#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
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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;
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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.
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Below are results from building and running this C++ console application.
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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 |