values of the short term interest rate and the slope of the yield curve could improve the calibration
of these models. This is intuitive if the future distribution of interest rates is not fully captured by
today’s yield curve, but, in addition, depends on the past values of interest rates. Our results also
have implications for the modeling of credit derivatives, whose payoffs depend on the default
spread, since we find that the shape of the smile can predict the default spread.
The structure of our paper is as follows. Section 2 describes the data set and presents summary
statistics. Section 3 presents the empirical patterns of the volatility smile that we observe in the
data. In section 4, we examine the impact of several macro-economic variables on these patterns.
Section 5 presents the results of the multivariate vector autoregression and the Granger-causality
tests. Section 6 concludes with a summary of the main results and directions for future research.
2. Data
The data for this study consist of prices of euro (€) caps and floors over the 29-month period,
January 1999 to May 2001, obtained from WestLB (Westdeutsche Landesbank Girozentrale)
Global Derivatives and Fixed Income Group. These are daily bid and offer quotes over 591
trading days for nine maturities (2 years to 10 years, in annual increments) across twelve different
strike rates ranging from 2% to 8%. This is an extensive set with price quotes for caps and floors
every day, reflecting the maturity-strike combinations that elicit market interest on that day.
WestLB is one of the dealers who subscribe to the interest rate option valuation service from
Totem. Totem is the leading industry source for asset valuation data and services, supporting
independent price verification and risk management in the global financial markets. Most leading
derivative dealers subscribe to their service. As part of this service, Totem collects data for the
entire range of caplets and floorlets across a series of maturities from these dealers. They
aggregate this information and return the consensus values back to the dealers who contribute
data to the service. The market consensus values supplied to the dealers include the underlying
5
term structure data, caplet and floorlet prices, as well as the prices and implied volatilities of the
reconstituted caps and floors across strikes and maturities. Hence, the prices quoted by dealers
such as WestLB, who are a part of this service, reflect the market-wide consensus information
about these products. This is especially true for plain-vanilla caps and floors, which are very
high-volume products with standardized structures, that are also used by dealers to calibrate their
models for pricing and hedging exotic derivatives. Therefore, it is extremely unlikely that any
large dealer, especially one that uses a market data integrator such as Totem, would deviate
systematically from market consensus prices for these vanilla products.
6
Our discussions with
market participants confirm that the prices quoted by different dealers (especially those that
subscribe to Totem) for vanilla caps and floors are generally similar.
Interest rate caps and floors are portfolios of European interest rate options on the 6-month
Euribor with a 6 monthly reset frequency.
7
In addition to the options data, we also collected data
on euro (€) swap rates and the daily term structure of euro interest rates curve from the same
source. These are the key inputs necessary for checking cap-floor parity, as well as for conducting
our subsequent empirical tests. We calculate the “moneyness” of the options by estimating the
Log Moneyness Ratio (LMR) for each cap/floor. The LMR is defined as the logarithm of the ratio
of the par swap rate to the strike rate of the option. Since the relevant swap rate changes every
day, the LMR of options at the same strike rate and maturity also changes each day.
6
The euro OTC interest rate derivatives market is extremely competitive, especially for plain-vanilla
contracts like caps and floors. The BIS estimates the Herfindahl index (sum of squares of market shares of
all participants) for euro interest rate options (which includes exotic options) at about 500-600 during the
period from 1999 to 2004, which is even lower than that for USD interest rate options (around 1,000).
Since a lower value of this index (away from the maximum possible value of 10,000) indicates a more
competitive market, it is safe to rely on option quotes from a top European derivatives dealer (reflecting the
best market consensus information available with them) like WestLB during our sample period. Thus, any
dealer-specific effects on price quotes are likely to be small and unsystematic across the over 30,000 bid
and ask price quotes each that are used in this paper.
7
For the details of the contract structure for caps and floors, please refer to Longstaff et al (2001) for the
US dollar market and to Deuskar, Gupta and Subrahmanyam (2007) for the Euro market.
6
We pool the data on caps and floors to obtain a wider range of strike rates, on both sides of the at-
the-money strike rate. Before doing so, we check for put-call parity between caps, floors and
swaps, using both bid and ask prices. We find that, on average, put-call parity holds in our
dataset, although there are deviations from parity for some individual observations.
8
These parity
computations are a consistency check, as well, to assure us about the integrity of our dataset.
3.
Shapes of the Volatility Smile in Interest Rate Option Markets
We use implied volatilities from the Black-BGM (Black (1976) and Brace, Gatarek and Musiela
(1997) (BGM)) model, throughout the analysis. We do so for two reasons. First, although there
may be an alternative complex model that explains at least part of the smile/skew or the term
structure of volatility, it is necessary to obtain an initial sense of the empirical regularities using
the standard model. In other words, we need to document the characteristics of the smile before
attempting to model it formally.
9
Furthermore, the evidence in the equity option markets suggests
that even such complex models may not explain the volatility smile adequately, without
considering the effect of market frictions. Second, Black-BGM implied volatilities are the
common market standard for dealer quotations for interest rate option prices.
We document volatility smiles in euro interest rate caps and floors across a range of maturities
using the implied “flat” volatilities of caps and floors over our sample period. The flat volatility is
a volatility number common to all the caplets (floorlets) in a cap (floor), which sets the sum of
their prices equal to the quoted price for the cap (floor). Thus the flat volatility is a weighted
average of the implied volatility of individual options included in a cap or a floor.
10
Furthermore,
8
Many of these deviations may not be actual violations from parity, given the difficulty in carrying out the
arbitrage using “off-market” swaps. Since the bid and ask prices of “off-market” swaps are not available,
we cannot examine which of these observations is a real violation of put-call parity.
9
The use of implied volatilities from the Black-Scholes model is in line with all prior studies in the
literature, including Bollen and Whaley (2004).
10
Our implied volatility estimation is likely to have much smaller errors than those generally encountered
in equity options (see, for example, Canina and Figlewski (1993)). We pool the data for caps and floors,
which reduces any error due to mis-estimation of the underlying yield curve. The options we consider have
7
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