option dealers. It could affect the prices of away-from-the-money options more than the prices of
ATM options, thus affecting the shape of the smile.
We also include a measure of the at-the-money relative bid-ask spreads of these options. The
objective of including this variable is to directly control for the explicit liquidity of these options,
while examining the relationship of the other economic variables to the volatility smile. The
relative bid-ask spreads of ATM options capture the general level of liquidity in the market.
The results from this regression analysis are presented in Table 2. The curvature of the smile is
positively and significantly related to the 6-month interest rate, with the effect being insignificant
for long maturity options. When interest rates are high, the away-from-the-money options,
especially the ones with shorter maturities, are priced relatively higher than during times when
interest rates are low. On the other hand, the curvature of the smile is negatively related to the
slope of the term structure; interestingly, this effect is significant only for the longer maturity
options. It appears that the volatility smiles in this market have more curvature when the term
structure is relatively flat. These results are consistent for the bid- as well as the ask-side
quotations.
The results also show that the degree of curvature is negatively related to the volatility of at-the-
money options, although this effect is significant mostly for short/medium maturity options.
Therefore, highly volatile periods tend to be associated with a lower curvature of the smile, which
is consistent with the evidence in the equity options literature (Pena, Rubio, and Serna (1999)).
These results suggest a stochastic volatility framework with the volatility itself exhibiting mean
reversion. In such a model, high volatility periods are likely to be followed by lower volatility
periods, which would result in a shallow smile when volatility is high. We also find weak
evidence of the curvature of the smile being positively related to the liquidity costs in the market,
but this effect is significant only for long maturity options on the ask-side. This is understandable,
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since higher liquidity costs i.e. higher costs of continuously hedging the options positions would
of more concern in case of away from the money options and longer maturities. Therefore,
especially for longer maturity options, it may be important to account for liquidity effects while
modeling the volatility smile.
The slope of the volatility smile (RR) exhibits somewhat different relationships to the
contemporaneous determinants examined in this section. When the short-term interest rate is
high, the RR appears to be more negative, especially for longer maturity options. Since the RR is
the difference between ScaledIVs at +0.25 LMR and -0.25 LMR, it is important to understand the
effects separately for caps and floors. A negative (positive) LMR refers to out-of-the-money caps
(floors). A negative relationship between 6-month rate and RR implies that when interest rates
increase (decrease), out-of-the-money caps (floors) become disproportionately expensive. These
results are quite intuitive. It is possible that the demand for out-of-the-money caps (floors) is
higher when interest rates go up (down). Then, consistent with the findings of Bollen and Whaley
(2004) and Garleanu, Pedersen and Poteshman (2006), demand pressure may affect the prices of
interest rate options at some strikes, thereby affecting the shape of the volatility smile. Similarly,
when the term structure becomes more steeply upward sloping, the smile becomes more negative.
An upward-sloping yield curve is a signal that interest rates will increase in the future, thereby
leading to higher demand for out-of-the-money caps, which would make the volatility smile more
negative. An alternate way of thinking about this effect is that the slope of the yield curve
captures the skew of the distribution of future interest rates, thus affecting the slope of the smile.
Finally, we find some evidence that the slope of the smile curve is related to the default spread.
However, this relationship is not consistent across all maturities. Perhaps there is a relation
between RR and the leads or lags of the default spread. The nature of such dynamic relationships
between the economic variables and the volatility smile is what we explore in the next section.
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5.
Multivariate Vector Autoregression
In the previous section, we show that economic variables are significantly related to the shape of
the contemporaneous smile. In this section, we examine the relationship between the lagged
values of economic variables and the shape of the smile, and vice-versa. We estimate a six-
equation, multivariate, vector autoregression separately for the butterfly spread and the risk
reversal, each of which includes the five economic and liquidity variables (ATM volatility, 6-
month rate, the slope of the term structure, the default spread, and the ATM bid-ask spreads).
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This framework can provide useful information on the linkages between the economic variables
and the volatility smile in a dynamic, predictive sense. We choose the appropriate number of lags
for the multivariate VAR estimation in each case, using the Akaike information criterion (AIC).
For most option maturities, this estimation results in two or three lags, with the maximum number
of lags in any system being five. We estimate 36 VAR models (9 option maturities each, for the
bid and ask sides, separately for BS and RR) that provide a comprehensive description of the
time-series movements in the shape of the smile and the economic and liquidity variables.
We first examine the cross-correlations of the innovations obtained from the VAR system.
Unexpected shocks to any of the economic variables may be related to the unexpected
fluctuations in the shape of the volatility smile. These correlations are presented in Table 3. The
most striking relationship noticed from the table is the negative correlation between the shocks to
the slope of the term structure and the shocks to the curvature and slope of the volatility smile,
which is consistent with our results in the previous section. It appears that unexpected twists in
the term structure, which may be proxies for unexpected changes in the higher moments of the
risk neutral distribution of interest rates, are related to unexpected changes in the shape of the
volatility smile curve. To a lesser degree, we find that the shocks to the 6-month interest rate are
positively correlated with the shocks to the shape of the smile, especially to the butterfly spread.
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