SABR. This study illustrates how SABR, alpha beta rho nu, is solved using NelderMead

This was a test case to work with SABR and NelderMead.

This C++ app does not require any external libraries. It does read in data from an excel s/h.
It is built with Visual C++ 2017 Community Edition. x64. Only debug mode has been tested.
It has two SABR related computations in function: SABR_BlackVol_InitialAlpha
- 'Obloj2008' Method in Obloj 2008
- 'hagan2002' Original Hagan's method in Hagan et. al. 2002
This SABR model does an initial fit of beta rho nu using method of a local calibration algorithm.
It has a '/doc' subdirectory with related pdfs embedded in spreadsheet, plus a few more.
It has an excel interface, to read in parameter data from a spreadsheet, needed for C++ calculations.

Original VB code by Changwei Xiong.

http://www.cs.utah.edu/~cxiong/
This C++ console app ported his VB code over to C++.
His Excel spreadshet is titled "Swaption_Volatility_SABR_Calibration.xls" See his worksheet titled "Copyright and Disclaimer"
This C++ console app reads in data from his spreadsheet. Then duplicates computations found on his worksheet 'SABR(Implied Alpha)'
This C++ console app has output that can be compared to Changwei Xiong's spreadsheet output on worksheet 'SABR(Implied Alpha)'

Issues

+ This code needs some cleanup. It was meant as a quick study.
+ Excel s/h was tested with Excel 2007 on Windows 10 Business
+ Excel s/h do not close if app is interrupted during debug.
+ Excel s/h is not that stable, with this xlsread....
+ Excel s/h has two addins:
	+ Solver
		+ If problems, google, how delete this Addin and then reinstall.
	+ EDate for 2007 is available via Analysis Tool Pack Add for Excel 2007,
+ Decimal accuracy between Excel and C++ is off...not sure why.
+ This app does not yet promise to always close excel s/h.  This has to be reviewed.
	+ If run of this app is interrupted, an excel s/h will not be cleaned up.  Have to add some code.

To run:

Swaption_Volatility_SABR_Calibration.xls MUST BE CLOSED.
Verify that this line is in 'Command Arguments" passed to app via VS properties, in debug mode. $(ProjectDir)Swaption_Volatility_SABR_Calibration.xls
If that has problems then uncomment line in main, and type in your path to that s/h.
- //wchar_t* XLSPATH = L"H:\github\workdir.2018\SABR.NelderMead_Studies_2\Swaption_Volatility_SABR_Calibration.xls";
- and comment out this line:
- wchar_t* XLSPATH = (wchar_t*)argv[1];

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Below is a partial output from this C++ application

=======

Fetching data from spreadsheet.... I added some formatting in Readme.md

=================Excel Input Local Calibration for NelderMead: Beta, Rho Nu ===========

           |  Beta                 |  Rho                     | Nu
           |---------------------- | ------------------------ | ---------------------
           |  0.56000000000000005  |   -0.01000000000000000   |   0.34599999999999997

=================Excel Input Fixed Rates ===========

           | Fixed Rates
           | -----------
           | 0.035257595 
           | 0.041738169
           | 0.042418148

================= Excel Input MktVol ( SABR Implied Alpha Worksheet) ===========

            |  -200               |  -100					
            |-------------------- | ------------------------ 
            | 0.27410000000000001 |  0.21270000000000000    
            | 0.20929999999999999 |  0.16990000000000000
            | 0.18420000000000000 |  0.14940000000000001
            | 0.18140000000000001 |  0.14760000000000001

================= Excel Input K ===========

              |  -200.00000000000000000 |  -100.00000000000000000

================= Excel Input StrikeRates_K ( Data Worksheet) ===========

              |  -200               |  -100					
              |-------------------- | ------------------------ 
              | 0.01525759459242056 |  0.02525759459242056   
              | 0.02173816907673853 |  0.03173816907673853 
              | 0.02241814840263004 |  0.03241814840263003  
              | 0.01925608289916845 |  0.02925608289916845

================= Excel Input ATMmktVol ( Data Worksheet) ===========

              |  1 yr               |  2 yr					
              |-------------------- | ------------------------ 
              | 0.21500000000000000 |  0.21900000000000000 
              | 0.21600000000000000 |  0.21299999999999999 
              | 0.20899999999999999 |  0.20499999999999999  
              | 0.20000000000000001 |  0.19600000000000001

================= SABR Model Output: NelderMead Iterations to Converge =========== ================= for each row of SABR parameters ===========

              | Loops per row for NelderMead to converge on values for alpha, beta, rho and nu.
              | -----------------------------------------
              | Row 0 Count 93
              | Row 1 Count 115
              | Row 2 Count 104
              | Row 3 Count 124

================= SABR Model Output: Beta Rho Nu (via NelderMead's RunFunction/MyFunction) ===========

              |  beta                |  rho                   |	Nu		
              |--------------------  | ---------------------- | ----------------------
              | 0.56321204711218043  |   -0.01319411793721832 |     0.34591921732199021
              | 0.56305676412278327  |   -0.02290658753951835 |     0.30525247364584918
              | 0.59977120409842088  |   -0.15816151757678021 |     0.25428353284912453
              | 0.62201488699163354  |   -0.17437854011038034 |     0.21155171237474185

================= SABR Model Output: Alpha (via NelderMead's RunFunction/MyFunction) ===========

              | alpha       | 
              | ----------- | 
              | 0.041294239 | 
              | 0.035201727 | 
              | 0.033891166 | 
              | 0.035117295 |

================= SABR Model Output: SABR volatilities (via NelderMead's RunFunction/MyFunction) ===========