The Economic Determinants of Interest Rate Option Smiles prachi deuskar

 

we scale the implied volatility of the cap/floor by the at-the-money volatility of the mid-price 

(average of bid and ask price) of the cap of the same maturity (and call it Scaled IV). This scaling 

accounts for the effect of changes in the level of implied volatilities over time. Scatter plots of the 

Scaled IV against the LMR for interest rate options in this market indicate a significant smile 

curve that is approximately quadratic and steeper for shorter maturity options than longer 

maturity ones.

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3.1 

Functional forms for implied volatility smiles 

Next, we estimate various functional forms for volatility smiles using pooled time-series and 

cross-sectional ordinary least squares regressions, in order to understand the overall form of the 

volatility smile over our entire sample period. The most common functional forms for the 

volatility smile used in the literature are quadratic functions of either moneyness or the logarithm 

of moneyness. In addition, the scatter plots of Scaled IV against LMR suggest a quadratic form. 

Therefore, we estimate the following functional form: 

2

*

3

*

2

1

 

LMR

c

LMR

c

c

IV

Scaled

+

+

=

 

   (1) 

 

We also estimate an asymmetric quadratic functional form, where the slope is allowed to differ 

for in-the-money and out-of-the-money options, with similar results. (Polynomials of higher order 

turn out to be statistically insignificant). In addition, we estimate the volatility smiles on the bid-

                                                                                                                                                                             

much longer maturities (the shortest cap/floor is 2 year maturity), which reduces this potential error further. 

In addition, for most of our empirical tests, we do not include deep ITM or deep OTM options, where 

estimation errors are likely to be larger. Furthermore, since we consider the implied flat volatilities, the 

errors are further reduced due to the implicit “averaging” in this computation. 

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 The scatter plots have not been presented in the paper to save space, and are available from the authors. 

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 In addition, we analyze the principal components of the changes in the Black volatility surface (across 

strike rates and maturities) for caps and floors. If away-from-the-money option prices were just mechanical 

transformations of ATM option prices, we would observe a very high proportion of the variation in these 

implied volatilities being explained by just one principal component. However, we find four significant 

principal components on the ask-side and two on the bid-side, indicating that the implied volatilities for 

away-from-the-money options are not just being adjusted by the dealer using a mechanical rule anchored 

by the at-the-money volatilities.  

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side and the ask-side separately. Using the mid-point of the bid-ask prices may not always 

accurately display the true smile in the implied volatility functions, given that bid-ask spreads 

differ across strike rates. 

 

Figure 1 presents the plots of fitted implied volatility functions based on specification (1) for caps 

and floors separately for different maturities. These plots clearly show a smile curve for these 

options and display some interesting patterns. Caps always display a smile, which flattens as the 

maturity of the cap increases. In-the-money caps (LMR>0) have a significantly steeper smile than 

out-of-the-money caps. More interestingly, the ask-side of the smile is steeper than the bid-side, 

the difference being significantly larger for in-the-money caps. Floors display somewhat similar 

patterns. The smile gets flatter as the maturity of the floor increases. In-the-money floors 

(LMR<0) exhibit a significantly steeper smile, especially for short-term floors. Long-term floors 

display almost a “smirk”, instead of a smile. As with caps, the smile curve for floors is steeper on 

the ask-side, as compared to that on the bid-side.  

In Table 1, we report the results for caps and floors pooled together for specification (1). The 

regression coefficients in almost all the maturities are highly significant. In addition, the quadratic 

functional form explains a high proportion of the variability in the scaled implied volatilities.

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The coefficient of the curvature of the smile decreases with the maturity of the options, indicating 

that as the maturity of these options increases, the smile flattens, and eventually converts into a 

“smirk” when we reach the 10-year maturity. In addition, we re-estimate these specifications 

using a volatility and maturity adjusted moneyness measure (log(Swap Rate/Strike Rate)/(ATM 

Volatility*(Maturity)

1/2

)) instead of LMR), similar to the one used in Carr and Wu (2003a, 2003b) 

and Li and Pearson (2004). We still observe similar smile patterns, with a flattening of the smile 

curve with maturity, consistent with the findings of Backus, Foresi, and Wu (1997) for currency 

                                                           

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 We also conducted the same exercise with spot volatilities i.e. using inferred prices of individual caplets 

and floorlets, obtained by bootstrapping from the flat volatilities of caps and floors. Model (1) fits well 

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