Table 4
Granger causality tests
This table presents results for the Granger causality tests based on the multivariate vector autoregression for
six variables - the level of volatility of at-the-money interest rate options (ATM Vol), the spot 6-month Euribor
(6 m rate), the slope of the term structure (5 yr rate - 6 m rate), the 6-month Treasury-Euribor spread (Default
Spread), the scaled ATM bid-ask spreads (ATM BA spread) and butterfly spread (BS) or risk reversal (RR)
separately for ask and bid sides for the period Jan 99 - May 01, based on data obtained from WestLB Global
Derivatives and Fixed Income Group and DataStream . The p-values for rejecting the null hypothesis of “No
Granger Causality” are given below. ** and * represent p-values less than or equal to 5% and 10%
respectively. The results are presented for three representative maturities - 2-year, 5-year, and 10-year.
Panel A: Null Hypothesis – presented variables do not individually Granger cause the butterfly spread (BS) /
risk reversal (RR) on the ask / bid side
Ask
Bid
ATM Vol. 6 m Rate
5 yr rate –
6 m Rate
Default
Spread
ATM BA
Spread
ATM Vol. 6 m Rate
5 yr rate –
6 m Rate
Default
Spread
ATM BA
Spread
BS
2-year 0.61 0.05** 0.98 0.73 0.65
0.43 0.05** 0.09* 0.44 0.97
5-year 0.01** 0.00** 0.00** 0.57 0.04**
0.38 0.22 0.06*
0.07*
0.74
10-year 0.29 0.33 0.12 0.04**
0.71
0.00** 0.70 0.00** 0.40 0.75
RR
2-year 0.58 0.92 0.74 0.81 0.16
0.89 0.18 0.81 0.46 0.11
5-year
0.00**
0.00** 0.30 0.47 0.22
0.26 0.00**
0.00** 0.06* 0.73
10-year 0.01** 0.09* 0.00** 0.69 0.42
0.85 0.13
0.00**
0.50
0.00**
Panel B: Null Hypothesis – Butterfly spread (BS) / risk reversal (RR) on the ask / bid side do not Granger
cause each of the presented variables
Ask
Bid
ATM Vol. 6 m Rate
5 yr rate –
6 m Rate
Default
Spread
ATM BA
Spread
ATM Vol. 6 m Rate
5 yr rate –
6 m Rate
Default
Spread
ATM BA
Spread
BS
2-year 0.58 0.28 0.69 0.15 0.80
0.92 0.12 0.34 0.27 0.72
5-year 0.58
0.28 0.01** 0.16 0.60
0.95 0.47 0.81 0.72 0.60
10-year 0.00**
0.77
0.92 0.01**
0.21
0.31 0.15 0.38 0.84
0.00**
RR
2-year 0.83 0.78 0.45 0.47 0.33
0.39 0.56 0.79
0.02**
0.13
5-year 0.14
0.39 0.02** 0.28 0.00**
0.67 0.04** 0.19 0.10 0.08*
10-year 0.81
0.22
0.16 0.00**
0.00**
0.19 0.97 0.37
0.02**
0.00**
28
Table 5
Variance decomposition
This table presents the variance decompositions (%) computed from the multivariate vector autoregression for
six variables - the level of volatility of at-the-money interest rate options (ATM Vol), the spot 6-month Euribor
(6 m rate), the slope of the term structure (5 yr rate - 6 m rate), the 6-month Treasury-Euribor spread (Default
Spread), the scaled ATM bid-ask spreads (ATM BA spread) and butterfly spread (BS) or risk reversal (RR)
separately for ask and bid sides for the period Jan 99 - May 01, based on data obtained from WestLB Global
Derivatives and Fixed Income Group and DataStream. The VAR is ordered as 6 m Rate, 5 yr rate - 6 m Rate,
Default Spread, ATM BA Spread, BS, ATM Vol in case of butterfly spread and as 6 m Rate, RR, 5 yr rate - 6
m Rate, Default Spread, ATM Vol, and ATM BA Spread in case of risk reversal. The results are presented for
three representative maturities - 2-year, 5-year, and 10-year.
Ask
Bid
Maturity
Forecast
Horizon
(Days)
Forecast
Standard
Error
ATM
Vol.
6 m
Rate
5 yr rate
– 6 m
Rate
Default
Spread
ATM
BA
Spread
BS /
RR
Forecast
Standard
Error
ATM
Vol.
6 m
Rate
5 yr rate
– 6 m
Rate
Default
Spread
ATM
BA
Spread
BS /
RR
Panel A: Variance decomposition of butterfly spread
2-year 1
0.4 0.0 2.1 0.0
0.7 0.0 97.2
0.4 0.0 8.2
0.0
1.2
0.3 90.3
10 1.1 1.7
4.4 0.1 0.3 0.2
93.3
1.1 1.2
11.8
6.2 3.9
0.2
76.7
5-year 1
0.2 0.0 2.8 0.0
0.0 2.2 95.5
0.2 0.0 0.2
0.9
0.8
0.3 97.9
10
0.6 3.5 7.6 3.0
2.3 1.9 81.8
0.6 0.7 0.5
1.2
5.5
0.4 91.7
10-year 1
0.1 0.0 2.0 0.9
0.1 1.6 95.4
0.1 0.0 1.5
0.1
0.1
2.2 96.1
10 0.3 4.5
1.1 0.2 5.6 0.8
87.8
0.4 11.9
1.5 1.1 0.4
2.2
82.9
Panel B: Variance decomposition of risk reversal
2-year 1
0.4 0.0 6.8 0.0
0.0 0.0 93.2
0.4 0.0 0.6
0.0
0.0
0.0 99.4
10
1.2 0.0 6.3 0.3
0.5 4.0 88.9
1.1 0.0 3.3
0.3
0.9
4.0 91.4
5-year 1
0.2 0.0 0.2 0.0
0.0 0.0 99.8
0.2 0.0 0.9
0.0
0.0
0.0 99.1
10
0.6 2.3 5.8 0.6
0.5 1.1 89.7
0.6 1.0 5.9
1.7
3.2
0.1 88.2
10-year 1
0.2 0.0 1.6 0.0
0.0 0.0 98.4
0.7 0.0 0.5
0.0
0.0
0.0 99.5
10
0.4 2.5 3.3 2.6
0.4 1.7 89.5
0.4 0.6 3.4
1.5
0.1
8.2 86.2
Panel C: Variance decomposition of default spread
2-year 1 0.4 0.0
9.6 0.2 88.9 0.0
1.4
0.4 0.0
16.5
0.0 79.0
0.0 4.6
10
1.2
2.6 17.7 2.1
74.4 0.6 2.7
1.1
1.0 27.7
0.3
57.3
1.2 12.5
5-year
1
0.2
0.0 12.4 0.3
85.4 0.0 2.0
0.2
0.0 12.6
0.0
86.8
0.0 0.6
10
0.6
1.4 19.7 0.8
73.5 0.3 4.5
0.6
1.5 20.0
0.4
73.0
0.3 4.7
10-year 1
0.2
0.0 12.6 0.0
87.3 0.0 0.2
0.2
0.0 13.0
0.1
85.2
0.0 1.7
10
0.4
0.3 17.5 0.5
73.5 1.5 6.7
0.4
0.4 17.7
0.6
68.8
4.5 8.1
29
Figure 1
Functional forms of implied volatility smiles in interest rate caps and floors
This figure presents the fitted smile functions for the bid and ask implied flat volatilities of euro interest rate
caps and floors separately, across different maturities. The horizontal axis in the plots corresponds to the
logarithm of the moneyness ratio (LMR), defined as the ratio of the par swap rate to the strike rate of the
option. The vertical axis in the plots corresponds to the implied flat volatility of the bid and ask prices of the
option, scaled by the at-the-money volatility for the option of similar maturity (Scaled IV). The fitted values
are calculated using a quadratic function of LMR as in specification (1). The plots are three representative
maturities - 2-year, 5-year, and 10-year for the period, Jan 99 - May 01, for various maturities, based on data
obtained from WestLB Global Derivatives and Fixed Income Group.
2 year caps
0
1
2
3
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
LMR
S
cal
ed
I
V
5 year caps
0
1
2
3
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
LMR
S
cal
ed
I
V
10 year caps
0
1
2
3
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
LMR
S
cal
ed
I
V
Ask
Bid
Ask
Bid
Ask
Bid
2 year floors
0
1
2
3
-0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
1.2
LMR
S
cal
ed
I
V
5 year floors
0
1
2
3
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
LMR
S
cal
ed
I
V
10 year floors
0
1
2
3
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
LMR
Sc
al
e
d
I
V
Ask
Bid
Ask
Bid
Ask
Bid
30
Figure 2
Time variation in volatility smiles and the Euro term structure
This figure presents surface plots showing the time variation in the implied flat volatilities of euro interest rate
caps and floors as well as the term structure of euro interest rates over the period Jan 99 - May 01. In figures A,
B, and C, The horizontal axes correspond to the logarithm of the moneyness ratio, LMR, (defined as the ratio
of the par swap rate to the strike rate of the option), and time. The vertical axis corresponds to the implied
volatility of the mid-price (average of bid and ask price) of the option scaled by the at-the-money volatility for
the option of similar maturity (Scaled IV). The values presented are the fitted values from a quadratic function
of LMR as specified in equation (1) estimated every day. Figure D depicts the Euro spot rate surface by
maturity (in years) over time (daily). The vertical axis corresponds to the spot rates. The horizontal axes
correspond to the maturity of the spot rate and time, based on data obtained from WestLB Global Derivatives
and Fixed Income Group.
A
B
C
D
31
Figure 3
Impulse responses
This figure presents impulse responses computed from the multivariate vector autoregression for six variables -
the level of volatility of at-the-money interest rate options (ATM Vol), the spot 6-month Euribor (6 m rate), the
slope of the term structure (5 yr rate - 6 m rate), the 6-month Treasury-Euribor spread (Default Spread), the
scaled ATM bid-ask spreads (ATM BA spread) and butterfly spread (BS) or risk reversal (RR) separately for
ask and bid sides for the period Jan 99 - May 01, based on data obtained from WestLB Global Derivatives and
Fixed Income Group and DataStream. The figure shows response for three representative maturities - 2-year,
5-year, and 10-year. VARs are ordered as follows: for BS 6 m Rate, 5 yr rate - 6 m Rate, Default Spread, ATM
BA Spread, BS, and ATM Vol and for RR 6 m Rate, RR, 5 yr rate - 6 m Rate, Default Spread, ATM Vol, and
ATM BA Spread. The solid line represents the ask side while the dashed line represents the bid side.
Panel A: Response of the butterfly spread to the 6-month interest rate
-.014
-.012
-.010
-.008
-.006
-.004
-.002
1
2
3
4
5
6
7
8
9
10
2-yr
-.0008
-.0004
.0000
.0004
.0008
.0012
1
2
3
4
5
6
7
8
9
10
5-yr
.0000
.0001
.0002
.0003
.0004
.0005
.0006
.0007
1
2
3
4
5
6
7
8
9
10
10-yr
Panel B: Response of the risk reversal to the 6-month interest rate
.000
.005
.010
.015
.020
.025
.030
1
2
3
4
5
6
7
8
9
10
2-yr
-.0045
-.0040
-.0035
-.0030
-.0025
-.0020
-.0015
-.0010
1
2
3
4
5
6
7
8
9
10
5-yr
-.0038
-.0036
-.0034
-.0032
-.0030
-.0028
-.0026
-.0024
1
2
3
4
5
6
7
8
9
10
10-yr
Panel C: Response of the default spread to the risk reversal
0.2
0.4
0.6
0.8
1.0
1.2
1
2
3
4
5
6
7
8
9
10
2-yr
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1
2
3
4
5
6
7
8
9
10
10-yr
5-yr
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
1
2
3
4
5
6
7
8
9
10
32
Document Outline - Panel A: Response of the butterfly spread to the 6-month interest rate
- Panel B: Response of the risk reversal to the 6-month interest rate
- Panel C: Response of the default spread to the risk reversal
Dostları ilə paylaş: |