Why Neither Time Homogeneity nor Time Dependence Will Do: Evidence from the US$ Swaption Market Cambridge, May 2005 Riccardo Rebonato Global Head of Quantitative Research, FM, RBS Global Head of Market Risk, CBFM, RBS December 2004
Goals of this talk 1. Providing an effective and concise description of the whole swaption matrix 2. Explaining its changes in a more parsimonious and financially more transparent way than using PCA 3. Highlighting some intrinsic problems of time-homogenous and timedependent approaches to derivatives pricing 4. Suggesting how these limitations may be overcome in a computationally tractable manner. 2
Why do we want to do this? Intrinsic interest in a parsimonious description of the static and dynamic features of the swaption matrix Describing patterns and regularities using few, physically-justifiable parameters means understanding what determines and drives a certain phenomenon. (By-product: some insight into what we really need for option pricing in general ) 3
Our setting Local fitting to the swaption matrix using a version of the LIBOR Market Model Drivers are forward rates (about which we have good intuition) Swap rates are by-products Definition: The (j,k) entry of the swaption matrix contains the Black implied percentage volatility of the swaption expiring at the i-th maturity into a swap of length k. I condense the whole swaption matrix into 60 numbers: 0.5, 1, 3, 5, 10-year-expiry options INTO 1, 2, 3, 5, 7, 10-year-maturity swaps 4
The dynamics of the forward rates df i f i i j, f j dt i t, T i dz i # (eq0) df i f i i j, f j dt i t,t i dz i # (eq10) 5
The forward rate (y-axis) for a given probability density (x-axis) for various values of the displacement coefficient (5-year horizon, 4%-forward rate) 0.08 0.06 0.04 0.02 a=0.01 a=0.5 a=2 a=5 a=100 0 0 0.2 0.4 0.6 0.8 1 1.2-0.02 6
A Functional Form for the Instantaneous-Volatility Function Let us begin by assuming that the instantaneous volatility at time t of the forward rate expir T should be given by the expression: t T a b T t exp c T t d a b exp c d T t The quantity therefore represents the residual time to maturity of a particular forward rate the functional form chosen, the presence of a linear term together with a decaying exponential a the existence of a hump in the curve, and the asymptotic instantaneous volatility is assumed to asymptotically to a finite value, d. A few typical shapes are displayed in Fig. 3 below. 7
Possible shapes of the volatility function. Note how both excited (Series 5) and normal states (series 1 to 4) can be obtained. Instantaneous Volatility Curves 44.00% 39.00% Instantaneous Volatility 34.00% 29.00% 24.00% 19.00% Series1 Series2 Series3 Series4 Series5 14.00% 9.00% 4.00% 0 1 2 3 4 5 6 7 8 9 10 Time 8
Moving to two regimes The possibility of two regimes can be accounted for by positing: i n t, T i a t n b t n T t exp c t n T t d t n i x t, T i a t x b t x T t exp c t x T t d t x 9
Our immediate goal Explaining the evolution of swaption prices using a description for the instantaneous volatility the instantaneous correlation of forward rates We must establish a link between the two 10
The time dependence of the instantaneous volatility of the six forward rates in a 3 x 3 semi-annual swaption. The time on the x-axis indicates the time to expiry of the swaption. 24.0% 22.0% 20.0% 18.0% 16.0% 14.0% Fwd1 Fwd2 Fwd3 Fwd4 Fwd5 Fwd6 12.0% 10.0% 0 0.5 1 1.5 2 2.5 3 11
The empirical evidence main features 1. Long periods when the swaption matrix remains largely self-similar, with minor moves up or down (normal periods) 2. Rare, sudden onset of excited periods, where the shape of the swaption surface changes abruptly 3. Reversion to the normal state after a relatively short period of time 12
Swaption prices during the 24-May-99 to 29-June-99 period 18.00% 17.00% 16.00% 15.00% 14.00% 13.00% 12.00% 11.00% 10.00% 11-Jun-99 0.5x1 0.5x10 1x7 3x5 5x3 10x2 24-May-99 13
Swaption prices during the 31-Jul-2000 to 5-Sep-2000 15.00% 14.00% 13.00% 12.00% 11.00% 10.00% 9.00% 18-Aug-00 0.5x1 0.5x10 1x7 3x5 5x3 10x2 31-Jul-00 14
Swaption prices during the 7-Sept-98 to 13-Oct-98 period (LTCM period) 20.00% 18.00% 16.00% 14.00% 12.00% 10.00% 25-Sep-98 0.5x1 0.5x10 1x7 3x5 5x3 10x2 07-Sep-98 15
Swaption prices during the 3-Jan-2001 to 8-Feb-2001 period (Fed cuts) 24.00% 22.00% 20.00% 18.00% 16.00% 14.00% 12.00% 10.00% 23-Jan-01 0.5x1 0.5x10 1x7 3x5 5x3 10x2 03-Jan-01 16
Swaption prices during the 24-Sep-2001 to 3-Oct-2001 period (post Sept 11) 35.00% 30.00% 25.00% 20.00% 15.00% 10.00% 12-Oct-01 0.5x1 0.5x10 1x7 3x5 5x3 10x2 24-Sep-01 17
Different shapes of non-excited swaption patterns (normalized to the 0.5 x 1 volatility for comparison) 130% 120% 110% 100% 90% 80% 70% 60% 50% 01-Mar-99 02-Feb-01 27-Jul-99 13-Aug-01 19-Nov-01 40% 30% 0.5x1 0.5x5 1x1 1x5 3x1 3x5 5x1 5x5 10x1 10x5 18
Same (non-excited) data as above organized by series instead of by expiries 130% 120% 110% 100% 90% 80% 70% 60% 01-Mar-99 02-Feb-01 27-Jul-99 13-Aug-01 19-Nov-01 50% 40% 30% 0.5x1 5x1 1x2 10x2 3x3 0.5x5 5x5 1x7 10x7 3x10 19
Different shapes of excited swaption patterns (normalized to the 0.5 x 1 volatility for comparison) 130% 120% 110% 100% 90% 80% 70% 60% 50% 26-Oct-98 22-Oct-01 23-Apr-01 14-Sep-99 40% 30% 0.5x1 0.5x5 1x1 1x5 3x1 3x5 5x1 5x5 10x1 10x5 20
Same (excited) data organized by series instead of by expiries 130% 120% 110% 100% 90% 80% 70% 60% 26-Oct-98 22-Oct-01 23-Apr-01 14-Sep-99 50% 40% 30% 0.5x1 5x1 1x2 10x2 3x3 0.5x5 5x5 1x7 10x7 3x10 21
A typical log-normal fit for a non-excited day (18-Feb-98). Implied volatility on the y-axis and swaption number on the x-axis: the first swaption is the 0.5 x 1 and the thirtieth the 10 x 10 0.19 0.17 0.15 0.13 0.11 0.09 0.07 0.05 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Model Real 22
The instantaneous volatility that produced the fit displayed above 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0 5 10 15 20 23
The log-normal fit for an excited day (5-Nov-01). 0.4 0.35 0.3 0.25 0.2 0.15 Model Real 0.1 0.05 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 24
The associated instantaneous volatility 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25
The displaced-diffusion fit for 5-Nov-2001 (compare with the log-normal fit for the same day shown before) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Model Real 26
The average absolute pricing errors for each swaption series in the log-normal and displaced-diffusion cases. 2.00% 1.80% 1.60% 1.40% 1.20% 1.00% 0.80% 0.60% 0.40% 0.20% 0.00% LOG DD 0.5x1 0.5x5 1x1 1x5 3x1 3x5 5x1 5x5 10x1 10x5 27
General features of the solutions The log-normal fits tend to find a humped volatility curve on normal days a decaying volatility curve on excited days the exception is the period before 09-11, where a decaying solution is found in the aftermath of excited periods a hybrid solution is found The implied instantaneous correlation is almost always found to be perfect how can we explain this? 28
The fit to the same data with constant (flat) instantaneous volatility and nonperfect instantaneous correlation 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Model Real 29
Typical fits: (6Feb 98) 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Model Real 30
Typical fits: 26-Jun-98 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Model Real 31
A hybrid instantaneous volatility curve in the aftermath (21 Dec 98) of the LTCM period 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 32
Checking the quality of the log-normal solution Desiderata: In a perfect world, the same volatility curve should describe the swaption matrix every day More realistically, we would hope that we can relate changes in the coefficients to changes in the swaption matrix changes in other observable financial quantities 33
Time series of the a coefficient for the log-normal fits a 0.4 0.3 0.2 0.1 a 0 22-Oct-97-0.1 06-Mar-99 18-Jul-00 30-Nov-01-0.2 34
Time series of the b coefficient for the log-normal fits b 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0-0.02 22-Oct-97 06-Mar-99 18-Jul-00 30-Nov-01-0.04 b 35
Time series of the c coefficient for the log-normal fits c 1.4 1.2 1 0.8 0.6 0.4 c 0.2 0 22-Oct-97 06-Mar-99 18-Jul-00 30-Nov-01 36
Time series of the d coefficient for the log-normal fits d 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 22-Oct-97 06-Mar-99 18-Jul-00 30-Nov-01 d 37
Do we need a displaced-diffusion solution? Does the displaced-diffusion solution do appreciably better on average? Does the displaced-diffusion solution do appreciably better during particular periods? Is it possible to associate period when the displaced-diffusion solution performs better with regimes of other economic variables? 38
The figure of merit for the log-normal and the displaced-diffusion fits (arbitrary units) 600 500 400 300 Log Displ 200 100-16-Nov-97 31-Mar-99 12-Aug-00 25-Dec-01 39
The ratio of the figures of merit in Fig 22 1.2 1 0.8 0.6 0.4 0.2 0 16-Nov- 97 04-Jun- 98 21-Dec- 98 09-Jul- 99 25-Jan- 00 12-Aug- 00 28-Feb- 01 16-Sep- 01 04-Apr- 02 40
A scatter-plot of the figures of merit for the log-normal and displaced-diffusion fits (arbitrary units, points linked by a timeline) 500 450 400 350 300 250 200 150 100 50 - - 100 200 300 400 500 600 41
Ratio of the log-normal to displaced diffusion figures of merit against the average level of rates and the level of the 6-month LIBOR rate (all the curves have been normalized to their initial value for ease of comparison) 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 26-Dec- 97 14-Jul- 98 30-Jan- 99 18-Aug- 99 05-Mar- 00 21-Sep- 00 09-Apr- 01 26-Oct- 01 42
Can we do better with displaced diffusion? What do we mean by better: Better fit Better financial story (discriminating between excited and normal states) Changes in coefficients more closely linked to changes in the empirical swaption matrix 43
The displaced-diffusion fit for 5-Nov-2001 (compare with the log-normal fit for the same day shown before) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Model Real 44
The log-normal fit for 10-Sep-2001, i.e. for the day before the events of September 11 0.3 0.25 0.2 0.15 Model Real 0.1 0.05 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 45
The instantaneous volatility function for the log-normal fit for 10-Sep-2001 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 46
The displaced-diffusion fit for 10-Sep-2001, ie for the day before the events of September 11 0.23 0.21 0.19 0.17 0.15 0.13 0.11 0.09 0.07 0.05 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Model Real 47
The instantaneous volatility function for the displaced-diffusion fit for 10-Sep- 2001 0.0004 0.00035 0.0003 0.00025 0.0002 0.00015 0.0001 0.00005 0 0 5 10 15 20 48
The location of the maxima for the log-normal and displaced-diffusion fits 4.00 3.50 3.00 2.50 2.00 1.50 LOG DD 1.00 0.50-49 05/01/1998 05/05/1998 05/09/1998 05/01/1999 05/05/1999 05/09/1999 05/01/2000 05/05/2000 05/09/2000 05/01/2001 05/05/2001 05/09/2001
Time series of the a coefficient for the displaced-diffusion fits (after normalization as discussed in the text) a 0.02 0.015 0.01 0.005 0-0.005-0.01 50 05/01/1998 05/04/1998 05/07/1998 05/10/1998 05/01/1999 05/04/1999 05/07/1999 05/10/1999 05/01/2000 05/04/2000 05/07/2000 05/10/2000 05/01/2001 05/04/2001 05/07/2001 05/10/2001
Time series of the b coefficient for the displaced-diffusion fits (after normalization as discussed in the text) b 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0-0.001 51 05/09/2001 05/01/1998 05/05/1998 05/09/1998 05/01/1999 05/05/1999 05/09/1999 05/01/2000 05/05/2000 05/09/2000 05/01/2001 05/05/2001
Time series of the c coefficient for the displaced-diffusion fits (after normalization as discussed in the text) c 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 52 05/10/2001 05/01/1998 05/04/1998 05/07/1998 05/10/1998 05/01/1999 05/04/1999 05/07/1999 05/10/1999 05/01/2000 05/04/2000 05/07/2000 05/10/2000 05/01/2001 05/04/2001 05/07/2001
Time series of the d coefficient for the displaced-diffusion fits (after normalization as discussed in the text) d 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 53 05/01/1998 05/04/1998 05/07/1998 05/10/1998 05/01/1999 05/04/1999 05/07/1999 05/10/1999 05/01/2000 05/04/2000 05/07/2000 05/10/2000 05/01/2001 05/04/2001 05/07/2001 05/10/2001
Remaining problems The aftermath of excited periods is still problematic No time-homogeneous (deterministic or stochastic) volatility function will do No deterministically time-dependent volatility function will do We need at least to regimes (two-state Markov chain) 54
Using these results to evolve the swaption matrix in the real-world measure The usual approach: PCA Poor in practice, wrong in principle Violation of static arbitrage conditions 55
Correlation between the model and observed changes in implied volatilities, for the whole data set (data labelled Whole ) and for the last period (data labelled Last ). 1 0.9 0.8 0.7 0.6 0.5 0.4 Log(Last) DD(Last) Log(Whole) DD(Whole) 0.3 0.2 0.5x1 0.5x5 1x1 1x5 3x1 3x5 5x1 5x5 10x1 10x5 56
Correlation between market and PCA-implied moves in implied volatility as a function of the number of principal components 1 0.5 0-0.5-1 0.5x1 1x1 3x1 5x1 10x1 1 16 57
Correlation as a function of the number of PC 0.5x1 0.5x2 0.5x3 0.5x5 0.5x7 0.5x10 1x1 1x2 1x3 1x5 1x7 1x10 3x1 3x2 3x3 3x5 3x7 3x10 1 0.694 0.678-0.683 0.787 0.803-0.759 0.817 0.842 0.850 0.878 0.861-0.785 0.816 0.863 0.842 0.832 0.788-0.767 2 0.374-0.729-0.304 0.863 0.834 0.518 0.774 0.837 0.855 0.896 0.696 0.246 0.822 0.878 0.879 0.862 0.813-0.817 3 0.276-0.737 0.049 0.909 0.870 0.651 0.742 0.821 0.893 0.910 0.618 0.104 0.844 0.890 0.884 0.828 0.293-0.811 4 0.441-0.613 0.010 0.864 0.881 0.621 0.784 0.850 0.892 0.910 0.625 0.066 0.820 0.890 0.890 0.848 0.273-0.798 5 0.586-0.570-0.030 0.865 0.888 0.793 0.820 0.850 0.905 0.910 0.634 0.117 0.825 0.891 0.892 0.834 0.193-0.798 6 0.745-0.374-0.100 0.816 0.888 0.775 0.845 0.841 0.910 0.905 0.465 0.542 0.827 0.885 0.884 0.824 0.159-0.764 7 0.775-0.375-0.103 0.879 0.936 0.770 0.844 0.850 0.896 0.912 0.522 0.257 0.770 0.919 0.866 0.880 0.485-0.732 8 0.644-0.476-0.080 0.881 0.939 0.796 0.878 0.841 0.895 0.910 0.512 0.165 0.795 0.911 0.852 0.886 0.648-0.650 9 0.572-0.429-0.079 0.880 0.936 0.743 0.908 0.855 0.901 0.908 0.494 0.088 0.824 0.875 0.840 0.814 0.639-0.521 10 0.828-0.132 0.036 0.901 0.936 0.700 0.920 0.867 0.910 0.908 0.066-0.195 0.824 0.847 0.855 0.877 0.596 0.263 11 0.933 0.220 0.257 0.925 0.935 0.652 0.920 0.872 0.907 0.907-0.425-0.501 0.837 0.776 0.855 0.884 0.536 0.653 12 0.948 0.411 0.435 0.934 0.935 0.641 0.911 0.886 0.890 0.900-0.450-0.538 0.848 0.765 0.907 0.908 0.669 0.686 13 0.960 0.364 0.306 0.948 0.967 0.752 0.904 0.877 0.891 0.886-0.453-0.446 0.879 0.765 0.929 0.902 0.860 0.828 14 0.956 0.496 0.507 0.962 0.968 0.730 0.904 0.873 0.907 0.887-0.261-0.398 0.877 0.824 0.936 0.900 0.861 0.784 15 0.945 0.633 0.826 0.975 0.968 0.759 0.908 0.868 0.922 0.890 0.375 0.073 0.867 0.887 0.934 0.917 0.879 0.284 16 0.936 0.683 0.928 0.979 0.968 0.785 0.912 0.869 0.942 0.891 0.608-0.271 0.865 0.879 0.933 0.914 0.898 0.449 17 0.933 0.719 0.975 0.982 0.972 0.830 0.918 0.879 0.964 0.892 0.674-0.364 0.859 0.926 0.941 0.950 0.962 0.490 18 0.924 0.753 0.998 0.984 0.972 0.855 0.915 0.881 0.965 0.891 0.775-0.162 0.862 0.925 0.951 0.984 0.963 0.535 19 0.940 0.815 0.997 0.979 0.972 0.819 0.905 0.949 0.946 0.940 0.736 0.312 0.967 0.938 0.955 0.981 0.966 0.535 20 0.951 0.854 0.997 0.975 0.973 0.793 0.904 0.961 0.946 0.942 0.766 0.729 0.972 0.957 0.952 0.975 0.967 0.704 21 0.961 0.887 0.996 0.968 0.977 0.780 0.906 0.941 0.947 0.942 0.772 0.731 0.976 0.956 0.955 0.975 0.966 0.753 22 0.973 0.924 0.996 0.959 0.979 0.848 0.919 0.928 0.944 0.942 0.833 0.556 0.980 0.956 0.949 0.973 0.964 0.779 23 0.982 0.950 0.996 0.948 0.979 0.893 0.927 0.951 0.939 0.943 0.864 0.697 0.980 0.959 0.895 0.955 0.939 0.734 24 0.988 0.968 0.996 0.941 0.984 0.917 0.933 0.973 0.979 0.925 0.913 0.658 0.981 0.958 0.942 0.958 0.953 0.725 25 0.989 0.969 0.996 0.943 0.986 0.976 0.985 0.950 0.978 0.976 0.909 0.692 0.983 0.959 0.941 0.957 0.956 0.719 26 0.988 0.970 0.996 0.943 0.986 0.979 0.987 0.961 0.981 0.978 0.916 0.699 0.985 0.956 0.953 0.948 0.958 0.814 27 0.991 0.974 0.996 0.949 0.985 0.982 0.989 0.973 0.989 0.980 0.932 0.713 0.992 0.964 0.947 0.969 0.966 0.822 28 0.992 0.979 0.997 0.960 0.986 0.984 0.990 0.973 0.988 0.996 0.937 0.793 0.998 0.959 0.944 0.984 0.967 0.854 29 0.999 0.998 0.999 0.996 0.999 0.999 1.000 0.998 0.992 0.996 0.990 1.000 1.000 0.952 0.980 1.000 0.981 0.998 30 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 58
5x1 5x2 5x3 5x5 5x7 5x10 10x1 10x2 10x3 10x5 10x7 10x10 Average 1-0.480-0.541-0.535-0.504-0.506 0.442-0.369 0.446 0.402 0.381 0.354 0.341 0.208 2-0.468-0.538 0.529 0.531-0.520-0.402 0.476 0.604 0.555 0.535 0.443 0.460 0.350 3-0.392-0.092 0.658 0.531-0.630-0.491 0.737 0.842 0.548 0.895 0.596 0.330 0.398 4-0.467-0.149 0.636 0.524 0.295-0.343 0.732 0.912 0.824 0.781 0.566 0.858 0.472 5-0.448-0.170 0.647 0.496 0.337-0.359 0.720 0.884 0.708 0.820 0.648 0.979 0.486 6-0.453-0.180 0.530 0.505 0.280-0.380 0.706 0.979 0.873 0.841 0.591 0.994 0.500 7-0.313-0.063 0.549 0.509 0.305-0.386 0.704 0.976 0.872 0.839 0.544 0.994 0.518 8-0.086 0.650 0.825 0.805 0.652-0.464 0.777 0.978 0.771 0.808 0.050 0.994 0.569 9 0.320 0.740 0.350 0.730 0.718-0.222 0.777 0.980 0.836 0.911 0.823 0.994 0.616 10-0.011 0.854-0.080 0.824 0.871-0.047 0.775 0.979 0.070 0.930 0.952 0.991 0.602 11 0.283 0.868 0.564 0.830 0.919-0.176 0.713 0.975 0.376 0.929 0.956 0.998 0.653 12 0.520 0.890 0.416 0.844 0.921-0.091 0.791 0.983 0.519 0.935 0.950 0.999 0.686 13 0.492 0.913 0.430 0.844 0.908-0.082 0.763 0.983 0.515 0.931 0.950 0.999 0.697 14 0.727 0.862 0.104 0.749 0.906-0.063 0.731 0.975 0.153 0.931 0.881 0.999 0.685 15 0.814 0.865 0.904 0.913 0.941-0.254 0.800 0.990 0.869 0.693 0.927 0.999 0.781 16 0.837 0.860 0.832 0.914 0.909 0.002 0.783 0.969 0.879 0.926 0.987 0.999 0.806 17 0.942 0.911 0.822 0.930 0.928 0.548 0.661 0.974 0.948 0.978 0.993 0.999 0.846 18 0.935 0.936 0.951 0.954 0.947 0.850 0.765 0.995 0.981 0.996 0.993 0.999 0.887 19 0.939 0.937 0.961 0.960 0.945 0.849 0.761 0.993 0.980 0.995 0.994 1.000 0.906 20 0.894 0.928 0.952 0.958 0.941 0.903 0.916 0.998 0.976 0.996 0.996 1.000 0.931 21 0.915 0.923 0.975 0.961 0.942 0.931 0.922 0.998 0.991 0.997 0.994 1.000 0.936 22 0.936 0.942 0.969 0.966 0.942 0.927 0.926 0.998 0.990 0.996 0.996 1.000 0.938 23 0.934 0.946 0.972 0.969 0.943 0.930 0.926 0.997 0.991 0.996 0.995 1.000 0.942 24 0.934 0.946 0.971 0.969 0.943 0.933 0.926 0.997 0.990 0.997 0.995 1.000 0.947 25 0.936 0.947 0.970 0.969 0.943 0.933 0.926 0.997 0.990 0.997 0.996 1.000 0.952 26 0.938 0.940 0.974 0.985 0.966 0.968 0.974 0.998 0.990 0.998 0.995 1.000 0.961 27 0.943 0.941 0.976 0.991 0.985 0.994 0.999 0.998 0.992 0.998 0.995 1.000 0.968 28 0.972 0.973 0.976 0.995 0.992 0.994 0.999 0.999 0.992 0.999 0.996 1.000 0.975 29 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.997 30 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 59
How to account for the observed features a two-state Markov chain 1. Choose a simple criterion to determine whether the swaption matrix is currently in the normal or excited state 2. The instantaneous volatility function for each forward rate can be described by either of these two functional forms: i n t,t i a t n b t n T t exp c t n T t d t n i x t,t i a tx b tx T t exp c tx T t d t x 60
The Two Basis States 0.5 0.45 0.4 0.35 0.3 Normal Excited 0.25 0.2 0.15 0 2 4 6 8 10 61
Two-State Transition Matrix between a Normal and an Excited Regime nn nx xn xx 62
Lower Lower Lower a_lower -0.04-0.04-0.04 b_lower 0.64 0.64 0.2 c_lower 1.2 1.2 1.2 d_lower 0.14 0.14 0.14 a_upper 0.41 0.41 0.41 b_upper -0.1 0.1-0.03 c_upper 1 1.1 0.7 d_upper 0.15 0.15 0.15 Probability Up Jump 0.750 0.750 0.750 Probability Down Jump 1.250 1.250 0.200 Sqr(ScalingUpper) 1.113342103 1.05675538 0.861562865 Sqr(ScalingNormal) 0.468818337 0.48352497 0.688019951 63
A Simple Fit 0.5 0.4 0.45 t = 1 year 0.35 t = 2 year 0.4 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.1 0. 02 0.03 0.04 0.05 0.06 0.07 0.08 0.35 0.35 t = 4 years t = 5 years 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.02 0.04 0.06 0.08 0.1 0.12 0.1 0. 02 0.04 0.06 0.08 0.1 0.12 0.3 0.3 t = 12 years t = 14 years 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.02 0.04 0.06 0.08 0.1 0.12 0.1 0. 02 0.04 0.06 0.08 0.1 0.12 64
Making it computationally feasible Conditional on a particular path having been realized, we simply are in a Black (displaced-diffusion) / LMM situation. To calculate caplet prices, average the analytical prices associated with a large number of volatility realizations. The analytical prices are obtained by using the (displaced-diffusion) Black formula with the appropriate conditional root-mean-square volatility. The appropriate conditional root-mean-squared volatility is obtained by integrating analytically over the segments of the two volatility states. Same approach can be used to calibrate to European swaption prices. 65
The variance integral for a particular realization of the Markov process 0 T u 2 du n i 1 0 t 1 u 2 t du 2 u 2 du... t 1 T t n 1 u 2 du 66
A particular path with state transitions 0.5 0.45 0.4 0.35 0.3 0.25 Normal Excited Mixed 0.2 0.15 0 2 4 6 8 10 67
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