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#SABR: Stochastic alpha beta rho

+   This is a port of Matlab code over to C++. Original matlab code is in matlab subdirectory.
+   This is a study in SABR.
+   This C++ code does not require any additional libraries.

#Issues: + (1) Results in Octave vs C++ VS2017 + Differences in decimals. Have not identified yet why?

#(1) This example first solves for SABR's alpha with the following constants:

* F = 100;            % Futures price
* T = 0.5;            % Maturity in years   
* b = 0.9999;         % beta SABR parameter
* v = 0.5;            % vol-of-vol SABR parameter
* ATMvol = 0.30;      % Market ATM volatility
  • Computation loops across these strikes and correlation levels

    • K = 70 80 90 100 110 120 130
    • rho = -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75;
  • Constants for Black 76 only

    • r = .02

#(2) Then SABR volatility is then computed with alpha's and constants

#(3) Black 76 Call/Put and PutCall Parity is computed with SABR vol.

  • =====================================================

  • Below are results from building and running this C++ console application.

  • =====================================================

  • SABR Black-76 Equivalent Volatilies (%). Agrees with results from Haug Book pg 270 (Agrees when rounded to two decimals):

p X=70 X=80 X=90 X=100 X=110 X=120 X=130
-0.75 36.678324 34.18337 31.969147 30.000406 28.260363 26.745429 25.460042
-0.5 35.17378 33.114696 31.399449 30.00007 28.898052 28.074855 27.505726
-0.25 33.509761 31.952084 30.795686 29.999994 29.520725 29.306531 29.302682
0 31.652567 30.682965 30.157598 30 30.126458 30.456642 30.92308
0.25 29.546048 29.288287 29.48468 30.000008 30.713715 31.53576 32.406472
0.5 27.086116 27.737585 28.776021 29.999951 31.281185 32.551056 33.776878
0.75 24.040084 25.97829 28.030273 29.999873 31.827825 33.507644 35.050007
  • Alpha Parameter Values used by SABR Black-76 Equivalent Volatilities, above:
p X=70 X=80 X=90 X=100 X=110 X=120 X=130
-0.75 0.305235 0.305235 0.305235 0.305235 0.305235 0.305235 0.305235
-0.5 0.302257 0.302257 0.302257 0.302257 0.302257 0.302257 0.302257
-0.25 0.299951 0.299951 0.299951 0.299951 0.299951 0.299951 0.299951
0 0.298278 0.298278 0.298278 0.298278 0.298278 0.298278 0.298278
0.25 0.297207 0.297207 0.297207 0.297207 0.297207 0.297207 0.297207
0.5 0.296717 0.296717 0.296717 0.296717 0.296717 0.296717 0.296717
0.75 0.296794 0.296794 0.296794 0.296794 0.296794 0.296794 0.296794
  • Black-76 Put Prices r:0.020000:
p X=70 X=80 X=90 X=100 X=110 X=120 X=130
-0.75 0.826769 2.050543 4.411993 8.362983 14.14351 21.624893 30.355513
-0.5 0.696645 1.879802 4.277698 8.36289 14.309785 21.884927 30.598439
-0.25 0.565428 1.699586 4.135938 8.362869 14.472881 22.135225 30.840911
0 0.435410 1.509967 3.986790 8.36287 14.632193 22.376342 31.081002
0.25 0.309673 1.310885 3.830296 8.362872 14.787226 22.608509 31.317215
0.5 0.192635 1.102052 3.666433 8.362857 14.937546 22.83177 31.548329
0.75 0.091258 0.882768 3.495115 8.362835 15.082796 23.046082 31.773332
  • Black-76 Call Prices r:0.020000:
p X=70 X=80 X=90 X=100 X=110 X=120 X=130
-0.75 30.528264 21.85154 14.312491 8.362983 4.243012 1.823896 0.654018
-0.5 30.39814 21.680799 14.178196 8.36289 4.409287 2.08393 0.896944
-0.25 30.266923 21.500583 14.036436 8.362869 4.572382 2.334228 1.139416
0 30.136905 21.310963 13.887288 8.36287 4.731695 2.575345 1.379507
0.25 30.011168 21.111882 13.730794 8.362872 4.886728 2.807512 1.61572
0.5 29.89413 20.903049 13.566931 8.362857 5.037047 3.030773 1.846834
0.75 29.792753 20.683765 13.395614 8.362835 5.182298 3.245086 2.071838
  • diff in Black-76 Put Call Parity Eqn: (C-P) - (F-K)exp(-rT) r:0.020000 F: 100.000000 T:0.500000:
p X=70 X=80 X=90 X=100 X=110 X=120 X=130
-0.75 0 0 0 0 0 0 0
-0.5 0 0 0 0 0 0 0
-0.25 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0.25 0 0 0 0 0 0 0
0.5 0 0 0 0 0 0 0
0.75 0 0 0 0 0 0 0

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