The Economic Determinants of Interest Rate Option Smiles prachi deuskar

 

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 

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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.

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 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.

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 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.

 

                                                           

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 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. 

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 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. 

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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.

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 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.

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 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.

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 Furthermore, 

                                                           

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 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. 

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 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). 

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 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 

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