The Economic Determinants of Interest Rate Option Smiles prachi deuskar

 

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 

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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 

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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 

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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 

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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