The Economic Determinants of Interest Rate Option Smiles prachi deuskar

 

options, where they find that the smile flattens with maturity even using the adjusted moneyness 

measure.

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3.2 

Time variation in volatility smiles 

In Figure 2, we present the surface plots for the fitted values of the scaled implied volatilities 

against moneyness represented by the LMR using specification (1) to fit a smile every day.

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 The 

shapes of these surface plots show similar trends – the 2-year maturity-contracts display a large 

curvature in the volatility smile, while the smile flattens out and turns into more of a skew as we 

move towards the longer maturity contracts, especially at the 10-year maturity. More importantly, 

both the curvature and the slope of the volatility smile show significant time-variation, sometimes 

even on a daily basis. The changes in the curvature and slope over time are more pronounced for 

the 2-year maturity contracts, although they are also perceptible for the longer maturity contracts.  

Figure 2 also presents the surface plot of the euro spot interest rates for maturities from one to ten 

years, which also shows significant time variation in level and slope over our sample period.  

Based on these figures, the natural question to ask is whether on a time-series basis, certain 

economic variables exhibit a significant relationship with the implied volatility smile patterns. In 

order to examine this question, we first need to define appropriate measures of the asymmetry and 

curvature of the smile curve each day. We can then determine empirical proxies for these 

attributes and estimate them using the volatility smile curve, each day. The measure of the 

asymmetry of the implied volatility curve, widely used by practitioners, is the “risk reversal,” 

                                                                                                                                                                             

there as well. Those results are not presented here to conserve space. 

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 We also plotted the scaled and unscaled implied volatilities respectively against the volatility and 

maturity adjusted moneyness measure. (These plots are not included in the paper to conserve space, and are 

available from the authors, upon request.)  Longer maturity caps and floors still have a flatter smile, so the 

transformation of the moneyness scale does not appear to change the pattern of the smiles across maturities. 

In addition, these scatter plots are very similar to the ones that use LMR as the moneyness measure. 

Therefore, in the Euro interest rate options markets, the shape of the smile appears to be the same 

regardless of the measure of moneyness, simple or adjusted. 

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 These plots are presented for representative maturities of 2-, 5-, and 10-years, since the plots for the other 

maturities are similar. In addition, we present the fitted volatility smiles over the LMR range from -0.3 to 

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which is the difference in the implied volatility of the in-the-money and out-of-the-money options 

(roughly equally above and below the at-the-money strike rate). The measure of the curvature is 

the “butterfly spread,” which is the difference between the average of the implied volatilities of 

two away-from-the-money volatilities and the at-the-money volatility.

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 The advantage of using 

these empirical measures is that they explicitly capture the slope and the curvature of the smile 

curve. Therefore, they can be interpreted as proxies for the skewness and kurtosis of the risk-

neutral distribution of interest rates. 

We fit a quadratic function of the LMR to the scaled implied volatilities each day and use the 

fitted values to construct the risk reversal (RR) and butterfly spread (BS), defined as follows:  

 

 

  (2) 

 

(

)

ATM

LMR

LMR

LMR

LMR

IV

Scaled

IV

Scaled

IV

Scaled

BS

IV

Scaled

IV

Scaled

RR

 

2

/

 

 

 

 

 

 

25

.

0

25

.

0

25

.

0

25

.

0

−

+

=

−

=

−

+

−

+

The butterfly spread captures the average scaled implied volatility at 0.25 LMR away-from-the-

money, on either side of 0. It is essentially a linear transformation of the curvature coefficient 

from the quadratic function. Hence, it is our proxy for the curvature of the smile. The risk reversal 

represents the difference between the implied volatility of in-the-money options and out-of-the-

money options. It is a linear transformation of the slope coefficient from the quadratic function. 

Thus, it is a proxy for the asymmetry in the slope of the smile.

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+0.3, which is the subset of strikes over which we have enough observations to estimate specification (1) 

over a substantial number of days in our dataset.  

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  These structures involve option-spread positions and are traded in the OTC interest rate and currency 

markets as explicit contracts.  These prices are often used in the industry for calibrating interest rate option 

models. See, for example, Wystup (2003). 

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Time-series plots of the risk reversal and the butterfly spread over our sample period show that both the 

slope and the curvature of the smile change almost on a daily basis, with the slope being more volatile than 

the curvature. The fluctuations in the slope of the smile are higher in the second half of our sample period, 

which is also one where interest rates increased. These variables could potentially be linked with each other 

through lead/lag relationships, which is one of the central issues that we examine in this paper. These plots 

have not been presented in the paper to conserve space, and are available from the authors. 

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