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.
11,12
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.
11
The scatter plots have not been presented in the paper to save space, and are available from the authors.
12
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.
8
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.
13
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
13
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
9