://github.com/ghais/LetsBeRational/releases
+ LICENSE view
@@ -0,0 +1,21 @@+MIT License++Copyright (c) 2021 Ghais Issa++Permission is hereby granted, free of charge, to any person obtaining a copy+of this software and associated documentation files (the "Software"), to deal+in the Software without restriction, including without limitation the rights+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell+copies of the Software, and to permit persons to whom the Software is+furnished to do so, subject to the following conditions:++The above copyright notice and this permission notice shall be included in all+copies or substantial portions of the Software.++THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE+SOFTWARE.
+ LetsBeRational.cabal view
@@ -0,0 +1,67 @@+cabal-version: 2.2+name: LetsBeRational+version: 1.0.0.0+synopsis: European option implied vol calculation+description: Haskell binding for Jaekel's "Let's be Rational" implied volatility calculation+homepage: https://github.com/ghais/LetsBeRational+bug-reports: https://github.com/ghais/LetsBeRational/issues+license: MIT+license-file: LICENSE+author: Ghais Issa+maintainer: Ghais Issa <0x47@0x49.dev>+copyright: 2021 Ghais Issa+category: Math, Quant, Finance, Numeric+build-type: Simple+extra-doc-files: README.md+ CHANGELOG.md+tested-with: GHC == 7.10.3+ GHC == 8.0.2+ GHC == 8.2.2+ GHC == 8.4.4+ GHC == 8.6.5+ GHC == 8.8.4+ GHC == 8.10.6+ GHC == 9.0.1++source-repository head+ type: git+ location: https://github.com/ghais/LetsBeRational.git++common common-options+ build-depends: base >= 4.8.0.2 && < 5+ + ghc-options: -Wall+ -Wcompat+ -Widentities+ -Wincomplete-uni-patterns+ -Wincomplete-record-updates+ if impl(ghc >= 8.0)+ ghc-options: -Wredundant-constraints+ if impl(ghc >= 8.2)+ ghc-options: -fhide-source-paths+ if impl(ghc >= 8.4)+ ghc-options: -Wmissing-export-lists+ -Wpartial-fields+ if impl(ghc >= 8.8)+ ghc-options: -Wmissing-deriving-strategies++ default-language: Haskell2010++library+ import: common-options+ hs-source-dirs: src+ exposed-modules: LetsBeRational+ include-dirs:+ external/include+ cxx-sources:+ external/src/lets_be_rational.cpp+ external/src/normaldistribution.cpp+ external/src/rationalcubic.cpp+ external/src/erf_cody.cpp+ install-includes:+ importexport.h+ lets_be_rational.h+ normaldistribution.h+ rationalcubic.h++
+ README.md view
@@ -0,0 +1,10 @@+# LetsBeRational++[](https://travis-ci.com/ghais/LetsBeRational)+[](https://github.com/ghais/LetsBeRational/actions/workflows/haskell.yml)+[](https://hackage.haskell.org/package/LetsBeRational)+[](http://stackage.org/lts/package/LetsBeRational)+[](http://stackage.org/nightly/package/LetsBeRational)+[](LICENSE)++Haskell binding for Jaekel's "Let's be Rational" implied volatility calculation
+ external/include/importexport.h view
@@ -0,0 +1,36 @@+//+// This source code resides at www.jaeckel.org/LetsBeRational.7z .+//+// ======================================================================================+// Copyright © 2013-2014 Peter Jäckel.+// +// Permission to use, copy, modify, and distribute this software is freely granted,+// provided that this notice is preserved.+//+// WARRANTY DISCLAIMER+// The Software is provided "as is" without warranty of any kind, either express or implied,+// including without limitation any implied warranties of condition, uninterrupted use,+// merchantability, fitness for a particular purpose, or non-infringement.+// ======================================================================================+//+#ifndef IMPORTEXPORT_H+#define IMPORTEXPORT_H++#if defined(_WIN32) || defined(_WIN64)+# define EXPORT __declspec(dllexport)+# define IMPORT __declspec(dllimport)+# else+# define EXPORT+# define IMPORT+#endif++#ifdef __cplusplus+# define EXTERN_C extern "C"+#else+# define EXTERN_C+#endif++# define EXPORT_EXTERN_C EXTERN_C EXPORT+# define IMPORT_EXTERN_C EXTERN_C IMPORT++#endif // IMPORTEXPORT_H
+ external/include/lets_be_rational.h view
@@ -0,0 +1,36 @@+//+// This source code resides at www.jaeckel.org/LetsBeRational.7z .+//+// ======================================================================================+// Copyright © 2013-2014 Peter Jäckel.+// +// Permission to use, copy, modify, and distribute this software is freely granted,+// provided that this notice is preserved.+//+// WARRANTY DISCLAIMER+// The Software is provided "as is" without warranty of any kind, either express or implied,+// including without limitation any implied warranties of condition, uninterrupted use,+// merchantability, fitness for a particular purpose, or non-infringement.+// ======================================================================================+//+#ifndef LETS_BE_RATIONAL_H+#define LETS_BE_RATIONAL_H++#include "importexport.h"++#define ENABLE_SWITCHING_THE_OUTPUT_TO_ITERATION_COUNT+#define ENABLE_CHANGING_THE_HOUSEHOLDER_METHOD_ORDER++EXPORT_EXTERN_C double set_implied_volatility_maximum_iterations(double n);+EXPORT_EXTERN_C double set_implied_volatility_output_type(double k);+EXPORT_EXTERN_C double set_implied_volatility_householder_method_order(double m);+EXPORT_EXTERN_C double normalised_black_call(double x, double s);+EXPORT_EXTERN_C double normalised_vega(double x, double s);+EXPORT_EXTERN_C double normalised_black(double x, double s, double q /* q=±1 */);+EXPORT_EXTERN_C double black(double F, double K, double sigma, double T, double q /* q=±1 */);+EXPORT_EXTERN_C double normalised_implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(double beta, double x, double q /* q=±1 */, int N);+EXPORT_EXTERN_C double normalised_implied_volatility_from_a_transformed_rational_guess(double beta, double x, double q /* q=±1 */);+EXPORT_EXTERN_C double implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(double price, double F, double K, double T, double q /* q=±1 */, int N);+EXPORT_EXTERN_C double implied_volatility_from_a_transformed_rational_guess(double price, double F, double K, double T, double q /* q=±1 */);++#endif // NORMAL_DISTRIBUTION_H
+ external/include/normaldistribution.h view
@@ -0,0 +1,34 @@+//+// This source code resides at www.jaeckel.org/LetsBeRational.7z .+//+// ======================================================================================+// Copyright © 2013-2014 Peter Jäckel.+// +// Permission to use, copy, modify, and distribute this software is freely granted,+// provided that this notice is preserved.+//+// WARRANTY DISCLAIMER+// The Software is provided "as is" without warranty of any kind, either express or implied,+// including without limitation any implied warranties of condition, uninterrupted use,+// merchantability, fitness for a particular purpose, or non-infringement.+// ======================================================================================+//+#ifndef NORMAL_DISTRIBUTION_H+#define NORMAL_DISTRIBUTION_H++#include <math.h>+#include <cmath>+#include "importexport.h"++#define ONE_OVER_SQRT_TWO 0.7071067811865475244008443621048490392848359376887+#define ONE_OVER_SQRT_TWO_PI 0.3989422804014326779399460599343818684758586311649+#define SQRT_TWO_PI 2.506628274631000502415765284811045253006986740610++EXPORT_EXTERN_C double erf_cody(double z);+EXPORT_EXTERN_C double erfc_cody(double z);+EXPORT_EXTERN_C double erfcx_cody(double z);+EXPORT_EXTERN_C double norm_cdf(double z);+inline double norm_pdf(double x){ return ONE_OVER_SQRT_TWO_PI*exp(-.5*x*x); }+EXPORT_EXTERN_C double inverse_norm_cdf(double u);++#endif // NORMAL_DISTRIBUTION_H
+ external/include/rationalcubic.h view
@@ -0,0 +1,34 @@+//+// This source code resides at www.jaeckel.org/LetsBeRational.7z .+//+// ======================================================================================+// Copyright © 2013-2014 Peter Jäckel.+// +// Permission to use, copy, modify, and distribute this software is freely granted,+// provided that this notice is preserved.+//+// WARRANTY DISCLAIMER+// The Software is provided "as is" without warranty of any kind, either express or implied,+// including without limitation any implied warranties of condition, uninterrupted use,+// merchantability, fitness for a particular purpose, or non-infringement.+// ======================================================================================+//+#ifndef RATIONAL_CUBIC_H+#define RATIONAL_CUBIC_H++// Based on+//+// “Shape preserving piecewise rational interpolation”, R. Delbourgo, J.A. Gregory - SIAM journal on scientific and statistical computing, 1985 - SIAM.+// http://dspace.brunel.ac.uk/bitstream/2438/2200/1/TR_10_83.pdf [caveat emptor: there are some typographical errors in that draft version]+//++#include "importexport.h"++EXPORT_EXTERN_C double rational_cubic_interpolation(double x, double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double r);+EXPORT_EXTERN_C double rational_cubic_control_parameter_to_fit_second_derivative_at_left_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_l);+EXPORT_EXTERN_C double rational_cubic_control_parameter_to_fit_second_derivative_at_right_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_r);+EXPORT_EXTERN_C double minimum_rational_cubic_control_parameter(double d_l, double d_r, double s, bool preferShapePreservationOverSmoothness);+EXPORT_EXTERN_C double convex_rational_cubic_control_parameter_to_fit_second_derivative_at_left_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_l, bool preferShapePreservationOverSmoothness);+EXPORT_EXTERN_C double convex_rational_cubic_control_parameter_to_fit_second_derivative_at_right_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_r, bool preferShapePreservationOverSmoothness);++#endif // RATIONAL_CUBIC_H
+ external/src/erf_cody.cpp view
@@ -0,0 +1,455 @@+//+// Original Fortran code taken from http://www.netlib.org/specfun/erf, compiled with f2c, and adapted by hand.+//+// Created with command line f2c -C++ -c -a -krd -r8 cody_erf.f+//+// Translated by f2c (version 20100827).+//++//+// This source code resides at www.jaeckel.org/LetsBeRational.7z .+//+// ======================================================================================+// WARRANTY DISCLAIMER+// The Software is provided "as is" without warranty of any kind, either express or implied,+// including without limitation any implied warranties of condition, uninterrupted use,+// merchantability, fitness for a particular purpose, or non-infringement.+// ======================================================================================+//++#if defined( _DEBUG ) || defined( BOUNDS_CHECK_STL_ARRAYS )+#define _SECURE_SCL 1+#define _SECURE_SCL_THROWS 1+#define _SCL_SECURE_NO_WARNINGS+#define _HAS_ITERATOR_DEBUGGING 0+#else+#define _SECURE_SCL 0+#endif+#if defined(_MSC_VER)+# define NOMINMAX // to suppress MSVC's definitions of min() and max()+// These four pragmas are the equivalent to /fp:fast.+# pragma float_control( except, off )+# pragma float_control( precise, off )+# pragma fp_contract( on )+# pragma fenv_access( off )+#endif++#include "normaldistribution.h"+#include <math.h>+#include <float.h>++namespace {+ inline double d_int(const double x){ return( (x>0) ? floor(x) : -floor(-x) ); }+}++/*< SUBROUTINE CALERF(ARG,RESULT,JINT) >*/+double calerf(double x, const int jint) {++ static const double a[5] = { 3.1611237438705656,113.864154151050156,377.485237685302021,3209.37758913846947,.185777706184603153 };+ static const double b[4] = { 23.6012909523441209,244.024637934444173,1282.61652607737228,2844.23683343917062 };+ static const double c__[9] = { .564188496988670089,8.88314979438837594,66.1191906371416295,298.635138197400131,881.95222124176909,1712.04761263407058,2051.07837782607147,1230.33935479799725,2.15311535474403846e-8 };+ static const double d__[8] = { 15.7449261107098347,117.693950891312499,537.181101862009858,1621.38957456669019,3290.79923573345963,4362.61909014324716,3439.36767414372164,1230.33935480374942 };+ static const double p[6] = { .305326634961232344,.360344899949804439,.125781726111229246,.0160837851487422766,6.58749161529837803e-4,.0163153871373020978 };+ static const double q[5] = { 2.56852019228982242,1.87295284992346047,.527905102951428412,.0605183413124413191,.00233520497626869185 };++ static const double zero = 0.;+ static const double half = .5;+ static const double one = 1.;+ static const double two = 2.;+ static const double four = 4.;+ static const double sqrpi = 0.56418958354775628695;+ static const double thresh = .46875;+ static const double sixten = 16.;++ double y, del, ysq, xden, xnum, result;++ /* ------------------------------------------------------------------ */+ /* This packet evaluates erf(x), erfc(x), and exp(x*x)*erfc(x) */+ /* for a real argument x. It contains three FUNCTION type */+ /* subprograms: ERF, ERFC, and ERFCX (or DERF, DERFC, and DERFCX), */+ /* and one SUBROUTINE type subprogram, CALERF. The calling */+ /* statements for the primary entries are: */+ /* Y=ERF(X) (or Y=DERF(X)), */+ /* Y=ERFC(X) (or Y=DERFC(X)), */+ /* and */+ /* Y=ERFCX(X) (or Y=DERFCX(X)). */+ /* The routine CALERF is intended for internal packet use only, */+ /* all computations within the packet being concentrated in this */+ /* routine. The function subprograms invoke CALERF with the */+ /* statement */+ /* CALL CALERF(ARG,RESULT,JINT) */+ /* where the parameter usage is as follows */+ /* Function Parameters for CALERF */+ /* call ARG Result JINT */+ /* ERF(ARG) ANY REAL ARGUMENT ERF(ARG) 0 */+ /* ERFC(ARG) ABS(ARG) .LT. XBIG ERFC(ARG) 1 */+ /* ERFCX(ARG) XNEG .LT. ARG .LT. XMAX ERFCX(ARG) 2 */+ /* The main computation evaluates near-minimax approximations */+ /* from "Rational Chebyshev approximations for the error function" */+ /* by W. J. Cody, Math. Comp., 1969, PP. 631-638. This */+ /* transportable program uses rational functions that theoretically */+ /* approximate erf(x) and erfc(x) to at least 18 significant */+ /* decimal digits. The accuracy achieved depends on the arithmetic */+ /* system, the compiler, the intrinsic functions, and proper */+ /* selection of the machine-dependent constants. */+ /* ******************************************************************* */+ /* ******************************************************************* */+ /* Explanation of machine-dependent constants */+ /* XMIN = the smallest positive floating-point number. */+ /* XINF = the largest positive finite floating-point number. */+ /* XNEG = the largest negative argument acceptable to ERFCX; */+ /* the negative of the solution to the equation */+ /* 2*exp(x*x) = XINF. */+ /* XSMALL = argument below which erf(x) may be represented by */+ /* 2*x/sqrt(pi) and above which x*x will not underflow. */+ /* A conservative value is the largest machine number X */+ /* such that 1.0 + X = 1.0 to machine precision. */+ /* XBIG = largest argument acceptable to ERFC; solution to */+ /* the equation: W(x) * (1-0.5/x**2) = XMIN, where */+ /* W(x) = exp(-x*x)/[x*sqrt(pi)]. */+ /* XHUGE = argument above which 1.0 - 1/(2*x*x) = 1.0 to */+ /* machine precision. A conservative value is */+ /* 1/[2*sqrt(XSMALL)] */+ /* XMAX = largest acceptable argument to ERFCX; the minimum */+ /* of XINF and 1/[sqrt(pi)*XMIN]. */+ // The numbers below were preselected for IEEE .+ static const double xinf = 1.79e308;+ static const double xneg = -26.628;+ static const double xsmall = 1.11e-16;+ static const double xbig = 26.543;+ static const double xhuge = 6.71e7;+ static const double xmax = 2.53e307;+ /* Approximate values for some important machines are: */+ /* XMIN XINF XNEG XSMALL */+ /* CDC 7600 (S.P.) 3.13E-294 1.26E+322 -27.220 7.11E-15 */+ /* CRAY-1 (S.P.) 4.58E-2467 5.45E+2465 -75.345 7.11E-15 */+ /* IEEE (IBM/XT, */+ /* SUN, etc.) (S.P.) 1.18E-38 3.40E+38 -9.382 5.96E-8 */+ /* IEEE (IBM/XT, */+ /* SUN, etc.) (D.P.) 2.23D-308 1.79D+308 -26.628 1.11D-16 */+ /* IBM 195 (D.P.) 5.40D-79 7.23E+75 -13.190 1.39D-17 */+ /* UNIVAC 1108 (D.P.) 2.78D-309 8.98D+307 -26.615 1.73D-18 */+ /* VAX D-Format (D.P.) 2.94D-39 1.70D+38 -9.345 1.39D-17 */+ /* VAX G-Format (D.P.) 5.56D-309 8.98D+307 -26.615 1.11D-16 */+ /* XBIG XHUGE XMAX */+ /* CDC 7600 (S.P.) 25.922 8.39E+6 1.80X+293 */+ /* CRAY-1 (S.P.) 75.326 8.39E+6 5.45E+2465 */+ /* IEEE (IBM/XT, */+ /* SUN, etc.) (S.P.) 9.194 2.90E+3 4.79E+37 */+ /* IEEE (IBM/XT, */+ /* SUN, etc.) (D.P.) 26.543 6.71D+7 2.53D+307 */+ /* IBM 195 (D.P.) 13.306 1.90D+8 7.23E+75 */+ /* UNIVAC 1108 (D.P.) 26.582 5.37D+8 8.98D+307 */+ /* VAX D-Format (D.P.) 9.269 1.90D+8 1.70D+38 */+ /* VAX G-Format (D.P.) 26.569 6.71D+7 8.98D+307 */+ /* ******************************************************************* */+ /* ******************************************************************* */+ /* Error returns */+ /* The program returns ERFC = 0 for ARG .GE. XBIG; */+ /* ERFCX = XINF for ARG .LT. XNEG; */+ /* and */+ /* ERFCX = 0 for ARG .GE. XMAX. */+ /* Intrinsic functions required are: */+ /* ABS, AINT, EXP */+ /* Author: W. J. Cody */+ /* Mathematics and Computer Science Division */+ /* Argonne National Laboratory */+ /* Argonne, IL 60439 */+ /* Latest modification: March 19, 1990 */+ /* ------------------------------------------------------------------ */+ /*< INTEGER I,JINT >*/+ /* S REAL */+ /*< >*/+ /*< DIMENSION A(5),B(4),C(9),D(8),P(6),Q(5) >*/+ /* ------------------------------------------------------------------ */+ /* Mathematical constants */+ /* ------------------------------------------------------------------ */+ /* S DATA FOUR,ONE,HALF,TWO,ZERO/4.0E0,1.0E0,0.5E0,2.0E0,0.0E0/, */+ /* S 1 SQRPI/5.6418958354775628695E-1/,THRESH/0.46875E0/, */+ /* S 2 SIXTEN/16.0E0/ */+ /*< >*/+ /* ------------------------------------------------------------------ */+ /* Machine-dependent constants */+ /* ------------------------------------------------------------------ */+ /* S DATA XINF,XNEG,XSMALL/3.40E+38,-9.382E0,5.96E-8/, */+ /* S 1 XBIG,XHUGE,XMAX/9.194E0,2.90E3,4.79E37/ */+ /*< >*/+ /* ------------------------------------------------------------------ */+ /* Coefficients for approximation to erf in first interval */+ /* ------------------------------------------------------------------ */+ /* S DATA A/3.16112374387056560E00,1.13864154151050156E02, */+ /* S 1 3.77485237685302021E02,3.20937758913846947E03, */+ /* S 2 1.85777706184603153E-1/ */+ /* S DATA B/2.36012909523441209E01,2.44024637934444173E02, */+ /* S 1 1.28261652607737228E03,2.84423683343917062E03/ */+ /*< >*/+ /*< >*/+ /* ------------------------------------------------------------------ */+ /* Coefficients for approximation to erfc in second interval */+ /* ------------------------------------------------------------------ */+ /* S DATA C/5.64188496988670089E-1,8.88314979438837594E0, */+ /* S 1 6.61191906371416295E01,2.98635138197400131E02, */+ /* S 2 8.81952221241769090E02,1.71204761263407058E03, */+ /* S 3 2.05107837782607147E03,1.23033935479799725E03, */+ /* S 4 2.15311535474403846E-8/ */+ /* S DATA D/1.57449261107098347E01,1.17693950891312499E02, */+ /* S 1 5.37181101862009858E02,1.62138957456669019E03, */+ /* S 2 3.29079923573345963E03,4.36261909014324716E03, */+ /* S 3 3.43936767414372164E03,1.23033935480374942E03/ */+ /*< >*/+ /*< >*/+ /* ------------------------------------------------------------------ */+ /* Coefficients for approximation to erfc in third interval */+ /* ------------------------------------------------------------------ */+ /* S DATA P/3.05326634961232344E-1,3.60344899949804439E-1, */+ /* S 1 1.25781726111229246E-1,1.60837851487422766E-2, */+ /* S 2 6.58749161529837803E-4,1.63153871373020978E-2/ */+ /* S DATA Q/2.56852019228982242E00,1.87295284992346047E00, */+ /* S 1 5.27905102951428412E-1,6.05183413124413191E-2, */+ /* S 2 2.33520497626869185E-3/ */+ /*< >*/+ /*< >*/+ /* ------------------------------------------------------------------ */+ /*< X = ARG >*/+ // x = *arg;+ /*< Y = ABS(X) >*/+ y = fabs(x);+ /*< IF (Y .LE. THRESH) THEN >*/+ if (y <= thresh) {+ /* ------------------------------------------------------------------ */+ /* Evaluate erf for |X| <= 0.46875 */+ /* ------------------------------------------------------------------ */+ /*< YSQ = ZERO >*/+ ysq = zero;+ /*< IF (Y .GT. XSMALL) YSQ = Y * Y >*/+ if (y > xsmall) {+ ysq = y * y;+ }+ /*< XNUM = A(5)*YSQ >*/+ xnum = a[4] * ysq;+ /*< XDEN = YSQ >*/+ xden = ysq;+ /*< DO 20 I = 1, 3 >*/+ for (int i__ = 1; i__ <= 3; ++i__) {+ /*< XNUM = (XNUM + A(I)) * YSQ >*/+ xnum = (xnum + a[i__ - 1]) * ysq;+ /*< XDEN = (XDEN + B(I)) * YSQ >*/+ xden = (xden + b[i__ - 1]) * ysq;+ /*< 20 CONTINUE >*/+ /* L20: */+ }+ /*< RESULT = X * (XNUM + A(4)) / (XDEN + B(4)) >*/+ result = x * (xnum + a[3]) / (xden + b[3]);+ /*< IF (JINT .NE. 0) RESULT = ONE - RESULT >*/+ if (jint != 0) {+ result = one - result;+ }+ /*< IF (JINT .EQ. 2) RESULT = EXP(YSQ) * RESULT >*/+ if (jint == 2) {+ result = exp(ysq) * result;+ }+ /*< GO TO 800 >*/+ goto L800;+ /* ------------------------------------------------------------------ */+ /* Evaluate erfc for 0.46875 <= |X| <= 4.0 */+ /* ------------------------------------------------------------------ */+ /*< ELSE IF (Y .LE. FOUR) THEN >*/+ } else if (y <= four) {+ /*< XNUM = C(9)*Y >*/+ xnum = c__[8] * y;+ /*< XDEN = Y >*/+ xden = y;+ /*< DO 120 I = 1, 7 >*/+ for (int i__ = 1; i__ <= 7; ++i__) {+ /*< XNUM = (XNUM + C(I)) * Y >*/+ xnum = (xnum + c__[i__ - 1]) * y;+ /*< XDEN = (XDEN + D(I)) * Y >*/+ xden = (xden + d__[i__ - 1]) * y;+ /*< 120 CONTINUE >*/+ /* L120: */+ }+ /*< RESULT = (XNUM + C(8)) / (XDEN + D(8)) >*/+ result = (xnum + c__[7]) / (xden + d__[7]);+ /*< IF (JINT .NE. 2) THEN >*/+ if (jint != 2) {+ /*< YSQ = AINT(Y*SIXTEN)/SIXTEN >*/+ double d__1 = y * sixten;+ ysq = d_int(d__1) / sixten;+ /*< DEL = (Y-YSQ)*(Y+YSQ) >*/+ del = (y - ysq) * (y + ysq);+ /*< RESULT = EXP(-YSQ*YSQ) * EXP(-DEL) * RESULT >*/+ d__1 = exp(-ysq * ysq) * exp(-del);+ result = d__1 * result;+ /*< END IF >*/+ }+ /* ------------------------------------------------------------------ */+ /* Evaluate erfc for |X| > 4.0 */+ /* ------------------------------------------------------------------ */+ /*< ELSE >*/+ } else {+ /*< RESULT = ZERO >*/+ result = zero;+ /*< IF (Y .GE. XBIG) THEN >*/+ if (y >= xbig) {+ /*< IF ((JINT .NE. 2) .OR. (Y .GE. XMAX)) GO TO 300 >*/+ if (jint != 2 || y >= xmax) {+ goto L300;+ }+ /*< IF (Y .GE. XHUGE) THEN >*/+ if (y >= xhuge) {+ /*< RESULT = SQRPI / Y >*/+ result = sqrpi / y;+ /*< GO TO 300 >*/+ goto L300;+ /*< END IF >*/+ }+ /*< END IF >*/+ }+ /*< YSQ = ONE / (Y * Y) >*/+ ysq = one / (y * y);+ /*< XNUM = P(6)*YSQ >*/+ xnum = p[5] * ysq;+ /*< XDEN = YSQ >*/+ xden = ysq;+ /*< DO 240 I = 1, 4 >*/+ for (int i__ = 1; i__ <= 4; ++i__) {+ /*< XNUM = (XNUM + P(I)) * YSQ >*/+ xnum = (xnum + p[i__ - 1]) * ysq;+ /*< XDEN = (XDEN + Q(I)) * YSQ >*/+ xden = (xden + q[i__ - 1]) * ysq;+ /*< 240 CONTINUE >*/+ /* L240: */+ }+ /*< RESULT = YSQ *(XNUM + P(5)) / (XDEN + Q(5)) >*/+ result = ysq * (xnum + p[4]) / (xden + q[4]);+ /*< RESULT = (SQRPI - RESULT) / Y >*/+ result = (sqrpi - result) / y;+ /*< IF (JINT .NE. 2) THEN >*/+ if (jint != 2) {+ /*< YSQ = AINT(Y*SIXTEN)/SIXTEN >*/+ double d__1 = y * sixten;+ ysq = d_int(d__1) / sixten;+ /*< DEL = (Y-YSQ)*(Y+YSQ) >*/+ del = (y - ysq) * (y + ysq);+ /*< RESULT = EXP(-YSQ*YSQ) * EXP(-DEL) * RESULT >*/+ d__1 = exp(-ysq * ysq) * exp(-del);+ result = d__1 * result;+ /*< END IF >*/+ }+ /*< END IF >*/+ }+ /* ------------------------------------------------------------------ */+ /* Fix up for negative argument, erf, etc. */+ /* ------------------------------------------------------------------ */+ /*< 300 IF (JINT .EQ. 0) THEN >*/+L300:+ if (jint == 0) {+ /*< RESULT = (HALF - RESULT) + HALF >*/+ result = (half - result) + half;+ /*< IF (X .LT. ZERO) RESULT = -RESULT >*/+ if (x < zero) {+ result = -(result);+ }+ /*< ELSE IF (JINT .EQ. 1) THEN >*/+ } else if (jint == 1) {+ /*< IF (X .LT. ZERO) RESULT = TWO - RESULT >*/+ if (x < zero) {+ result = two - result;+ }+ /*< ELSE >*/+ } else {+ /*< IF (X .LT. ZERO) THEN >*/+ if (x < zero) {+ /*< IF (X .LT. XNEG) THEN >*/+ if (x < xneg) {+ /*< RESULT = XINF >*/+ result = xinf;+ /*< ELSE >*/+ } else {+ /*< YSQ = AINT(X*SIXTEN)/SIXTEN >*/+ double d__1 = x * sixten;+ ysq = d_int(d__1) / sixten;+ /*< DEL = (X-YSQ)*(X+YSQ) >*/+ del = (x - ysq) * (x + ysq);+ /*< Y = EXP(YSQ*YSQ) * EXP(DEL) >*/+ y = exp(ysq * ysq) * exp(del);+ /*< RESULT = (Y+Y) - RESULT >*/+ result = y + y - result;+ /*< END IF >*/+ }+ /*< END IF >*/+ }+ /*< END IF >*/+ }+ /*< 800 RETURN >*/+L800:+ return result;+ /* ---------- Last card of CALERF ---------- */+ /*< END >*/+} /* calerf_ */++/* S REAL FUNCTION ERF(X) */+/*< DOUBLE PRECISION FUNCTION DERF(X) >*/+double erf_cody(double x){+ /* -------------------------------------------------------------------- */+ /* This subprogram computes approximate values for erf(x). */+ /* (see comments heading CALERF). */+ /* Author/date: W. J. Cody, January 8, 1985 */+ /* -------------------------------------------------------------------- */+ /*< INTEGER JINT >*/+ /* S REAL X, RESULT */+ /*< DOUBLE PRECISION X, RESULT >*/+ /* ------------------------------------------------------------------ */+ /*< JINT = 0 >*/+ /*< CALL CALERF(X,RESULT,JINT) >*/+ return calerf(x, 0);+ /* S ERF = RESULT */+ /*< DERF = RESULT >*/+ /*< RETURN >*/+ /* ---------- Last card of DERF ---------- */+ /*< END >*/+} /* derf_ */++/* S REAL FUNCTION ERFC(X) */+/*< DOUBLE PRECISION FUNCTION DERFC(X) >*/+double erfc_cody(double x) {+ /* -------------------------------------------------------------------- */+ /* This subprogram computes approximate values for erfc(x). */+ /* (see comments heading CALERF). */+ /* Author/date: W. J. Cody, January 8, 1985 */+ /* -------------------------------------------------------------------- */+ /*< INTEGER JINT >*/+ /* S REAL X, RESULT */+ /*< DOUBLE PRECISION X, RESULT >*/+ /* ------------------------------------------------------------------ */+ /*< JINT = 1 >*/+ /*< CALL CALERF(X,RESULT,JINT) >*/+ return calerf(x, 1);+ /* S ERFC = RESULT */+ /*< DERFC = RESULT >*/+ /*< RETURN >*/+ /* ---------- Last card of DERFC ---------- */+ /*< END >*/+} /* derfc_ */++/* S REAL FUNCTION ERFCX(X) */+/*< DOUBLE PRECISION FUNCTION DERFCX(X) >*/+double erfcx_cody(double x) {+ /* ------------------------------------------------------------------ */+ /* This subprogram computes approximate values for exp(x*x) * erfc(x). */+ /* (see comments heading CALERF). */+ /* Author/date: W. J. Cody, March 30, 1987 */+ /* ------------------------------------------------------------------ */+ /*< INTEGER JINT >*/+ /* S REAL X, RESULT */+ /*< DOUBLE PRECISION X, RESULT >*/+ /* ------------------------------------------------------------------ */+ /*< JINT = 2 >*/+ /*< CALL CALERF(X,RESULT,JINT) >*/+ return calerf(x, 2);+ /* S ERFCX = RESULT */+ /*< DERFCX = RESULT >*/+ /*< RETURN >*/+ /* ---------- Last card of DERFCX ---------- */+ /*< END >*/+} /* derfcx_ */
+ external/src/lets_be_rational.cpp view
@@ -0,0 +1,639 @@+//+// This source code resides at www.jaeckel.org/LetsBeRational.7z .+//+// ======================================================================================+// Copyright © 2013-2017 Peter Jäckel.+// +// Permission to use, copy, modify, and distribute this software is freely granted,+// provided that this notice is preserved.+//+// WARRANTY DISCLAIMER+// The Software is provided "as is" without warranty of any kind, either express or implied,+// including without limitation any implied warranties of condition, uninterrupted use,+// merchantability, fitness for a particular purpose, or non-infringement.+// ======================================================================================+//++#include "lets_be_rational.h"+// To cross-compile on a command line, you could just use something like+//+// i686-w64-mingw32-g++ -w -fpermissive -shared -DNDEBUG -O3 erf_cody.cpp rationalcubic.cpp normaldistribution.cpp lets_be_rational.cpp xlcall.cpp excel_registration.cpp xlcall32.lib -o lets_be_rational.xll -static-libstdc++ -static-libgcc -s+//+// To compile into a shared library on non-Windows systems, you could try+//+// g++ -fPIC -shared -DNDEBUG -Ofast erf_cody.cpp rationalcubic.cpp normaldistribution.cpp lets_be_rational.cpp -o lets_be_rational.so+//++#if defined(_MSC_VER)+# define NOMINMAX // to suppress MSVC's definitions of min() and max()+// These four pragmas are the equivalent to /fp:fast.+# pragma float_control( except, off )+# pragma float_control( precise, off )+# pragma fp_contract( on )+# pragma fenv_access( off )+#endif++#include "normaldistribution.h"+#include "rationalcubic.h"+#include <float.h>+#include <cmath>+#include <algorithm>+#if defined(_WIN32) || defined(_WIN64)+# include <windows.h>+#endif++#define TWO_PI 6.283185307179586476925286766559005768394338798750+#define SQRT_PI_OVER_TWO 1.253314137315500251207882642405522626503493370305 // sqrt(pi/2) to avoid misinterpretation.+#define SQRT_THREE 1.732050807568877293527446341505872366942805253810+#define SQRT_ONE_OVER_THREE 0.577350269189625764509148780501957455647601751270+#define TWO_PI_OVER_SQRT_TWENTY_SEVEN 1.209199576156145233729385505094770488189377498728 // 2*pi/sqrt(27)+#define PI_OVER_SIX 0.523598775598298873077107230546583814032861566563++namespace {+ static const double SQRT_DBL_EPSILON = sqrt(DBL_EPSILON);+ static const double FOURTH_ROOT_DBL_EPSILON = sqrt(SQRT_DBL_EPSILON);+ static const double EIGHTH_ROOT_DBL_EPSILON = sqrt(FOURTH_ROOT_DBL_EPSILON);+ static const double SIXTEENTH_ROOT_DBL_EPSILON = sqrt(EIGHTH_ROOT_DBL_EPSILON);+ static const double SQRT_DBL_MIN = sqrt(DBL_MIN);+ static const double SQRT_DBL_MAX = sqrt(DBL_MAX);++ // Set this to 0 if you want positive results for (positive) denormalised inputs, else to DBL_MIN.+ // Note that you cannot achieve full machine accuracy from denormalised inputs!+ static const double DENORMALISATION_CUTOFF = 0; ++ static const double VOLATILITY_VALUE_TO_SIGNAL_PRICE_IS_BELOW_INTRINSIC = -DBL_MAX;+ static const double VOLATILITY_VALUE_TO_SIGNAL_PRICE_IS_ABOVE_MAXIMUM = DBL_MAX;++ inline bool is_below_horizon(double x){ return fabs(x) < DENORMALISATION_CUTOFF; } // This weeds out denormalised (a.k.a. 'subnormal') numbers.++ // See https://www.kernel.org/doc/Documentation/atomic_ops.txt for further details on this simplistic implementation of an atomic flag that is *not* volatile.+ typedef struct { +#if defined(_MSC_VER) || defined(_WIN32) || defined(_WIN64)+ long data;+#else+ int data;+#endif+ } atomic_t;++ static atomic_t implied_volatility_maximum_iterations = { 2 }; // (DBL_DIG*20)/3 ≈ 100 . Only needed when the iteration effectively alternates Householder/Halley/Newton steps and binary nesting due to roundoff truncation.++#ifdef ENABLE_SWITCHING_THE_OUTPUT_TO_ITERATION_COUNT+ static atomic_t implied_volatility_output_type = { 0 };+ inline double implied_volatility_output(int count, double volatility){ return implied_volatility_output_type.data>0 ? count : volatility; }+#else+ inline double implied_volatility_output(int count, double volatility){ return volatility; }+#endif++#ifdef ENABLE_CHANGING_THE_HOUSEHOLDER_METHOD_ORDER+ static atomic_t implied_volatility_householder_method_order = { 4 };+ inline double householder_factor(double newton, double halley, double hh3){+ return implied_volatility_householder_method_order.data > 3 ? (1+0.5*halley*newton)/(1+newton*(halley+hh3*newton/6)) : ( implied_volatility_householder_method_order.data > 2 ? 1/(1+0.5*halley*newton) : 1 );+ }+#else+ inline double householder_factor(double newton, double halley, double hh3){ return (1+0.5*halley*newton)/(1+newton*(halley+hh3*newton/6)); }+#endif++}++EXPORT_EXTERN_C double set_implied_volatility_maximum_iterations(double t){+ int i = (int)t;+ if (i>=0) {+#if defined(_MSC_VER) || defined(_WIN32) || defined(_WIN64)+ InterlockedExchange(&(implied_volatility_maximum_iterations.data),i);+#elif defined( __x86__ ) || defined( __x86_64__ )+ implied_volatility_maximum_iterations.data = i;+#elif defined ( __aarch64__ ) || defined (__aarch32__)+ implied_volatility_householder_method_order.data = i;+#else+# error Atomic operations not implemented for this platform.+#endif+ }+ return implied_volatility_maximum_iterations.data;+}++#ifdef ENABLE_SWITCHING_THE_OUTPUT_TO_ITERATION_COUNT+EXPORT_EXTERN_C double set_implied_volatility_output_type(double t){+ int i = (int)t;+#if defined(_MSC_VER) || defined(_WIN32) || defined(_WIN64)+ InterlockedExchange(&(implied_volatility_output_type.data),i);+#elif defined( __x86__ ) || defined( __x86_64__ )+ implied_volatility_output_type.data = i;+#elif defined ( __aarch64__ ) || defined (__arm__)+ implied_volatility_output_type.data = i;+#else+# error Atomic operations not implemented for this platform.+#endif+ return implied_volatility_output_type.data;+}+#endif ++#ifdef ENABLE_CHANGING_THE_HOUSEHOLDER_METHOD_ORDER+EXPORT_EXTERN_C double set_implied_volatility_householder_method_order(double t){+ int i = (int)t;+ if (i>=0) {+#if defined(_MSC_VER) || defined(_WIN32) || defined(_WIN64)+ InterlockedExchange(&(implied_volatility_householder_method_order.data),i);+#elif defined( __x86__ ) || defined( __x86_64__ )+ implied_volatility_householder_method_order.data = i;+#elif defined ( __aarch64__ ) || defined (__aarch32__)+ implied_volatility_householder_method_order.data = i;+#else+# error Atomic operations not implemented for this platform.+#endif+ }+ return implied_volatility_householder_method_order.data;+}+#endif ++double normalised_intrinsic(double x, double q /* q=±1 */){+ if (q*x<=0)+ return 0;+ const double x2=x*x;+ if (x2<98*FOURTH_ROOT_DBL_EPSILON ) // The factor 98 is computed from last coefficient: √√92897280 = 98.1749+ return fabs( std::max( (q<0?-1:1)*x*(1+x2*((1.0/24.0)+x2*((1.0/1920.0)+x2*((1.0/322560.0)+(1.0/92897280.0)*x2)))) , 0.0 ) );+ const double b_max = exp(0.5*x), one_over_b_max = 1 / b_max;+ return fabs(std::max((q<0?-1:1)*(b_max-one_over_b_max),0.));+}++double normalised_intrinsic_call(double x){ return normalised_intrinsic(x,1); }++// Asymptotic expansion of+//+// b = Φ(h+t)·exp(x/2) - Φ(h-t)·exp(-x/2)+// with+// h = x/s and t = s/2+// which makes+// b = Φ(h+t)·exp(h·t) - Φ(h-t)·exp(-h·t)+//+// exp(-(h²+t²)/2)+// = --------------- · [ Y(h+t) - Y(h-t) ]+// √(2π)+// with+// Y(z) := Φ(z)/φ(z)+//+// for large negative (t-|h|) by the aid of Abramowitz & Stegun (26.2.12) where Φ(z) = φ(z)/|z|·[1-1/z^2+...].+// We define+// r+// A(h,t) := --- · [ Y(h+t) - Y(h-t) ]+// t+//+// with r := (h+t)·(h-t) and give an expansion for A(h,t) in q:=(h/r)² expressed in terms of e:=(t/h)² .+double asymptotic_expansion_of_normalised_black_call(double h, double t){+ const double e=(t/h)*(t/h), r=((h+t)*(h-t)), q=(h/r)*(h/r);+ // 17th order asymptotic expansion of A(h,t) in q, sufficient for Φ(h) [and thus y(h)] to have relative accuracy of 1.64E-16 for h <= η with η:=-10.+ const double asymptotic_expansion_sum = (2.0+q*(-6.0E0-2.0*e+3.0*q*(1.0E1+e*(2.0E1+2.0*e)+5.0*q*(-1.4E1+e*(-7.0E1+e*(-4.2E1-2.0*e))+7.0*q*(1.8E1+e*(1.68E2+e*(2.52E2+e*(7.2E1+2.0*e)))+9.0*q*(-2.2E1+e*(-3.3E2+e*(-9.24E2+e*(-6.6E2+e*(-1.1E2-2.0*e))))+1.1E1*q*(2.6E1+e*(5.72E2+e*(2.574E3+e*(3.432E3+e*(1.43E3+e*(1.56E2+2.0*e)))))+1.3E1*q*(-3.0E1+e*(-9.1E2+e*(-6.006E3+e*(-1.287E4+e*(-1.001E4+e*(-2.73E3+e*(-2.1E2-2.0*e))))))+1.5E1*q*(3.4E1+e*(1.36E3+e*(1.2376E4+e*(3.8896E4+e*(4.862E4+e*(2.4752E4+e*(4.76E3+e*(2.72E2+2.0*e)))))))+1.7E1*q*(-3.8E1+e*(-1.938E3+e*(-2.3256E4+e*(-1.00776E5+e*(-1.84756E5+e*(-1.51164E5+e*(-5.4264E4+e*(-7.752E3+e*(-3.42E2-2.0*e))))))))+1.9E1*q*(4.2E1+e*(2.66E3+e*(4.0698E4+e*(2.3256E5+e*(5.8786E5+e*(7.05432E5+e*(4.0698E5+e*(1.08528E5+e*(1.197E4+e*(4.2E2+2.0*e)))))))))+2.1E1*q*(-4.6E1+e*(-3.542E3+e*(-6.7298E4+e*(-4.90314E5+e*(-1.63438E6+e*(-2.704156E6+e*(-2.288132E6+e*(-9.80628E5+e*(-2.01894E5+e*(-1.771E4+e*(-5.06E2-2.0*e))))))))))+2.3E1*q*(5.0E1+e*(4.6E3+e*(1.0626E5+e*(9.614E5+e*(4.08595E6+e*(8.9148E6+e*(1.04006E7+e*(6.53752E6+e*(2.16315E6+e*(3.542E5+e*(2.53E4+e*(6.0E2+2.0*e)))))))))))+2.5E1*q*(-5.4E1+e*(-5.85E3+e*(-1.6146E5+e*(-1.77606E6+e*(-9.37365E6+e*(-2.607579E7+e*(-4.01166E7+e*(-3.476772E7+e*(-1.687257E7+e*(-4.44015E6+e*(-5.9202E5+e*(-3.51E4+e*(-7.02E2-2.0*e))))))))))))+2.7E1*q*(5.8E1+e*(7.308E3+e*(2.3751E5+e*(3.12156E6+e*(2.003001E7+e*(6.919458E7+e*(1.3572783E8+e*(1.5511752E8+e*(1.0379187E8+e*(4.006002E7+e*(8.58429E6+e*(9.5004E5+e*(4.7502E4+e*(8.12E2+2.0*e)))))))))))))+2.9E1*q*(-6.2E1+e*(-8.99E3+e*(-3.39822E5+e*(-5.25915E6+e*(-4.032015E7+e*(-1.6934463E8+e*(-4.1250615E8+e*(-6.0108039E8+e*(-5.3036505E8+e*(-2.8224105E8+e*(-8.870433E7+e*(-1.577745E7+e*(-1.472562E6+e*(-6.293E4+e*(-9.3E2-2.0*e))))))))))))))+3.1E1*q*(6.6E1+e*(1.0912E4+e*(4.74672E5+e*(8.544096E6+e*(7.71342E7+e*(3.8707344E8+e*(1.14633288E9+e*(2.07431664E9+e*(2.33360622E9+e*(1.6376184E9+e*(7.0963464E8+e*(1.8512208E8+e*(2.7768312E7+e*(2.215136E6+e*(8.184E4+e*(1.056E3+2.0*e)))))))))))))))+3.3E1*(-7.0E1+e*(-1.309E4+e*(-6.49264E5+e*(-1.344904E7+e*(-1.4121492E8+e*(-8.344518E8+e*(-2.9526756E9+e*(-6.49588632E9+e*(-9.0751353E9+e*(-8.1198579E9+e*(-4.6399188E9+e*(-1.6689036E9+e*(-3.67158792E8+e*(-4.707164E7+e*(-3.24632E6+e*(-1.0472E5+e*(-1.19E3-2.0*e)))))))))))))))))*q)))))))))))))))));+ const double b = ONE_OVER_SQRT_TWO_PI*exp((-0.5*(h*h+t*t)))*(t/r)*asymptotic_expansion_sum;+ return fabs(std::max(b , 0.));+}++namespace { /* η */ static const double asymptotic_expansion_accuracy_threshold = -10; }++double normalised_black_call_using_erfcx(double h, double t) {+ // Given h = x/s and t = s/2, the normalised Black function can be written as+ //+ // b(x,s) = Φ(x/s+s/2)·exp(x/2) - Φ(x/s-s/2)·exp(-x/2)+ // = Φ(h+t)·exp(h·t) - Φ(h-t)·exp(-h·t) . (*)+ //+ // It is mentioned in section 4 (and discussion of figures 2 and 3) of George Marsaglia's article "Evaluating the+ // Normal Distribution" (available at http://www.jstatsoft.org/v11/a05/paper) that the error of any cumulative normal+ // function Φ(z) is dominated by the hardware (or compiler implementation) accuracy of exp(-z²/2) which is not+ // reliably more than 14 digits when z is large. The accuracy of Φ(z) typically starts coming down to 14 digits when+ // z is around -8. For the (normalised) Black function, as above in (*), this means that we are subtracting two terms+ // that are each products of terms with about 14 digits of accuracy. The net result, in each of the products, is even+ // less accuracy, and then we are taking the difference of these terms, resulting in even less accuracy. When we are+ // using the asymptotic expansion asymptotic_expansion_of_normalised_black_call() invoked in the second branch at the+ // beginning of this function, we are using only *one* exponential instead of 4, and this improves accuracy. It+ // actually improves it a bit more than you would expect from the above logic, namely, almost the full two missing+ // digits (in 64 bit IEEE floating point). Unfortunately, going higher order in the asymptotic expansion will not+ // enable us to gain more accuracy (by extending the range in which we could use the expansion) since the asymptotic+ // expansion, being a divergent series, can never gain 16 digits of accuracy for z=-8 or just below. The best you can+ // get is about 15 digits (just), for about 35 terms in the series (26.2.12), which would result in an prohibitively+ // long expression in function asymptotic expansion asymptotic_expansion_of_normalised_black_call(). In this last branch,+ // here, we therefore take a different tack as follows.+ // The "scaled complementary error function" is defined as erfcx(z) = exp(z²)·erfc(z). Cody's implementation of this+ // function as published in "Rational Chebyshev approximations for the error function", W. J. Cody, Math. Comp., 1969, pp.+ // 631-638, uses rational functions that theoretically approximates erfcx(x) to at least 18 significant decimal digits,+ // *without* the use of the exponential function when x>4, which translates to about z<-5.66 in Φ(z). To make use of it,+ // we write+ // Φ(z) = exp(-z²/2)·erfcx(-z/√2)/2+ //+ // to transform the normalised black function to+ //+ // b = ½ · exp(-½(h²+t²)) · [ erfcx(-(h+t)/√2) - erfcx(-(h-t)/√2) ]+ //+ // which now involves only one exponential, instead of three, when |h|+|t| > 5.66 , and the difference inside the+ // square bracket is between the evaluation of two rational functions, which, typically, according to Marsaglia,+ // retains the full 16 digits of accuracy (or just a little less than that).+ //+ const double b = 0.5 * exp(-0.5*(h*h+t*t)) * ( erfcx_cody(-ONE_OVER_SQRT_TWO*(h+t)) - erfcx_cody(-ONE_OVER_SQRT_TWO*(h-t)) );+ return fabs(std::max(b,0.0));+}++// Calculation of+//+// b = Φ(h+t)·exp(h·t) - Φ(h-t)·exp(-h·t)+//+// exp(-(h²+t²)/2)+// = --------------- · [ Y(h+t) - Y(h-t) ]+// √(2π)+// with+// Y(z) := Φ(z)/φ(z)+//+// using an expansion of Y(h+t)-Y(h-t) for small t to twelvth order in t.+// Theoretically accurate to (better than) precision ε = 2.23E-16 when h<=0 and t < τ with τ := 2·ε^(1/16) ≈ 0.21.+// The main bottleneck for precision is the coefficient a:=1+h·Y(h) when |h|>1 .+double small_t_expansion_of_normalised_black_call(double h, double t){+ // Y(h) := Φ(h)/φ(h) = √(π/2)·erfcx(-h/√2)+ // a := 1+h·Y(h) --- Note that due to h<0, and h·Y(h) -> -1 (from above) as h -> -∞, we also have that a>0 and a -> 0 as h -> -∞+ // w := t² , h2 := h²+ const double a = 1+h*(0.5*SQRT_TWO_PI)*erfcx_cody(-ONE_OVER_SQRT_TWO*h), w=t*t, h2=h*h;+ const double expansion = 2*t*(a+w*((-1+3*a+a*h2)/6+w*((-7+15*a+h2*(-1+10*a+a*h2))/120+w*((-57+105*a+h2*(-18+105*a+h2*(-1+21*a+a*h2)))/5040+w*((-561+945*a+h2*(-285+1260*a+h2*(-33+378*a+h2*(-1+36*a+a*h2))))/362880+w*((-6555+10395*a+h2*(-4680+17325*a+h2*(-840+6930*a+h2*(-52+990*a+h2*(-1+55*a+a*h2)))))/39916800+((-89055+135135*a+h2*(-82845+270270*a+h2*(-20370+135135*a+h2*(-1926+25740*a+h2*(-75+2145*a+h2*(-1+78*a+a*h2))))))*w)/6227020800.0))))));+ const double b = ONE_OVER_SQRT_TWO_PI*exp((-0.5*(h*h+t*t)))*expansion;+ return fabs(std::max(b,0.0));+}++namespace { /* τ */ static const double small_t_expansion_of_normalised_black_threshold = 2*SIXTEENTH_ROOT_DBL_EPSILON; }++// b(x,s) = Φ(x/s+s/2)·exp(x/2) - Φ(x/s-s/2)·exp(-x/2)+// = Φ(h+t)·exp(x/2) - Φ(h-t)·exp(-x/2)+// with+// h = x/s and t = s/2+double normalised_black_call_using_norm_cdf(double x, double s){+ const double h = x/s, t = 0.5*s, b_max = exp(0.5*x), b = norm_cdf(h + t) * b_max - norm_cdf(h - t) / b_max;+ return fabs(std::max(b,0.0));+}++//+// Introduced on 2017-02-18+//+// b(x,s) = Φ(x/s+s/2)·exp(x/2) - Φ(x/s-s/2)·exp(-x/2)+// = Φ(h+t)·exp(x/2) - Φ(h-t)·exp(-x/2)+// = ½ · exp(-u²-v²) · [ erfcx(u-v) - erfcx(u+v) ]+// = ½ · [ exp(x/2)·erfc(u-v) - exp(-x/2)·erfc(u+v) ]+// = ½ · [ exp(x/2)·erfc(u-v) - exp(-u²-v²)·erfcx(u+v) ]+// = ½ · [ exp(-u²-v²)·erfcx(u-v) - exp(-x/2)·erfc(u+v) ]+// with+// h = x/s , t = s/2 ,+// and+// u = -h/√2 and v = t/√2 .+//+// Cody's erfc() and erfcx() functions each, for some values of their argument, involve the evaluation+// of the exponential function exp(). The normalised Black function requires additional evaluation(s)+// of the exponential function irrespective of which of the above formulations is used. However, the total+// number of exponential function evaluations can be minimised by a judicious choice of one of the above+// formulations depending on the input values and the branch logic in Cody's erfc() and erfcx().+//+double normalised_black_call_with_optimal_use_of_codys_functions(double x, double s){+ const double codys_threshold = 0.46875, h = x/s, t = 0.5*s, q1 = -ONE_OVER_SQRT_TWO*(h+t), q2 = -ONE_OVER_SQRT_TWO*(h-t);+ double two_b;+ if ( q1 < codys_threshold )+ if ( q2 < codys_threshold )+ two_b = exp(0.5*x)*erfc_cody(q1) - exp(-0.5*x)*erfc_cody(q2);+ else+ two_b = exp(0.5*x)*erfc_cody(q1) - exp(-0.5*(h*h+t*t))*erfcx_cody(q2);+ else+ if ( q2 < codys_threshold )+ two_b = exp(-0.5*(h*h+t*t))*erfcx_cody(q1) - exp(-0.5*x)*erfc_cody(q2);+ else+ two_b = exp(-0.5*(h*h+t*t)) * ( erfcx_cody(q1) - erfcx_cody(q2) );+ return fabs(std::max(0.5*two_b,0.0));+}++EXPORT_EXTERN_C double normalised_black_call(double x, double s) {+ if (x>0)+ return normalised_intrinsic_call(x)+normalised_black_call(-x,s); // In the money.+ if (s<=fabs(x)*DENORMALISATION_CUTOFF)+ return normalised_intrinsic_call(x); // sigma=0 -> intrinsic value.+ // Denote h := x/s and t := s/2.+ // We evaluate the condition |h|>|η|, i.e., h<η && t < τ+|h|-|η| avoiding any divisions by s , where η = asymptotic_expansion_accuracy_threshold and τ = small_t_expansion_of_normalised_black_threshold .+ if ( x < s*asymptotic_expansion_accuracy_threshold && 0.5*s*s+x < s*(small_t_expansion_of_normalised_black_threshold+asymptotic_expansion_accuracy_threshold) )+ return asymptotic_expansion_of_normalised_black_call(x/s,0.5*s);+ if ( 0.5*s < small_t_expansion_of_normalised_black_threshold )+ return small_t_expansion_of_normalised_black_call(x/s,0.5*s);+#ifdef DO_NOT_OPTIMISE_NORMALISED_BLACK_IN_REGIONS_3_AND_4_FOR_CODYS_FUNCTIONS+ // When b is more than, say, about 85% of b_max=exp(x/2), then b is dominated by the first of the two terms in the Black formula, and we retain more accuracy by not attempting to combine the two terms in any way.+ // We evaluate the condition h+t>0.85 avoiding any divisions by s.+ if ( x+0.5*s*s > s*0.85 )+ return normalised_black_call_using_norm_cdf(x,s);+ return normalised_black_call_using_erfcx(x/s,0.5*s);+#else+ return normalised_black_call_with_optimal_use_of_codys_functions(x,s);+#endif+}++inline double square(double x){ return x*x; }++EXPORT_EXTERN_C double normalised_vega(double x, double s) {+ const double ax = fabs(x);+ return (ax<=0) ? ONE_OVER_SQRT_TWO_PI*exp(-0.125*s*s) : ( (s<=0 || s<=ax*SQRT_DBL_MIN) ? 0 : ONE_OVER_SQRT_TWO_PI*exp(-0.5*(square(x/s)+square(0.5*s))) );+}++EXPORT_EXTERN_C double normalised_black(double x, double s, double q /* q=±1 */) { return normalised_black_call(q<0?-x:x,s); /* Reciprocal-strike call-put equivalence */ }++EXPORT_EXTERN_C double black(double F, double K, double sigma, double T, double q /* q=±1 */) {+ const double intrinsic = fabs(std::max((q<0?K-F:F-K),0.0));+ // Map in-the-money to out-of-the-money+ if (q*(F-K)>0)+ return intrinsic + black(F,K,sigma,T,-q);+ return std::max(intrinsic,(sqrt(F)*sqrt(K))*normalised_black(log(F/K),sigma*sqrt(T),q));+}++#ifdef COMPUTE_LOWER_MAP_DERIVATIVES_INDIVIDUALLY+double f_lower_map(const double x,const double s){ + if (is_below_horizon(x))+ return 0;+ if (is_below_horizon(s))+ return 0;+ const double z=SQRT_ONE_OVER_THREE*fabs(x)/s, Phi=norm_cdf(-z);+ return TWO_PI_OVER_SQRT_TWENTY_SEVEN*fabs(x)*(Phi*Phi*Phi);+}+double d_f_lower_map_d_beta(const double x,const double s){+ if (is_below_horizon(s))+ return 1;+ const double z=SQRT_ONE_OVER_THREE*fabs(x)/s, y = z*z, Phi=norm_cdf(-z);+ return TWO_PI*y*(Phi*Phi) * exp(y+0.125*s*s);+}+double d2_f_lower_map_d_beta2(const double x,const double s){+ const double ax=fabs(x), z=SQRT_ONE_OVER_THREE*ax/s, y = z*z, s2=s*s, Phi=norm_cdf(-z), phi=norm_pdf(z);+ return PI_OVER_SIX * y/(s2*s) * Phi * ( 8*SQRT_THREE*s*ax + (3*s2*(s2-8)-8*x*x)*Phi/phi ) * exp(2*y+0.25*s2);+}+void compute_f_lower_map_and_first_two_derivatives(const double x,const double s,double &f,double &fp,double &fpp){+ f = f_lower_map(x,s);+ fp = d_f_lower_map_d_beta(x,s);+ fpp = d2_f_lower_map_d_beta2(x,s);+}+#else+void compute_f_lower_map_and_first_two_derivatives(const double x,const double s,double &f,double &fp,double &fpp){+ const double ax=fabs(x), z=SQRT_ONE_OVER_THREE*ax/s, y = z*z, s2=s*s, Phi=norm_cdf(-z), phi=norm_pdf(z);+ fpp = PI_OVER_SIX * y/(s2*s) * Phi * ( 8*SQRT_THREE*s*ax + (3*s2*(s2-8)-8*x*x)*Phi/phi ) * exp(2*y+0.25*s2);+ if (is_below_horizon(s)) {+ fp = 1;+ f = 0;+ } else {+ const double Phi2=Phi*Phi;+ fp = TWO_PI*y*Phi2*exp(y+0.125*s*s);+ if (is_below_horizon(x))+ f = 0;+ else+ f = TWO_PI_OVER_SQRT_TWENTY_SEVEN*ax*(Phi2*Phi);+ }+}+#endif++double inverse_f_lower_map(const double x,const double f){+ return is_below_horizon(f) ? 0 : fabs(x/(SQRT_THREE*inverse_norm_cdf( std::pow( f/(TWO_PI_OVER_SQRT_TWENTY_SEVEN*fabs(x)) , 1./3.) ))); +}++#ifdef COMPUTE_UPPER_MAP_DERIVATIVES_INDIVIDUALLY+double f_upper_map(const double s){+ return norm_cdf(-0.5*s);+}+double d_f_upper_map_d_beta(const double x,const double s){+ return is_below_horizon(x) ? -0.5 : -0.5*exp(0.5*square(x/s));+}+double d2_f_upper_map_d_beta2(const double x,const double s){+ if (is_below_horizon(x))+ return 0;+ const double w = square(x/s);+ return SQRT_PI_OVER_TWO*exp(w+0.125*s*s)*w/s;+}+void compute_f_upper_map_and_first_two_derivatives(const double x,const double s,double &f,double &fp,double &fpp){+ f = f_upper_map(s);+ fp = d_f_upper_map_d_beta(x,s);+ fpp = d2_f_upper_map_d_beta2(x,s);+}+#else+void compute_f_upper_map_and_first_two_derivatives(const double x,const double s,double &f,double &fp,double &fpp){+ f = norm_cdf(-0.5*s);+ if (is_below_horizon(x)) {+ fp = -0.5;+ fpp = 0;+ } else {+ const double w = square(x/s);+ fp = -0.5*exp(0.5*w);+ fpp = SQRT_PI_OVER_TWO*exp(w+0.125*s*s)*w/s;+ }+}+#endif++double inverse_f_upper_map(double f){+ return -2.*inverse_norm_cdf(f);+}++// See http://en.wikipedia.org/wiki/Householder%27s_method for a detailed explanation of the third order Householder iteration.+//+// Given the objective function g(s) whose root x such that 0 = g(s) we seek, iterate+//+// s_n+1 = s_n - (g/g') · [ 1 - (g''/g')·(g/g') ] / [ 1 - (g/g')·( (g''/g') - (g'''/g')·(g/g')/6 ) ]+//+// Denoting newton:=-(g/g'), halley:=(g''/g'), and hh3:=(g'''/g'), this reads+//+// s_n+1 = s_n + newton · [ 1 + halley·newton/2 ] / [ 1 + newton·( halley + hh3·newton/6 ) ]+//+//+// NOTE that this function returns 0 when beta<intrinsic without any safety checks.+//+double unchecked_normalised_implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(double beta, double x, double q /* q=±1 */, int N){+ // Subtract intrinsic.+ if (q*x>0) {+ beta = fabs(std::max(beta-normalised_intrinsic(x, q),0.));+ q = -q;+ }+ // Map puts to calls+ if (q<0){+ x = -x;+ q = -q;+ }+ if (beta<=0) // For negative or zero prices we return 0.+ return implied_volatility_output(0,0);+ if (beta<DENORMALISATION_CUTOFF) // For positive but denormalised (a.k.a. 'subnormal') prices, we return 0 since it would be impossible to converge to full machine accuracy anyway.+ return implied_volatility_output(0,0);+ const double b_max = exp(0.5*x);+ if (beta>=b_max)+ return implied_volatility_output(0,VOLATILITY_VALUE_TO_SIGNAL_PRICE_IS_ABOVE_MAXIMUM);+ int iterations=0, direction_reversal_count = 0;+ double f=-DBL_MAX, s=-DBL_MAX, ds=s, ds_previous=0, s_left=DBL_MIN, s_right=DBL_MAX;+ // The temptation is great to use the optimised form b_c = exp(x/2)/2-exp(-x/2)·Phi(sqrt(-2·x)) but that would require implementing all of the above types of round-off and over/underflow handling for this expression, too.+ const double s_c=sqrt(fabs(2*x)), b_c = normalised_black_call(x,s_c), v_c = normalised_vega(x, s_c);+ // Four branches.+ if ( beta<b_c ) {+ const double s_l = s_c - b_c/v_c, b_l = normalised_black_call(x,s_l);+ if (beta<b_l){+ double f_lower_map_l, d_f_lower_map_l_d_beta, d2_f_lower_map_l_d_beta2;+ compute_f_lower_map_and_first_two_derivatives(x,s_l,f_lower_map_l,d_f_lower_map_l_d_beta,d2_f_lower_map_l_d_beta2);+ const double r_ll=convex_rational_cubic_control_parameter_to_fit_second_derivative_at_right_side(0.,b_l,0.,f_lower_map_l,1.,d_f_lower_map_l_d_beta,d2_f_lower_map_l_d_beta2,true);+ f = rational_cubic_interpolation(beta,0.,b_l,0.,f_lower_map_l,1.,d_f_lower_map_l_d_beta,r_ll);+ if (!(f>0)) { // This can happen due to roundoff truncation for extreme values such as |x|>500.+ // We switch to quadratic interpolation using f(0)≡0, f(b_l), and f'(0)≡1 to specify the quadratic.+ const double t = beta/b_l;+ f = (f_lower_map_l*t + b_l*(1-t)) * t;+ }+ s = inverse_f_lower_map(x,f);+ s_right = s_l;+ //+ // In this branch, which comprises the lowest segment, the objective function is+ // g(s) = 1/ln(b(x,s)) - 1/ln(beta)+ // ≡ 1/ln(b(s)) - 1/ln(beta)+ // This makes+ // g' = -b'/(b·ln(b)²)+ // newton = -g/g' = (ln(beta)-ln(b))·ln(b)/ln(beta)·b/b'+ // halley = g''/g' = b''/b' - b'/b·(1+2/ln(b))+ // hh3 = g'''/g' = b'''/b' + 2(b'/b)²·(1+3/ln(b)·(1+1/ln(b))) - 3(b''/b)·(1+2/ln(b))+ //+ // The Householder(3) iteration is+ // s_n+1 = s_n + newton · [ 1 + halley·newton/2 ] / [ 1 + newton·( halley + hh3·newton/6 ) ]+ //+ for (; iterations<N && fabs(ds)>DBL_EPSILON*s; ++iterations){+ if (ds*ds_previous<0)+ ++direction_reversal_count;+ if ( iterations>0 && ( 3==direction_reversal_count || !(s>s_left && s<s_right) ) ) {+ // If looping inefficently, or the forecast step takes us outside the bracket, or onto its edges, switch to binary nesting.+ // NOTE that this can only really happen for very extreme values of |x|, such as |x| = |ln(F/K)| > 500.+ s = 0.5*(s_left+s_right);+ if (s_right-s_left<=DBL_EPSILON*s) break;+ direction_reversal_count = 0;+ ds = 0;+ }+ ds_previous=ds;+ const double b = normalised_black_call(x,s), bp = normalised_vega(x, s);+ if ( b>beta && s<s_right ) s_right=s; else if ( b<beta && s>s_left ) s_left=s; // Tighten the bracket if applicable.+ if (b<=0||bp<=0) // Numerical underflow. Switch to binary nesting for this iteration.+ ds = 0.5*(s_left+s_right)-s;+ else {+ const double ln_b=log(b), ln_beta=log(beta), bpob=bp/b, h=x/s, b_halley = h*h/s-s/4, newton = (ln_beta-ln_b)*ln_b/ln_beta/bpob, halley = b_halley-bpob*(1+2/ln_b);+ const double b_hh3 = b_halley*b_halley-3*square(h/s)-0.25, hh3 = b_hh3+2*square(bpob)*(1+3/ln_b*(1+1/ln_b))-3*b_halley*bpob*(1+2/ln_b);+ ds = newton * householder_factor(newton,halley,hh3);+ }+ s += ds = std::max(-0.5*s , ds );+ }+ return implied_volatility_output(iterations,s);+ } else {+ const double v_l = normalised_vega(x, s_l), r_lm = convex_rational_cubic_control_parameter_to_fit_second_derivative_at_right_side(b_l,b_c,s_l,s_c,1/v_l,1/v_c,0.0,false);+ s = rational_cubic_interpolation(beta,b_l,b_c,s_l,s_c,1/v_l,1/v_c,r_lm);+ s_left = s_l;+ s_right = s_c;+ }+ } else {+ const double s_h = v_c>DBL_MIN ? s_c+(b_max-b_c)/v_c : s_c, b_h = normalised_black_call(x,s_h);+ if(beta<=b_h){+ const double v_h = normalised_vega(x, s_h), r_hm = convex_rational_cubic_control_parameter_to_fit_second_derivative_at_left_side(b_c,b_h,s_c,s_h,1/v_c,1/v_h,0.0,false);+ s = rational_cubic_interpolation(beta,b_c,b_h,s_c,s_h,1/v_c,1/v_h,r_hm);+ s_left = s_c;+ s_right = s_h;+ } else {+ double f_upper_map_h, d_f_upper_map_h_d_beta, d2_f_upper_map_h_d_beta2;+ compute_f_upper_map_and_first_two_derivatives(x,s_h,f_upper_map_h,d_f_upper_map_h_d_beta,d2_f_upper_map_h_d_beta2);+ if ( d2_f_upper_map_h_d_beta2>-SQRT_DBL_MAX && d2_f_upper_map_h_d_beta2<SQRT_DBL_MAX ){+ const double r_hh = convex_rational_cubic_control_parameter_to_fit_second_derivative_at_left_side(b_h,b_max,f_upper_map_h,0.,d_f_upper_map_h_d_beta,-0.5,d2_f_upper_map_h_d_beta2,true);+ f = rational_cubic_interpolation(beta,b_h,b_max,f_upper_map_h,0.,d_f_upper_map_h_d_beta,-0.5,r_hh);+ }+ if (f<=0) {+ const double h=b_max-b_h, t=(beta-b_h)/h;+ f = (f_upper_map_h*(1-t) + 0.5*h*t) * (1-t); // We switch to quadratic interpolation using f(b_h), f(b_max)≡0, and f'(b_max)≡-1/2 to specify the quadratic.+ }+ s = inverse_f_upper_map(f);+ s_left = s_h;+ if (beta>0.5*b_max) { // Else we better drop through and let the objective function be g(s) = b(x,s)-beta. + //+ // In this branch, which comprises the upper segment, the objective function is+ // g(s) = ln(b_max-beta)-ln(b_max-b(x,s))+ // ≡ ln((b_max-beta)/(b_max-b(s)))+ // This makes+ // g' = b'/(b_max-b)+ // newton = -g/g' = ln((b_max-b)/(b_max-beta))·(b_max-b)/b'+ // halley = g''/g' = b''/b' + b'/(b_max-b)+ // hh3 = g'''/g' = b'''/b' + g'·(2g'+3b''/b')+ // and the iteration is+ // s_n+1 = s_n + newton · [ 1 + halley·newton/2 ] / [ 1 + newton·( halley + hh3·newton/6 ) ].+ //+ for (; iterations<N && fabs(ds)>DBL_EPSILON*s; ++iterations){+ if (ds*ds_previous<0)+ ++direction_reversal_count;+ if ( iterations>0 && ( 3==direction_reversal_count || !(s>s_left && s<s_right) ) ) {+ // If looping inefficently, or the forecast step takes us outside the bracket, or onto its edges, switch to binary nesting.+ // NOTE that this can only really happen for very extreme values of |x|, such as |x| = |ln(F/K)| > 500.+ s = 0.5*(s_left+s_right);+ if (s_right-s_left<=DBL_EPSILON*s) break;+ direction_reversal_count = 0;+ ds = 0;+ }+ ds_previous=ds;+ const double b = normalised_black_call(x,s), bp = normalised_vega(x, s);+ if ( b>beta && s<s_right ) s_right=s; else if ( b<beta && s>s_left ) s_left=s; // Tighten the bracket if applicable.+ if (b>=b_max||bp<=DBL_MIN) // Numerical underflow. Switch to binary nesting for this iteration.+ ds = 0.5*(s_left+s_right)-s;+ else {+ const double b_max_minus_b = b_max-b, g = log((b_max-beta)/b_max_minus_b), gp = bp/b_max_minus_b;+ const double b_halley = square(x/s)/s-s/4, b_hh3 = b_halley*b_halley-3*square(x/(s*s))-0.25;+ const double newton = -g/gp, halley = b_halley+gp, hh3 = b_hh3+gp*(2*gp+3*b_halley);+ ds = newton * householder_factor(newton,halley,hh3);+ }+ s += ds = std::max(-0.5*s , ds );+ }+ return implied_volatility_output(iterations,s);+ }+ }+ }+ // In this branch, which comprises the two middle segments, the objective function is g(s) = b(x,s)-beta, or g(s) = b(s) - beta, for short.+ // This makes+ // newton = -g/g' = -(b-beta)/b'+ // halley = g''/g' = b''/b' = x²/s³-s/4+ // hh3 = g'''/g' = b'''/b' = halley² - 3·(x/s²)² - 1/4+ // and the iteration is+ // s_n+1 = s_n + newton · [ 1 + halley·newton/2 ] / [ 1 + newton·( halley + hh3·newton/6 ) ].+ //+ for (; iterations<N && fabs(ds)>DBL_EPSILON*s; ++iterations){+ if (ds*ds_previous<0)+ ++direction_reversal_count;+ if ( iterations>0 && ( 3==direction_reversal_count || !(s>s_left && s<s_right) ) ) {+ // If looping inefficently, or the forecast step takes us outside the bracket, or onto its edges, switch to binary nesting.+ // NOTE that this can only really happen for very extreme values of |x|, such as |x| = |ln(F/K)| > 500.+ s = 0.5*(s_left+s_right);+ if (s_right-s_left<=DBL_EPSILON*s) break;+ direction_reversal_count = 0;+ ds = 0;+ }+ ds_previous=ds;+ const double b = normalised_black_call(x,s), bp = normalised_vega(x, s);+ if ( b>beta && s<s_right ) s_right=s; else if ( b<beta && s>s_left ) s_left=s; // Tighten the bracket if applicable.+ const double newton = (beta-b)/bp, halley = square(x/s)/s-s/4, hh3 = halley*halley-3*square(x/(s*s))-0.25;+ s += ds = std::max(-0.5*s , newton * householder_factor(newton,halley,hh3) );+ }+ return implied_volatility_output(iterations,s);+}++EXPORT_EXTERN_C double implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(double price, double F, double K, double T, double q /* q=±1 */, int N){+ const double intrinsic = fabs(std::max((q<0?K-F:F-K),0.0));+ if (price<intrinsic)+ return implied_volatility_output(0,VOLATILITY_VALUE_TO_SIGNAL_PRICE_IS_BELOW_INTRINSIC);+ const double max_price = (q<0?K:F);+ if (price>=max_price)+ return implied_volatility_output(0,VOLATILITY_VALUE_TO_SIGNAL_PRICE_IS_ABOVE_MAXIMUM);+ const double x = log(F/K);+ // Map in-the-money to out-of-the-money+ if (q*x>0) {+ price = fabs(std::max(price-intrinsic,0.0));+ q = -q;+ }+ return unchecked_normalised_implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(price/(sqrt(F)*sqrt(K)), x, q, N)/sqrt(T);+}++EXPORT_EXTERN_C double implied_volatility_from_a_transformed_rational_guess(double price, double F, double K, double T, double q /* q=±1 */){+ return implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(price,F,K,T,q,implied_volatility_maximum_iterations.data);+}++EXPORT_EXTERN_C double normalised_implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(double beta, double x, double q /* q=±1 */, int N){+ // Map in-the-money to out-of-the-money+ if (q*x>0) {+ beta -= normalised_intrinsic(x, q);+ q = -q;+ }+ if (beta<0)+ return implied_volatility_output(0,VOLATILITY_VALUE_TO_SIGNAL_PRICE_IS_BELOW_INTRINSIC);+ return unchecked_normalised_implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(beta, x, q, N);+}++EXPORT_EXTERN_C double normalised_implied_volatility_from_a_transformed_rational_guess(double beta, double x, double q /* q=±1 */){+ return normalised_implied_volatility_from_a_transformed_rational_guess_with_limited_iterations(beta,x,q,implied_volatility_maximum_iterations.data);+}+
+ external/src/normaldistribution.cpp view
@@ -0,0 +1,147 @@+//+// normaldistribution.cpp+//++#if defined(_MSC_VER)+# define NOMINMAX // to suppress MSVC's definitions of min() and max()+// These four pragmas are the equivalent to /fp:fast.+# pragma float_control( except, off )+# pragma float_control( precise, off )+# pragma fp_contract( on )+# pragma fenv_access( off )+#endif++#include "normaldistribution.h"+#include <float.h>++namespace {+ // The asymptotic expansion Φ(z) = φ(z)/|z|·[1-1/z^2+...], Abramowitz & Stegun (26.2.12), suffices for Φ(z) to have+ // relative accuracy of 1.64E-16 for z<=-10 with 17 terms inside the square brackets (not counting the leading 1).+ // This translates to a maximum of about 9 iterations below, which is competitive with a call to erfc() and never+ // less accurate when z<=-10. Note that, as mentioned in section 4 (and discussion of figures 2 and 3) of George+ // Marsaglia's article "Evaluating the Normal Distribution" (available at http://www.jstatsoft.org/v11/a05/paper),+ // for values of x approaching -8 and below, the error of any cumulative normal function is actually dominated by+ // the hardware (or compiler implementation) accuracy of exp(-x²/2) which is not reliably more than 14 digits when+ // x becomes large. Still, we should switch to the asymptotic only when it is beneficial to do so.+ const double norm_cdf_asymptotic_expansion_first_threshold = -10.0;+ const double norm_cdf_asymptotic_expansion_second_threshold = -1/sqrt(DBL_EPSILON);+}++double norm_cdf(double z){+ if (z <= norm_cdf_asymptotic_expansion_first_threshold) {+ // Asymptotic expansion for very negative z following (26.2.12) on page 408+ // in M. Abramowitz and A. Stegun, Pocketbook of Mathematical Functions, ISBN 3-87144818-4.+ double sum = 1;+ if (z >= norm_cdf_asymptotic_expansion_second_threshold) {+ double zsqr = z * z, i = 1, g = 1, x, y, a = DBL_MAX, lasta;+ do {+ lasta = a;+ x = (4 * i - 3) / zsqr;+ y = x * ((4 * i - 1) / zsqr);+ a = g * (x - y);+ sum -= a;+ g *= y;+ ++i;+ a = fabs(a);+ } while (lasta > a && a >= fabs(sum * DBL_EPSILON));+ }+ return -norm_pdf(z) * sum / z;+ }+ return 0.5*erfc_cody( -z*ONE_OVER_SQRT_TWO );+}++double inverse_norm_cdf(double u){+ //+ // ALGORITHM AS241 APPL. STATIST. (1988) VOL. 37, NO. 3+ //+ // Produces the normal deviate Z corresponding to a given lower+ // tail area of u; Z is accurate to about 1 part in 10**16.+ // see http://lib.stat.cmu.edu/apstat/241+ //+ const double split1 = 0.425;+ const double split2 = 5.0;+ const double const1 = 0.180625;+ const double const2 = 1.6;++ // Coefficients for P close to 0.5+ const double A0 = 3.3871328727963666080E0;+ const double A1 = 1.3314166789178437745E+2;+ const double A2 = 1.9715909503065514427E+3;+ const double A3 = 1.3731693765509461125E+4;+ const double A4 = 4.5921953931549871457E+4;+ const double A5 = 6.7265770927008700853E+4;+ const double A6 = 3.3430575583588128105E+4;+ const double A7 = 2.5090809287301226727E+3;+ const double B1 = 4.2313330701600911252E+1;+ const double B2 = 6.8718700749205790830E+2;+ const double B3 = 5.3941960214247511077E+3;+ const double B4 = 2.1213794301586595867E+4;+ const double B5 = 3.9307895800092710610E+4;+ const double B6 = 2.8729085735721942674E+4;+ const double B7 = 5.2264952788528545610E+3;+ // Coefficients for P not close to 0, 0.5 or 1.+ const double C0 = 1.42343711074968357734E0;+ const double C1 = 4.63033784615654529590E0;+ const double C2 = 5.76949722146069140550E0;+ const double C3 = 3.64784832476320460504E0;+ const double C4 = 1.27045825245236838258E0;+ const double C5 = 2.41780725177450611770E-1;+ const double C6 = 2.27238449892691845833E-2;+ const double C7 = 7.74545014278341407640E-4;+ const double D1 = 2.05319162663775882187E0;+ const double D2 = 1.67638483018380384940E0;+ const double D3 = 6.89767334985100004550E-1;+ const double D4 = 1.48103976427480074590E-1;+ const double D5 = 1.51986665636164571966E-2;+ const double D6 = 5.47593808499534494600E-4;+ const double D7 = 1.05075007164441684324E-9;+ // Coefficients for P very close to 0 or 1+ const double E0 = 6.65790464350110377720E0;+ const double E1 = 5.46378491116411436990E0;+ const double E2 = 1.78482653991729133580E0;+ const double E3 = 2.96560571828504891230E-1;+ const double E4 = 2.65321895265761230930E-2;+ const double E5 = 1.24266094738807843860E-3;+ const double E6 = 2.71155556874348757815E-5;+ const double E7 = 2.01033439929228813265E-7;+ const double F1 = 5.99832206555887937690E-1;+ const double F2 = 1.36929880922735805310E-1;+ const double F3 = 1.48753612908506148525E-2;+ const double F4 = 7.86869131145613259100E-4;+ const double F5 = 1.84631831751005468180E-5;+ const double F6 = 1.42151175831644588870E-7;+ const double F7 = 2.04426310338993978564E-15;++ if (u<=0)+ return log(u);+ if (u>=1)+ return log(1-u);++ const double q = u-0.5;+ if (fabs(q) <= split1)+ {+ const double r = const1 - q*q;+ return q * (((((((A7 * r + A6) * r + A5) * r + A4) * r + A3) * r + A2) * r + A1) * r + A0) /+ (((((((B7 * r + B6) * r + B5) * r + B4) * r + B3) * r + B2) * r + B1) * r + 1.0);+ }+ else+ {+ double r = q<0.0 ? u : 1.0-u;+ r = sqrt(-log(r));+ double ret;+ if (r < split2)+ {+ r = r - const2;+ ret = (((((((C7 * r + C6) * r + C5) * r + C4) * r + C3) * r + C2) * r + C1) * r + C0) /+ (((((((D7 * r + D6) * r + D5) * r + D4) * r + D3) * r + D2) * r + D1) * r + 1.0);+ }+ else+ {+ r = r - split2;+ ret = (((((((E7 * r + E6) * r + E5) * r + E4) * r + E3) * r + E2) * r + E1) * r + E0) /+ (((((((F7 * r + F6) * r + F5) * r + F4) * r + F3) * r + F2) * r + F1) * r + 1.0);+ }+ return q<0.0 ? -ret : ret;+ }+}+
+ external/src/rationalcubic.cpp view
@@ -0,0 +1,115 @@+//+// This source code resides at www.jaeckel.org/LetsBeRational.7z .+//+// ======================================================================================+// Copyright © 2013-2014 Peter Jäckel.+// +// Permission to use, copy, modify, and distribute this software is freely granted,+// provided that this notice is preserved.+//+// WARRANTY DISCLAIMER+// The Software is provided "as is" without warranty of any kind, either express or implied,+// including without limitation any implied warranties of condition, uninterrupted use,+// merchantability, fitness for a particular purpose, or non-infringement.+// ======================================================================================+//++#include "rationalcubic.h"++#if defined(_MSC_VER)+# define NOMINMAX // to suppress MSVC's definitions of min() and max()+// These four pragmas are the equivalent to /fp:fast.+// YOU NEED THESE FOR THE SAKE OF *ACCURACY* WHEN |x| IS LARGE, say, |x|>50.+// This is because they effectively enable the evaluation of certain+// expressions in 80 bit registers without loss of intermediate accuracy.+# pragma float_control( except, off )+# pragma float_control( precise, off )+# pragma fp_contract( on )+# pragma fenv_access( off )+#endif++#include <float.h>+#include <cmath>+#include <algorithm>++// Based on+//+// “Shape preserving piecewise rational interpolation”, R. Delbourgo, J.A. Gregory - SIAM journal on scientific and statistical computing, 1985 - SIAM.+// http://dspace.brunel.ac.uk/bitstream/2438/2200/1/TR_10_83.pdf [caveat emptor: there are some typographical errors in that draft version]+//++namespace {+ const double minimum_rational_cubic_control_parameter_value = -(1 - sqrt(DBL_EPSILON));+ const double maximum_rational_cubic_control_parameter_value = 2 / (DBL_EPSILON * DBL_EPSILON);+ inline bool is_zero(double x){ return fabs(x) < DBL_MIN; }+}++double rational_cubic_interpolation(double x, double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double r) {+ const double h = (x_r - x_l);+ if (fabs(h)<=0)+ return 0.5 * (y_l + y_r);+ // r should be greater than -1. We do not use assert(r > -1) here in order to allow values such as NaN to be propagated as they should.+ const double t = (x - x_l) / h;+ if ( ! (r >= maximum_rational_cubic_control_parameter_value) ) {+ const double t = (x - x_l) / h, omt = 1 - t, t2 = t * t, omt2 = omt * omt;+ // Formula (2.4) divided by formula (2.5)+ return (y_r * t2 * t + (r * y_r - h * d_r) * t2 * omt + (r * y_l + h * d_l) * t * omt2 + y_l * omt2 * omt) / (1 + (r - 3) * t * omt);+ }+ // Linear interpolation without over-or underflow.+ return y_r * t + y_l * (1 - t);+}++double rational_cubic_control_parameter_to_fit_second_derivative_at_left_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_l) {+ const double h = (x_r-x_l), numerator = 0.5*h*second_derivative_l+(d_r-d_l);+ if (is_zero(numerator))+ return 0;+ const double denominator = (y_r-y_l)/h-d_l;+ if (is_zero(denominator))+ return numerator>0 ? maximum_rational_cubic_control_parameter_value : minimum_rational_cubic_control_parameter_value;+ return numerator/denominator;+}++double rational_cubic_control_parameter_to_fit_second_derivative_at_right_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_r) {+ const double h = (x_r-x_l), numerator = 0.5*h*second_derivative_r+(d_r-d_l);+ if (is_zero(numerator))+ return 0;+ const double denominator = d_r-(y_r-y_l)/h;+ if (is_zero(denominator))+ return numerator>0 ? maximum_rational_cubic_control_parameter_value : minimum_rational_cubic_control_parameter_value;+ return numerator/denominator;+}++double minimum_rational_cubic_control_parameter(double d_l, double d_r, double s, bool preferShapePreservationOverSmoothness) {+ const bool monotonic = d_l * s >= 0 && d_r * s >= 0, convex = d_l <= s && s <= d_r, concave = d_l >= s && s >= d_r;+ if (!monotonic && !convex && !concave) // If 3==r_non_shape_preserving_target, this means revert to standard cubic.+ return minimum_rational_cubic_control_parameter_value;+ const double d_r_m_d_l = d_r - d_l, d_r_m_s = d_r - s, s_m_d_l = s - d_l;+ double r1 = -DBL_MAX, r2 = r1;+ // If monotonicity on this interval is possible, set r1 to satisfy the monotonicity condition (3.8).+ if (monotonic){+ if (!is_zero(s)) // (3.8), avoiding division by zero.+ r1 = (d_r + d_l) / s; // (3.8)+ else if (preferShapePreservationOverSmoothness) // If division by zero would occur, and shape preservation is preferred, set value to enforce linear interpolation.+ r1 = maximum_rational_cubic_control_parameter_value; // This value enforces linear interpolation.+ }+ if (convex || concave) {+ if (!(is_zero(s_m_d_l) || is_zero(d_r_m_s))) // (3.18), avoiding division by zero.+ r2 = std::max(fabs(d_r_m_d_l / d_r_m_s), fabs(d_r_m_d_l / s_m_d_l));+ else if (preferShapePreservationOverSmoothness)+ r2 = maximum_rational_cubic_control_parameter_value; // This value enforces linear interpolation.+ } else if (monotonic && preferShapePreservationOverSmoothness)+ r2 = maximum_rational_cubic_control_parameter_value; // This enforces linear interpolation along segments that are inconsistent with the slopes on the boundaries, e.g., a perfectly horizontal segment that has negative slopes on either edge.+ return std::max(minimum_rational_cubic_control_parameter_value, std::max(r1, r2));+}++double convex_rational_cubic_control_parameter_to_fit_second_derivative_at_left_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_l, bool preferShapePreservationOverSmoothness) {+ const double r = rational_cubic_control_parameter_to_fit_second_derivative_at_left_side(x_l, x_r, y_l, y_r, d_l, d_r, second_derivative_l);+ const double r_min = minimum_rational_cubic_control_parameter(d_l, d_r, (y_r-y_l)/(x_r-x_l), preferShapePreservationOverSmoothness);+ return std::max(r,r_min);+}++double convex_rational_cubic_control_parameter_to_fit_second_derivative_at_right_side(double x_l, double x_r, double y_l, double y_r, double d_l, double d_r, double second_derivative_r, bool preferShapePreservationOverSmoothness) {+ const double r = rational_cubic_control_parameter_to_fit_second_derivative_at_right_side(x_l, x_r, y_l, y_r, d_l, d_r, second_derivative_r);+ const double r_min = minimum_rational_cubic_control_parameter(d_l, d_r, (y_r-y_l)/(x_r-x_l), preferShapePreservationOverSmoothness);+ return std::max(r,r_min);+}
+ src/LetsBeRational.hs view
@@ -0,0 +1,27 @@+{- |+Copyright: (c) 2021 Ghais Issa+SPDX-License-Identifier: MIT+Maintainer: Ghais Issa <0x47@0x49.dev>++Haskell binding for Jaekel's "Let's be Rational" implied volatility calculation+-}++module LetsBeRational+ ( lbr+ ) where+import Foreign.C.Types+import Data.Coerce (coerce)+++foreign import ccall+ "lets_be_rational.h implied_volatility_from_a_transformed_rational_guess" c_lbr ::+ CDouble -> CDouble -> CDouble -> CDouble -> CDouble -> CDouble++-- | Calculate implied volatility for a European option using Let's Be Rational.+lbr :: Int -- ^ 1 for CALL -1 for PUT.+ -> Double -- ^ Forward+ -> Double -- ^ Strike+ -> Double -- ^ Time to maturity+ -> Double -- ^ Premium+ -> Double -- ^ Implied vol.+lbr cp f k t p = coerce $ c_lbr (coerce p) (coerce f) (coerce k) (coerce t) (coerce (fromIntegral cp::Double))