We report simple regressions and Granger causality tests in order to understand the pattern of implied volatilities across exercise prices. We employ all calls and puts transacted between 16:00 and 16:45 on the Spanish IBEX-35 index from January 1994 to April 1996. Transaction costs, proxied by the bid±ask spread, seem to be a key determinant of the curvature of the volatility smile. Moreover, time to expiration, the uncertainty associated with the market and the relative market momentum are also important variables in explaining the smile. Ó

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Previous studies have explored the seasonal behaviour of commodity prices as a deterministic factor. This paper goes further by proposing a general (n+2m)-factor model for the stochastic behaviour of commodity prices, which nests the deterministic seasonal model by Sorensen (2002). We consider seasonality as a stochastic factor, with n non-seasonal and m seasonal factors. The nonseasonal factors are as defined in Schwartz (1997), Schwartz and Smith (2000) and Cortazar and Schwartz (2003). The seasonal factors are trigonometric components generated by stochastic processes. The model has been applied to the Henry Hub natural gas futures contracts listed by NYMEX. We find that models allowing for stochastic seasonality outperform standard models with deterministic seasonality. We obtain similar results with other energy commodities. Moreover, we find that stochastic seasonality implies that the volatility of futures returns follows a seasonal pattern. This result has important implications in terms of option pricing. This paper is the sole responsibility of its authors. The views represented here do not necessarily reflect those of the Banco de España. We thank John Doukas (the editor), an anonymous referee,

Given the evidence provided by Longstaff (1995), and PenÄa, Rubio and Serna (1999) a serious candidate to explain the pronounced pattern of volatility estimates across exercise prices might be related to liquidity costs. Using all calls and puts transacted between 16:00 and 16:45 on the Spanish IBEX-35 index futures from January 1994 to October 1998 we extend previous papers to study the influence of liquidity costs, as proxied by the relative bid-ask spread, on the pricing of options. Surprisingly, alternative parametric option pricing models incorporating the bid-ask spread seem to perform poorly relative to Black-Scholes.

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