Understanding Cascade Calibration of Exact Swaptions Volatilities
School
Roger Williams University**We aren't endorsed by this school
Course
AAH 122
Subject
Economics
Date
Dec 10, 2024
Pages
2
Uploaded by sttlbxv
7.4 Exact Swaptions “Cascade” Calibration with Volatilities as in TABLE 1 335 18.46 - - - - - ‘ - - - - 14.09 22.03 - - - - - - - - 12.84 13.11 24.71 - - - - - - - 12.14 11.17 13.00 25.94 - - - - - - 11.64 10.11 10.59 12.54 27.10 - - - - - 11.19 9.51 9.44 9.87 12.73 28.06 - - - - 10.94 8.88 8.47 8.53 9.82 13.01 28.58 - - - 10.59 8.61 7.82 7.57 8.58 10.06 1292 29.62 - - 10.37 8.25 7.53 6.81 7.52 8.61 9.74 13.51 30.20 - 10.26 7.73 7.21 6.43 7.14 7.65 8.31 1045 13.56 30.35 8.89 8.89 7.08 6.31 6.39 7.23 7.38 8.73 1040 1341 8.07 8.07 8.07 6.23 6.30 6.82 6.79 7.96 8.63 10.10 7.35 7.35 7.35 7.35 6.27 6.43 6.29 7.38 7.96 8.44 7.01 7.01 7.01 7.01 7.01 6.39 5.85 6.89 6.70 7.46 6.53 6.53 6.53 6.53 6.53 6.53 6.29 596 6.92 6.68 6.23 6.23 6.23 6.23 6.23 6.23 6.23 6.97 5.68 6.57 6.06 6.06 6.06 6.06 6.06 6.06 6.06 6.06 6.57 5.77 5.76 5.76 5.76 5.76 5.76 5.76 5.76 5.76 5.76 6.35 5.62 5.62 5.62 5.62 5.62 5.62 5.62 5.62 5.62 5.62 Table 7.8. 100x LFM TABLE 1 GPC volatilities o following calibration of smoothed swaptions 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 10y | 1.000 0.677 0.695 0.640 0.544 0.817 0.666 0.762 0.753 0.740 11y | 0.677 1.000 0.614 0.617 0.665 0.768 0.696 0.760 0.752 0.740 12y | 0.695 0.614 1.000 0.758 0.716 0.938 0.848 0.870 0.862 0.850 13y | 0.640 0.617 0.758 1.000 0.740 0.866 0.914 0.894 0.885 0.875 14y | 0.544 0.665 0.716 0.740 1.000 0.771 0919 0.885 0.879 0.868 15y | 0.817 0.768 0.938 0.866 0.771 1.000 0.923 0.965 0.960 0.953 16y | 0.666 0.696 0.848 0.914 0.919 0923 1.000 0.983 0.980 0.975 17y | 0.762 0.760 0.870 0.894 0.885 0.965 0.983 1.000 0.999 0.995 18y | 0.753 0.752 0.862 0.885 0.879 0.960 0.980 0.999 1.000 0.999 19y | 0.740 0.740 0.850 0.875 0.868 0.953 0.975 0.995 0.999 1.000 Table 7.9. Ten-year terminal correlations on non-smoothed swaption data 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 10y | 1.000 0.939 0.898 0.872 0.851 0.838 0.823 0.809 0.817 0.787 11y | 0.939 1.000 0.992 0.980 0.969 0.962 0.947 0.941 0.936 0.915 12y | 0.898 0.992 1.000 0.996 0.990 0.986 0975 0972 0.966 0.950 13y | 0.872 0980 0.996 1.000 0.997 0.995 0.986 0.984 0.979 0.966 14y | 0.851 0.969 0.990 0.997 1.000 0.997 0.992 0.989 0.984 0.973 15y | 0.838 0.962 0.986 0.995 0997 1.000 0.994 0.995 0.990 0.982 16y | 0.823 0.947 0.975 0.986 0.992 0.994 1.000 0.997 0.997 0.992 17y | 0.809 0.941 0.972 0.984 0989 0.995 0.997 1.000 0.998 0.995 18y | 0.817 0.936 0.966 0.979 0.984 0.990 0.997 0.998 1.000 0.998 19y | 0.787 0.915 0.950 0.966 0.973 0.982 0.992 0.995 0.998 1.000 Table 7.10. Ten-year terminal correlations on smoothed swaptions data
336 7. Cases of Calibration of the LIBOR Market Model also a slightly erratic evolution of the term structure of volatilities compared to the smoothed case. 0.3 — T —— — ; - — . - 0.28| g \ b4 026} 4 % \ y \ 3 \ \ \ 0.24- \ \ \ \ \ \ 0.22 i 0 \ \ K. % 0.2} SN e re_ ol N\ N 0.18 - ) h Sy ~. -~ = ~ 016} ~ < 0.14 Ty | P | A . pre =Bt o . 1 2 3 a 5 6 Fig. 7.4. Term structure evolution corresponding to the smoothed volatility swap- tion data As we can see in Figure 7.4, with the smoothed swaption prices in input, the evolution of the term structure of volatility resulting from the calibration is interesting, and the related terminal correlations given in Table 7.10 are decreasing and non-negative. Figure 7.4, although being interesting and displaying a smooth evolution of the term structure of volatilities following the calibration, presents the worrying feature of caplet-like swaption volatilities increasing up to a level of 30%, when the maximum initial value is below 19%. To see whether this undesirable feature is typical of the decomposition of swaptions volatilities into LIBOR ¢’s implicit in the cascade calibration, or rather it is due to peculiar characteristics of this swaption matrix (featuring a first column much higher than those who follow), we need to perform more tests. This is done in Sections 7.6 and 7.7, where we also tackle the issue of discarding the numerical problems encountered, starting from the considerations in the next section.