ارز، هم انباشتگی جزء به جزء و رابطه نوسانات ضمنی تحقق یافته

Almost all relevant literature has characterized implied volatility as a biased predictor of realized volatility. In this paper we provide new time series techniques to investigate the validity of this finding in several foreign exchange options markets, including the Euro market. First, we develop a new fractional cointegration test that is shown to be robust to both stationary and non-stationary regions. Second, we employ both intra-day and daily data to measure realized volatility in order to assess the relevance of data frequency in resolving the bias. Third, we use data on implied volatility traded on the market. In contrast to previous studies, we show that the frequency of data used for measuring realized volatility within a fractionally cointegrating framework is important for the results of unbiasedness tests. Significantly, for many popular exchange rates, the use of intra-day rather than daily data affects the emergence of a different bias, as the possibility of a fractionally integrated risk premium admits itself!

Market efficiency in options markets is typically examined by estimating the following regression: equation(1) View the MathML sourceσt+τRV=α+βσtIV+ut+τ, Turn MathJax on where View the MathML sourceσtIV is the implied volatility (IV) over a period of time τ and View the MathML sourceσt+τRV is the realized volatility (RV). Unbiasedness holds in (1) when α = 0, β = 1 and ut+τ is serially uncorrelated. Of course, unbiasedness is a sufficient condition for market efficiency but is not necessary in the presence of either a constant or a time-varying option market risk premium. Conventional tests in the previous literature have generally led to the conclusion that IV is a biased forecast of RV in the sense that the slope parameter in (1) is not equal to unity (see, inter alia, Christensen and Prabhala, 1998 and Poteshman, 2000). This conclusion is found to be robust across a variety of asset markets (see Neely, 2009) and has thus provided the motivation for several attempted explanations of this common finding. Popular suggestions include computing RV with low frequency data ( Poteshman, 2000); that the standard estimation with overlapping observations produces inconsistent parameter estimates ( Dunis and Keller, 1995 and Christensen et al., 2001); and that volatility risk is not priced ( Poteshman, 2000 and Chernov, 2007). However, Neely (2009) evaluates these possible solutions and finds that the bias in IV is not removed. Of course, the optimality of the estimation procedure applied to (1) depends critically on the order of integration of the component variables. Given the acknowledged persistence in individual volatility series, the recent literature suggests they are well represented as fractionally integrated processes (see, inter alia, Andersen et al., 2001a and Andersen et al., 2001b). Notably Bandi and Perron, 2006, Christensen and Nielson, 2006 and Nielsen, 2007 have begun to examine the consequences of this approach for regression (1). Employing stock market data, Bandi and Perron, 2006, Christensen and Nielson, 2006 and Nielsen, 2007 suggest that IV and RV are fractionally cointegrated series.1 Interestingly, Bandi and Perron (2006) stress the fractional order of volatility is found in the non-stationary region whereas Christensen and Nielson, 2006 and Nielsen, 2007 indicate the stationary region. However, allowing for 95% confidence intervals, the estimates could plausibly lie in either region. In any case, Marinucci and Robinson (2001) stress that it is typically difficult to determine the integration order of fractional variables because a smooth transition exists between stationary and non-stationary regions. Christensen and Nielson, 2006 and Nielsen, 2007 note that when the fractional nature of the data is accounted for a slope parameter of unity in Eq. (1) cannot be rejected. Bandi and Perron (2006), noting the non-standard asymptotic distribution of conventional estimators in the non-stationary region, cannot formally test the relevant null hypothesis. However, subsampling shows their results also give support to the unbiasedness hypothesis. This paper builds on the empirical work of Bandi and Perron, 2006, Christensen and Nielson, 2006 and Nielsen, 2007 in five steps. Firstly, we employ data for several foreign exchange markets including the relatively new Euro market. Importantly, the IV data collected is traded on the market (and hence is directly observable). Since these data are directly quoted from brokers, they avoid the potential measurement errors associated with the more common approach (see, inter alia, Christensen and Prabhala, 1998) of backing out implied volatilities from a specific option-pricing model. Secondly, it is important to note that in the recent literature, RV is constructed either from (i) high frequency intra-day return data (see, for example, Nielsen, 2007) or (ii) daily return data (see Bandi and Perron, 2006). Neely (2009) suggests that, at least in the context of least squares regression, the use of intra-day instead of daily data, does not resolve the biased slope coefficient. However, to our knowledge, this comparison has not been formally drawn in a fractionally cointegrated setting. Additionally, given that RV constructed from intra-day data is likely to be a less noisy proxy2 for the unobserved but true volatility, the key to detecting (small) time-varying risk premia might be the use of such high frequency data. For example, consider augmenting regression (1) with a time-varying risk premium term rpt equation(2) View the MathML sourceσt+τRV=α+βσtIV+δrpt+ut+τ. Turn MathJax on Bivariate fractional cointegration between RV and IV implies any risk premium will be of a lower order of (fractional) integration than the original regressors. As a result, and as noted by Bandi and Perron (2006), the use of spectral methods like narrow band least squares will estimate regression (1) consistently, even in the presence of the risk premium. Re-arranging (2) leads to equation(3) View the MathML sourceσt+τRV-α-βσtIV=δrpt+ut+τ. Turn MathJax on Given that daily data is relatively noisy, it might be that any long memory behaviour of the risk premium 3 is swamped 4 by ut+τ in finite samples. In other words, a potential pitfall of employing daily data to construct RV is that it might render the risk premium undetectable. On the other hand, the use of a less noisy intra-day derived RV may lead to a smaller ut+τ and therefore the revealing of a time-varying risk premium. Following Bandi and Perron (2006), we deliberately eschew modelling a specific functional form for a risk premium, simply suggesting that fractionally integrated behaviour in the residual of (1) provides prima facie evidence for latent risk premia. To examine these issues, we construct two RV series from intra-day 5 and daily data. Thirdly, the possibility of fractional cointegration is examined formally using a new adaptation of the recently developed semi-parametric technique of Hassler et al. (2006) [hereafter HMV]. Under certain assumptions HMV prove that a residual-based log periodogram estimator, where the first few harmonic frequencies have been trimmed, has a limiting normality property. In particular, this methodology provides an asymptotically reliable testing procedure for fractional cointegration when the fractional order of regressors presents a particular type of non-stationarity. However, given the noted empirical uncertainty, (foreign exchange) volatility may present an integration order that violates the assumptions for the HMV test, as well as other fractional cointegration tests. To circumvent this uncertainty, we suggest, examine and apply an adapted fractional cointegration test robust to both stationary and non-stationary regions. Fourthly, given the non-standard asymptotic distribution of conventional estimators when using fractionally integrated data, we employ a wild bootstrap procedure as suggested by Gerolimetto (2006) to compute appropriate confidence intervals in (1). Again, this specifically overcomes the difficulties encountered when estimators are applied in the non-stationary region. Fifthly, we stress that the existence of fractional cointegration and that α = 0 and β = 1 in (1) are only necessary conditions for unbiasedness. The important condition, that ut+τ in (1) is serially uncorrelated, is required but such tests have been neglected by the recent extant literature. For completeness therefore, we employ an appropriate portmanteau test to the fractionally cointegrating residual. The paper is divided into five sections: Section 2 presents the empirical methodology; Section 3 describes the data; Section 4 analyses the empirical results and, finally, Section 5 concludes.

Almost all relevant literature has characterized implied volatility (IV) in the foreign exchange options markets as a biased predictor of realized volatility (RV). The cause of this bias has been the subject of much debate but in a recent paper, Neely (2009), the popular suggestions (i.e. overlapping data; use of low frequency data; and the non-pricing of volatility premia) are rejected. In this paper we examine the unbiasedness hypothesis by employing data for several very liquid foreign exchange options markets, including the relatively new Euro market. We make three contributions to the existing literature. Firstly, we develop, examine and apply a new test for fractional cointegration which is shown to be robust to both the stationary and non-stationary regions, including the weakly non-stationary region. Secondly, given that RV constructed from intra-day data is a less noisy proxy for the unobserved but true volatility, we posit that any (small) time-varying risk premia are more likely to be detected using the higher frequency data. To examine this issue, for each currency we construct two RV series from intra-day and daily data respectively and their respective relation with IV is compared throughout the empirical analysis. Thirdly, we collect data on IV that is traded on the market (and hence is directly observable). Since these data are directly quoted from brokers, they avoid the potential measurement errors associated with the more common approach of backing out implied volatilities from a specific option-pricing model. In contrast to previous studies, we find that the frequency of data used for the construction of RV is important for both the results of unbiasedness tests and the detection of time-varying risk premia in foreign exchange options markets. Employing the new fractional cointegration test, we show that foreign exchange RV and IV are fractionally cointegrated across a range of currencies and data frequencies. Moreover, tests using bootstrapped estimates and confidence intervals are not typically able to reject the hypothesis that the slope parameter in the RV–IV relation is unity. Contrary to the widely held view derived from previous research, the slope coefficient in the RV–IV relation is therefore shown not to be the primary source of bias. However, serial correlation tests of the forecast error in the RV–IV relation, which are frequently neglected in the extant literature, reveal a different picture. Results based on the RV measure derived from intra-day data show significant serial correlation in the forecast error for four out of six currencies, which suggests rejection of the unbiasedness hypothesis. In contrast, results obtained from the daily RV typically indicate absence of serial correlation in the forecast error and hence they support unbiasedness. It would appear that intra-day data reveal structures in the forecast error, while the more noisy and hence less reliable daily data veils this structure. Furthermore, when we employ the RV measure derived from intra-day data, the forecast errors for currencies like the Sterling/US$, Euro/Yen and US$/Swiss Franc, are shown to be fractionally integrated, suggesting the possibility of a time-varying risk premium with long memory. This is a new finding in the literature on options markets. This result is not found when the RV measure is derived from daily data and provides further testimony that the use of less noisy proxies for RV has significant time series implications.