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Swaption Calibration

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Swaption Calibration

This page provides an example fit to the entire matrix of ATM JPY Swaption vols. Our fitting approach considers the volatility of each Libor forward rate and the correlations of Libor forward rates.

We assume piece-wise constant forward rate volatility. The volatility function is a product of two components:

  • a time-homogeneous component $ \psi(T_i - t) = (a + b (T_{i-1} - t))e^{-c(T_{i-1} - t)} + d $
  • a time-dependent scaling factor $ \phi_i(t) $

The advantage of using the linear exponential volatility function $ \psi(T-t) $ is that the integral of its cross product can be obtained closed form and that the estimated forward rate volatility has a humped shape. This approach also provides much better fitting quality to the entire volatility surface and ensures stable time evolution of the volatility term structure.

The closed form formula for the volatility integral is given as follows:

$ \int_{0}^{ T_{\alpha} } \sigma_i(t) \sigma_j(t) dt $
$ = \int_{0}^{ T_{\alpha} } \left ( [a(T_{i-1}-t)+d]e^{-b (T_{i-1}-t)} + c \right ) \left ( [a(T_{j-1}-t)+d]e^{-b (T_{j-1}-t)} + c \right ) dt $
$ = e^{-b T_{j-1} - b T_{i-1}} / (4 b^3 ) \Bigg [ (2a^2 b^2 T_{\alpha}^2 + (-2a^2 b^2 T_{j-1} -2a^2 b^2 T_{i-1} -4ab^2 d - 2a^2 b)T_{\alpha} $
$ + (2a^2 b^2 T_{i-1} + 2ab^2 d + a^2 b) T_{j-1} +(2ab^2 d + a^2 b) T_{i-1} +2b^2 d^2 +2abd +a^2 ) e^{2b T_{\alpha}} $
$ +((-4ab^2 c \ e^{b T_{j-1}} -4ab^2 c \ e^{b T_{i-1}} ) T_{\alpha} $
$ +(4ab^2 c \ T_{i-1} + 4b^2 cd + 4abc) e^{b T_{j-1}} $
$ + 4ab^2 c \ e^{b T_{i-1}} T_{j-1} + (4b^2 cd + 4abc) e^{b T_{i-1}} ) e^{b T_{\alpha}} $
$ + 4b^3 c^2 \ e^{b T_{j-1} + b T_{i-1}} T_{\alpha} \Bigg ] $
$ - e^{-b T_{j-1} - b T_{i-1}} / (4 b^3 ) \Bigg [ ( 4ab^2 c \ T_{i-1} + 4b^2 cd + 4abc ) e^{b T_{j-1}} $
$ + (4ab^2 c \ e^{b T_{i-1}} + 2a^2 b^2 \ T_{i-1} + 2ab^2 d + a^2 b) T_{j-1} $
$ + (4b^2 cd + 4abc) e^{b T_{i-1}} $
$ + (2ab^2 d + a^2 b) T_{i-1} + 2 b^2 d^2 + 2abd + a^2 \Bigg ] $

Example fit to JPY swaption volatility surface

The historical one-year JPY Libor forward rate correlations are fitted using Rebonato's angle formulation:

Historical Libor Forward Rate Correlations Fitted Libor Forward Rate Correlations

The 3/30/2012 JPY Swaption ATM market vols are fitted using a parametric forward rate volatility function. The volatility function are assumed being the product of a linear exponential function and a set of scale factors:

JPY Swaption ATM Market Vols JPY Swaption ATM Model Vols

The following graphs show a plot of the initial forward rate vols using the fitted parameters and the forward rate vols in 1 year, in 5 years, in 10 and 15 years:

JPY Libor Forward Rate Vols Time Evolution of Libor Forward Rate Vols