Let us see the end product first. SABR coefficients have been calibrated for 16 different volatility skews. By using four calibrated parameters within SABR model, we should get extremely close replica of original volatility skew. Original forward volatility surface and its SABR-calibrated replica is presented in the screenshot below. Source data has been taken from this great book (Excel files included) written by Richard Flavell. The essence of SABR model calibration has been chewed in the chapter 10.
In order to validate this C# implementation, I have been replicating example (Non-shifted SABR calibration results) found in MathWorks page on SABR model calibration. Original example data and calibration results of this C# implementation are presented in the screenshot below.
Additional results have been presented for shifted SABR model calibrations. On the current low rate environment (in some currencies) we may enter into negative strike/forward rate territory. As this happens, log-normal model will crash. For such cases, using shifted log-normal SABR model is deceptively simple : constant shift will be added into strike prices and forward rates, in order to be able to calibrate the model.
It should be noted, that (in this implementation) beta coefficient is assumed to be a given estimated constant as this is usually a market practice. Moreover, the program is able to calibrate the model by solving three remaining coefficients (alpha, rho, nu) directly or then solving alpha implicitly from two remaining coefficients (rho, nu). These two calibration schemes have been presented in MathWorks example (Method 1 and Method 2). Finally, calibration speed (this should not be a problem anyway) and especially accuracy can be greatly improved by using Vector-Jacobian scheme (done in my previous post) and explicitly defined analytical partial derivatives instead of their numerical equivalents (finite difference).
In order to test this calibration program, create a new C# project (SABRCalibrationTester), copy-paste the following code into a new CS file and add reference to Alglib DLL file.
Finally, thanks for reading my blog.