WORKING PAPER SERIES NO. 366 / JUNE 24 THE INFORMATIONAL CONTENT OF OVER-THE-COUNTER CURRENCY OPTIONS by Peter Cristoffersen and Stefano Mazzotta
WORKING PAPER SERIES NO. 366 / JUNE 24 THE INFORMATIONAL CONTENT OF OVER-THE-COUNTER CURRENCY OPTIONS 1 by Peter Cristoffersen 2 and Stefano Mazzotta 3 In 24 all publications will carry a motif taken from te 1 banknote. Tis paper can be downloaded witout carge from ttp://www.ecb.int or from te Social Science Researc Network electronic library at ttp://ssrn.com/abstract_id=533126. 1 We ave benefited from several visits to te External Division of te European Central Bank wose ospitality is gratefully acknowledged. Very useful comments were provided by Torben Andersen, Lorenzo Cappiello, Olli Castren, Bruce Lemann, Filippo di Mauro, Stelios Makrydakis, Nour Meddai and Neil Separd. Te OTC volatilities used in tis paper were provided by Citibank N.A.Te usual disclaimer applies. 2 McGill University, CIRANO and CIREQ McGill University: peter.cristoffersen@mcgill.ca. 3 McGill University.
European Central Bank, 24 Address Kaiserstrasse 29 6311 Frankfurt am Main, Germany Postal address Postfac 16 3 19 666 Frankfurt am Main, Germany Telepone +49 69 1344 Internet ttp://www.ecb.int Fax +49 69 1344 6 Telex 411 144 ecb d All rigts reserved. Reproduction for educational and noncommercial purposes is permitted provided tat te source is acknowledged. Te views expressed in tis paper do not necessarily reflect tose of te European Central Bank. Te statement of purpose for te Working Paper Series is available from te website, ttp://www.ecb.int. ISSN 1561-81 (print) ISSN 1725-286 (online)
CONTENTS Abstract 4 Non-tecnical summary 5 1 Introduction 7 2 Volatility forecast evaluation 9 3 Volatility forecast evaluation results 12 4 Interval forecast evaluation 15 5 Density forecast evaluation 18 5a Grapical density forecast evaluation 19 5b Tests of te unconditional normal distribution 2 5c Tests of te conditional normal distribution 22 6 Conclusion and directions for future work 23 References 25 Figures 28 Tables 36 European Central Bank working paper series 47 3
Abstract Financial decision makers often consider te information in currency option valuations wen making assessments about future excange rates. Te purpose of tis paper is to systematically assess te quality of option based volatility, interval and density forecasts. We use a unique dataset consisting of over 1 years of daily data on over-te-counter currency option prices. We find tat te OTC implied volatilities explain a muc larger sare of te variation in realized volatility tan previously found using market-traded options. Finally, we find tat wide-range interval and density forecasts are often misspecified wereas narrow-range interval forecasts are well specified. JEL Classifications: G13, G14, C22, C53 Keywords: FX, Volatility, Interval, Density, Forecasting 4
Non-Tecnical Summary Financial decision makers often consider te forward-looking information in currency option valuations wen making assessments about future developments in foreign excange rates. Option implied volatilities can be used as forecasts of realised volatility and interval and density forecasts can be extracted from strangles and risk-reversals. Te purpose of tis paper is to assess te quality of suc volatility, interval and density forecasts. Our work is based on a very unique database consisting of more tan ten years of daily quotes on European currency options from te OTC market. Te OTC quotes include at-te-money implied volatilities, strangles and risk-reversals on te dollar, yen and pound per euro as well as on te yen per dollar. From tis data we ave constructed daily 1-mont interval and density forecasts using te metodology in Malz (1997). Te main findings of te paper are as follows: First and foremos we find tat te OTC implied volatilities explain a muc larger sare of te variation in realized volatility tan as been found previously in studies relying on market-traded options. Second, we find tat widerange interval forecasts are often misspecified wereas narrow-range interval forecasts are well specified. Tird, we find tat te option-based density forecasts are rejected in general. Grapical inspection of te density forecasts suggests tat wile te sources of rejections vary from currency to currency misspecification of te distribution tails is a common source of error. Our paper contributes in two areas of te literature. Firs to our knowledge, te empirical performance of option-based interval and density forecasts as not been systematically explored so far. Second, wile tere is a considerable literature on implied volatility forecasts from market-traded options, OTC data ave only recently been employed. One of our contributions consists of analyzing a unique dataset of OTC options wic turns out to ave impressive volatility prediction properties. OTC options are quoted daily wit a fixed maturity (say one mont) wereas market-traded options ave rolling maturities wic in turn complicate teir use in fixed-orizon volatility forecasting. In addition to volatility forecasts we evaluate option-based interval and density forecasts wic are widely used by practitioners but wic ave not been systematically assessed so far. OTC options are quoted daily wit fixed moneyness in contrast wit market-traded options wic ave fixed strike prices and tus time-varying moneyness as te spot price canges. Tis time-varying moneyness complicates te use of market-traded options for interval and density forecasting in tat te effective support of te distribution is canging over time. Finally, te 5
trading volume in OTC options is often muc larger tan in te corresponding market traded contracts wic in turn is likely to render te OTC quotes more reliable for information extraction. Several promising directions for future researc exist. Firs policy makers may be interested in assessing speculative pressures on a given excange rate. Te option implied densities can be used in tis regard by constructing daily option-implied probabilities of say a 3% appreciation or depreciation during te next mont. Second, te accuracy of te left and rigt tail interval forecast could be analyzed separately in order to gain furter insigt on te probability of a sizable appreciation or depreciation. Tird, relying on te triangular arbitrage condition linking te JPY/EUR, te USD/EUR, and te JPY/USD, one can construct option implied covariances and correlations from te option implied volatilities. Tese implied covariances can ten be used to forecast realized covariances as done for volatilities in Tables 1-4. Four te misspecification found in te option-implied density forecasts may be rectified by assuming different tail-sapes in te density estimation or by incorporating return-based information. Converting te risk-neutral densities to teir statistical counterparts may be useful as well but will require furter assumptions, wic may or may not be empirically valid. Bliss and Panigirtzoglou (24) present promising results in tis direction. Finally, one could consider making density forecasts combining option-implied and return based densities. We leave tese important issues for future work. 6
1. Introduction Financial decision makers often consider te forward-looking information in currency option valuations wen making assessments about future developments in foreign excange rates. 1 Option implied volatilities can be used as forecasts of realised volatility and interval and density forecasts can be extracted from strangles and risk-reversals. Te purpose of tis paper is to assess te quality of suc volatility, interval and density forecasts. Our work is based on a very unique database consisting of more tan ten years of daily quotes on European currency options from te OTC market. 2 Te OTC quotes include at-te-money implied volatilities, strangles and risk-reversals on te dollar, yen and pound per euro 3 as well as on te yen per dollar. From tis data we ave constructed daily 1-mont interval and density forecasts using te metodology in Malz (1997). Te main findings of te paper are as follows: First and foremos we find tat te OTC implied volatilities explain a muc larger sare of te variation in realized volatility tan as been found previously in studies relying on market-traded options. Second, we find tat widerange interval forecasts are often misspecified wereas narrow-range interval forecasts are well specified. Tird, we find tat te option-based density forecasts are rejected in general. Grapical inspection of te density forecasts suggests tat wile te sources of rejections vary from currency to currency misspecification of te distribution tails is a common source of error. Several early contributions use market-based options data wit mixed results. Beckers (1981) finds tat not all available information is reflected in te current option price and question te efficiency of te option markets. Canina and Figlewski (1993) find implied volatility to be a poor forecast of subsequent realized volatility. Lamoureux and Lastrapes (1993) provide evidence against restrictions of option pricing models wic assume tat variance risk is not priced. Jorion (1995) finds tat statistical models of volatility based on returns are outperformed by implied volatility forecasts even wen te former are given te advantage of ex post in sample parameter estimation. He also finds evidence of bias. More recently, Cristensen and Prabala (1998) using longer time series and non overlapping data find tat implied volatility outperforms 1 See for example Bank for International Settlements (23), Bank of England (2), International Monetary Fund (22), and OECD (1999). 2 Te OTC volatilities used in tis paper were provided by Citibank N.A 3 Prior to January 1, 1999 tese were denoted in DEM. 7
past volatility in forecasting future volatility. Fleming (1998) finds tat implied volatility dominates te istorical volatility in terms of ex ante forecasting power and suggests tat a linear model wic corrects for te implied volatility s bias can provide a useful market-based estimator of conditional volatility. Blair, Poon, and Taylor (21), find tat nearly all relevant information is provided by te VIX index and tere is not muc incremental information in igfrequency index returns. Neely (23) finds tat econometric projections supplement implied volatility in out-of-sample forecasting and delta edging. He also provides some explanations for te bias and inefficiency pointing to autocorrelation and measurement error in implied volatility. In work concurrent wit ours, Pong, Sackleton, Taylor and Xu (24) find tat igfrequency istorical forecasts are superior to implied volatilities using OTC data for orizons up to one week. Covrig and Low (23) use OTC data to find tat quoted implied volatility subsumes te information content of istorically based forecasts at sorter orizons, and te former is as good as te latter at longer orizons. Our paper contributes in two areas of te literature. Firs to our knowledge, te empirical performance of option-based interval and density forecasts as not been systematically explored so far. Second, wile tere is a considerable literature on implied volatility forecasts from market-traded options, OTC data ave only recently been employed. One of our contributions consists of analyzing a unique dataset of OTC options wic turns out to ave impressive volatility prediction properties. OTC options are quoted daily wit a fixed maturity (say one mont) wereas market-traded options ave rolling maturities wic in turn complicate teir use in fixed-orizon volatility forecasting. In addition to volatility forecasts we evaluate option-based interval and density forecasts wic are widely used by practitioners but wic ave not been systematically assessed so far. OTC options are quoted daily wit fixed moneyness in contrast wit market-traded options wic ave fixed strike prices and tus time-varying moneyness as te spot price canges. Tis time-varying moneyness complicates te use of market-traded options for interval and density forecasting in tat te effective support of te distribution is canging over time. Finally, te trading volume in OTC options is often muc larger tan in te corresponding market traded contracts wic in turn is likely to render te OTC quotes more reliable for information extraction. Te remainder of te paper is structured as follows. Section 2 defines te competing volatility forecasts we consider and describes te standard regression-based framework for 8
volatility forecast evaluation. Section 3 presents results on te option-implied and istorical return-based volatility forecasts of realized volatility. Section 4 suggests a metod for evaluating interval forecasts from option prices and present results from tis metod. Section 5 suggests metods for evaluating density forecasts from option prices and present results from tese metods. Finally, Section 6 discusses potential points for future researc. 2. Volatility Forecast Evaluation In order to evaluate te informational content of te volatilities implied from currency options, we define te realized 4 future volatility for te next days to be RV 252 i 1 R 2 t i in annualized terms, were R t+i = ln(s t+i /S t+i-1 ) is te FX spot return on day t+i. Tis realized volatility (and its logaritm) will be our forecasting object of interest in tis section. 5 We will consider four competing forecasts of realized volatility. First and most importantly te implied volatility from at-te-money OTC currency options wit maturity, were is eiter 1 mont or 3 monts corresponding to rougly 21 and 63 trading days respectively. Denote tis options-implied volatility by. IV Te oter tree volatility forecasts are derived from istorical FX returns only. Te simplest possible forecast is te istorical -day volatility, defined as HV 252 2 Rt i i 1 RV t, Te istorical volatility is a simple equal weigted average of past squared returns. We can instead consider volatilities tat apply an exponential weigting sceme putting progressively less weigt on distant observations. Te simplest suc volatility is te Exponential Smooter or RiskMetrics volatility, were daily variance evolves as ~ i 1 2 2 2 1 R R 2 t 1 t i 1 t 1 i 1 ~ t 4 See Andersen, T., T. Bollerslev, F. X. Diebold, and P. Labys, 23. 5 Later on we will consider realized volatilities calculated from 3-minute rater tan daily returns. 9
Following JP Morgan we simply fix =.94 for all te daily FX returns. Te fact tat te coefficients on past variance and past squared returns sum to one makes tis model akin to a random walk in variance. Te annualized forecast for -day volatility is terefore simply 252 ~ t RM, Finally we consider a simple, symmetric GARCH(1,1) model, were te daily variance evolves as ˆ 2 2 ˆ t 1 t 2 t 1 R In contrast wit te RiskMetrics model, te GARCH model implies a non-constant term structure of volatility. Te unconditional variance in te model can be computed as ˆ 2 1 Te conditional variance for day t+ can be derived as ˆ 2 t 1 2 2 ˆ 2 2 ˆ ˆ t t t 1 And te annualized GARCH volatility forecast for day t+1 troug t+ is tus 252 GH 2 ˆ ˆ t 1 i 1 i 1 2 2 ˆ Te GARCH model will ave a downward sloping volatility term structure wen te current variance is above te long orizon variance and vice versa. 6 Notice tat wile we take te standard approac of estimating a daily volatility model to forecast montly and quarterly volatility, tis is not te only way to proceed. Gysels, Santa- Clara and Valkanov (23) ave recently explored ways to estimate volatility forecasting models directly on te relevant orizon of interest. Rater tan estimating say a montly forecasting model on montly data tey make use of te information in daily observations in a orizonspecific forecasting model imposing a parsimonious lag structure on te daily observations. Figure 1 sows te spot rates of te four FX rates analysed in tis paper. Prior to te euro introduction in 1999 we observe FX options denoted against te Deutscmark (DEM) and we 6 Te GARCH model contains parameters wic must be estimated. We do tis on rolling 1-year samples starting in January 1982 and using QMLE. Eac year we forecast volatility one-year out-of-sample before updating te estimation sample by anoter calendar year of daily returns. Te euro volatility forecasts are constructed using syntetic euro rates in te period prior to te introduction of te euro. 1
will terefore work wit te DEM spot rates prior to te euro introduction as well. Prior to January 1, 1999 we use DEM options to forecast DEM volatility and afterwards we use euro options to forecast euro volatility. Te five volatility specifications including te realized volatility are plotted in Figures 2-5. Eac page corresponds to a particular volatility specification and eac column on a page represents an FX rate. Te top row sows te 1-mont volatility and te bottom row te 3-mont volatility. Notice tat te RiskMetrics volatilities in Figure 4 are identical for 1-mont and 3- mont maturities as te random-walk nature of tis specification implies a flat volatility term structure. We are now ready to assess te quality of te different volatility forecasts. Tis will be done in simple linear predictability regressions. We first run four univariate regressions for eac currency RV a b j j, for j IV, HV, Te purpose of tese univariate regressions is to assess te fit troug te adjusted R 2 and to ceck ow close te estimates of a are to and ow close te estimate of b are to 1. Bollerslev and Zou (23) 7 point out tat if te volatility risk is priced in te options markets ten we sould expect to find a positive intercept and a slope less tan one in te above regression. In a standard stocastic volatility set up, it can be sown tat if te price of volatility risk is zero, te process followed by te volatility is identical under te objective and te risk neutral measures. In suc a case tere would be no bias. However, te volatility risk premium is generally estimated to be negative wic in turn implies tat te volatility process under te risk neutral measure will ave iger drift. Tis is also consistent wit te fact tat implied volatilities are empirically found to be upward biased estimates of te objective volatility. Aside tese considerations, for someone using implied volatility in te real time monitoring of FX movements, te intercept and slope coefficients are informative of te size of te bias and efficiency respectively of te forecasts. In addition we will run tree bivariate regressions including te implied volatility forecast as well as eac of te tree return-based volatility forecasts in turn. Tus we ave RV a b IV c j IV, j RM, GH, for j HV, RM, GH 7 See also Bandi and Perron (23), Cernov (23), Bates (22), and Benzoni (21). 11
Te purpose of te bivariate regressions is to assess if te return-based volatility forecasts add anyting to te market-based forecasts implied from currency options. Finally, we run a regression including all te four volatility forecasts in te same equation. Te purpose of tis regression is to assess te relative merits of te different volatility forecasts. We will run all regressions for =21 and 63 corresponding to te 1-mont and 3-mont option maturities. We will also run all regressions in levels of volatility as above as well as in logaritms. Due to te volatilities being strictly positive, te log specification may ave error terms, wic are better beaved tan tose from te level regressions. 3. Volatility Forecast Evaluation Results Tables 1 and 2 report te regression point estimates as well as standard errors corrected for eteroskedasticity and autocorrelation. Trougout tis paper we apply GMM using te Newey- West 8 weigting matrix wit a prespecified bandwidt equal to 21 days for te 1-mont orizon (Table 1) and 63 days for te 3-mont orizon (Table 2). Te bandwidt is cosen as to eliminate te influence of te autocorrelation induced by te overlapping observation. We also report te regression fit using te adjusted R 2. Several strong and interesting empirical regularities emerge. Firs te regression fit is very good in all cases. Jorion (1995) reports R 2 in te region.1-.15 for te USD/JPY, USD/DEM and USD/CHF using implied volatility forecasts. We get instead R 2 of.3-.38 for te 1-mont maturity and.16-.35 for te 3-mont maturity case. Second, comparing te R 2 across te univariate forecast regressions we see tat te implied volatility is te best volatility forecast. Tis result olds across currencies and orizons. Tird, comparing te slope estimates across te bivariate forecast regressions were te implied volatility forecast is included along wit eac of te oter tree forecasts, te implied volatility always as te igest slope. Tus, in te cases wen GARCH as a iger slope in te univariate regression te bivariate regressions including te IV and GARCH forecasts always assign a larger slope to te IV forecast. Te fact tat GARCH-based forecasts sometimes ave a slope closer to one tan do te implied volatility forecasts is not surprising given te price of 8 See Newey and West (1987). 12
volatility risk argument in Bollerslev and Zou (23) and oters. 9 Neverteless, it is interesting to note tat te R 2 is iger for te implied volatility forecasts even in te cases were its slope is lower tan tat of te GARCH-based forecasts. Four comparing te slope estimates across te multivariate forecast regressions were all four forecasts are included simultaneously te implied volatility as te igest slope. Tis result olds across currencies and orizons. Fif comparing across te orizon forecasts it appears peraps not surprisingly tat te 1-mont forecasts ave iger R 2 tan te 3-mont forecasts. Finally, te slope coefficient is often insignificantly different from one for te IV forecasts, and its intercept is often insignificantly different from zero. Tables 3 and 4 contain te same set of regressions as Tables 1 and 2, but now run on te euro sample (i.e. post January 1, 1999) only, and furtermore relying on 3-minute intraday returns rater tan daily returns to compute te one and tree mont realized volatilities. We also report te euro sample estimates using daily data in Table 3a and 4a. Te objective of Tables 3 and 4 is to see if te post-euro sample is different from te full sample period wic straddles te introduction, and furtermore to assess te value of using ig-frequency returns in volatility forecast evaluation. Te teoretical benefits of doing so ave been documented in Andersen and Bollerslev (1998) and Andersen, Bollerslev and Meddai (23) wo sow tat te R 2 in te regressions we run will be significantly iger wen proxying for true volatility using an intraday rater tan daily return-based volatility measure. As pointed out by Alizade, Brandt and Diebold (22), and Brandt and Diebold (23) tis teoretical benefit may in practice be outweiged by market microstructure noise, but relying on 3-minutes returns in very liquid markets as we do ere sould mitigate tese problems. Te results in Table 3 and 4 are broadly similar to tose from te full sample but using ig-frequency returns does lead to some new interesting findings. Firs for te tree euro cross currencies te regression fit is typically muc better now. Due to te obvious structural break in 1999 tis is peraps not surprising. But it is still interesting tat we now get R 2 as ig as 65% in te univariate regressions. Note tat te R 2 for te 3-mont JPY/USD case is now sligtly lower tan before. It is terefore not simply te case te FX volatility as become more predictable as of late. 9 Weter volatility risk is priced is of course an empirical question: some of our results indirectly support te conjecture tat volatility risk is priced in te currency options markets. 13
Second, comparing te R 2 across te univariate forecast regressions te implied volatility is typically te best volatility forecast. 1 Te exception is te EUR/JPY rate. Tird, comparing te slope estimates across te bivariate and multivariate forecast regressions te implied volatility typically as te igest slope. It is interesting tat te simple istorical realized volatility forecast now sometimes as te igest slope. 11 Tis is results is exclusively due to te use of ig frequency data as it is easy to infer from te comparison of Table 3-4 to 3a-4a. Te added accuracy in tis forecast from te intra-day returns is tus evident. In order to give assess te importance of te coice of estimation period we ave also run te same regressions using in sample GARCH estimates. 12 Altoug te explanatory power as measured by te adjusted R 2 of te in sample GARCH is substantially iger compared to tat of te out of sample regressions, in most cases te cange does not affect te results in te 1-mont regressions for any currency bot in te full sample and in te post 1999 sample. Te same is true for te 3-mont orizon wit te only exception of te GBP in te full sample: ere te in sample GARCH get te igest coefficient in te multivariate regression. In summary, we find strong evidence tat te implied volatility from FX options is useful in predicting future realized volatility at te one and tree mont orizons. Te predictability is particularly strong for te euro cross rates in te recent period. In spite of te potential bias from volatility risk being priced in te options, te regression slope on te volatility forecasts are often quite close to one. Peraps te most striking finding in Tables 1-4 is te ig level of R 2 found in te implied volatility regressions. It appears tat te volatility implied in te OTC options offer muc more precise forecasts tan te volatility implied from market-traded options, wic ave been analyzed in previous studies. We suspect tat te so-called telescoping bias arising from te rolling-maturity structure of market-traded options (see Cristensen, Hansen, and Prabala, 21) could be part of te reason. Furtermore, te fact tat OTC options are quoted daily wit a fixed moneyness, as opposed to a fixed strike price, wic ensures tat te options used for 11 Te istorical volatility forecast could potentially be improved furter by estimating a slope coefficient tus allowing for mean reversion in te forecast. 12 Results are not sown ere to conserve space. 14
volatility forecasting are exactly at-te-money eac day. Finally, te large volume of transaction in OTC currency options compared wit market traded options may offer additional explanation. 4. Interval Forecast Evaluation Te information in currency options may be useful not only for volatility forecasting but for spot rate distribution forecasting more generally. In te following sections we abstract from te difference between risk neutral and objective distributions. Te empirical question we want to ask is: How well can risk neutral intervals and densities computed using standard metodologies forecast pysical interval and densities. Te legitimacy of te question stems from te fact tat financial decision makers often consider te information in currency option valuations wen making assessments about future excange rates witout worrying about tis important teoretical difference. In addition, tis pragmatic approac can be justified by considering tat for currencies te risk premium, i.e. te conditional mean, wic would largely determine te difference between risk neutral and pysical, may not be as important as te iger order moments and particularly te conditional variance. 13 In oter words, te tests in te following sections can be considered as joint tests of te metodology used to extract densities and intervals under te additional ypotesis tat te objective and risk neutral distributions are not very different. 14 Tese tests may ave low power wit respect to generic alternative ypoteses but tey can elp assessing weter certain specific pieces of information ave been duly taken into account in te construction of tese intervals and densities. Rejection may come from te presence of some currency risk premium: tis situation will sift apart te mean under te two probabilistic measures. Rejection could also come from metodological and data sortcoming in te construction of te interval and densities: tis is likely to be te case for te widest intervals and te tails of te distributions wen tey are based on te extrapolation of market data. 13 It is also te case tat tere is no metodology to transform risk neutral distributions into teir objective counterparts witout making several, possibly very restrictive assumptions. 14 In order to save space we do not compare te predictive performance of our option implied densities wit tat of densities based on istorical FX returns only. Tis is an interesting exercise wic we leave for future studies. 15
In substance, te results of tese tests sould be seen as suggestion for improvement of te prevalent metodologies and caveats wit regard to ow muc trust sould one put in tese forecasts. In tis section we study te performance of one-mont interval forecasts calculated from option prices and forward rates. Te intervals are constructed from te option-implied densities wic in turn are calculated using te estimation metod in Malz (1997). Te Malz metodology is based on a second order Taylor approximation to te volatility smile. Te procedure forces te approximation of te implied volatility function to be exactly equal to te observed implied volatility for te tree values of te Black-Scoles delta, namely.25,.5, and.75. It uses tis interpolated approximation of te smile to compute a continuous option price function. It ten uses te classical result in Breeden and Litzenberger (1978) to compute te risk neutral density. We ave computed conditional interval forecasts for te {.45,.55} probability interval, as well as te {.35,.65}, {.25,.75}, {.15,.85}, and te {.5,.95} intervals. Tese forecasts are sown in Figure 6. Notice tat te intervals for te GBP/DEM look excessively jagged in a large part of te pre euro sample. We now set out to evaluate te usefulness of te interval forecasts following te metodology developed first in Cristoffersen (1998). Let te generic interval forecast be defined as L ( pl), U ( pu ) were p L and p U are te percentages associated wit te lower and upper conditional quantiles making up te interval forecast. Consider now te indicator variable defined as I, if St {L(pL ),U(pU )} 1, if not Ten if te interval forecast is correctly calibrated, we must ave tat E I X 1 p p p t were X t denotes a vector of information variables (and functions tereof) available on day t. If te interval forecast is correctly calibrated ten te expected outcome of te future FX rate U L 16
falling outside te predicted interval must be a constant equal to te pre-specified interval probability p. Tis ypotesis will be tested in a linear regression setup, but binary regression metods could ave been used as well. Under te alternative ypotesis we ave I p a bx t and te null ypotesis corresponds to te restrictions a b Running tese regressions on daily data we again ave to worry about overlapping observations, wic we allow for using GMM estimation. Table 5 sows te results for te regression-based tests of te interval forecasts. Te interval forecasts for te {.45,.55}, {.35,.65}, {.25,.75}, {.15,.85}, and te {.5,.95} intervals are denoted by te probability of an observation outside te interval, i.e. p=.9,.7,.5,.3 and.1 respectively. We refer to tese outside observations as its. Te zero/one it sequence (less its expected value p) is regressed on a constan te 21-day lagged it and te 21- day lagged 1-mont implied volatility. Te lagged it is included to capture any dependence in te outside observations. Te implied volatility is included to assess if it is incorporated optimally in te construction of te interval forecast. If te interval forecast is correctly specified ten te intercept and slopes sould all be equal to zero. Table 5 reports coefficient estimates along wit t-statistics again calculated using GMM. Below te solid line in eac subsection of te table te average it rate, wic sould be equal to p, is reported along wit te t-statistic from te test tat te average it rate indeed equals p. All t-statistics larger tan two in absolute value are denoted in boldface type. We also include Wald tests of te joint ypotesis tat all te estimated coefficients are zero. Te results in Table 5 can be summarized as follows. Firs for te pound te average it rate is significantly different from te pre-specified p for all but te narrowest interval (wit outside probability equal to.9). Te jagged pound intervals evident from Figure 6 are probably te culprit ere. Second, for te oter tree FX rates, te average it rate is typically not significantly different from te pre-specified p. Te only notably exception is te wide-range intervals (wit outside probability.1) were all but te JPY/EUR intervals are rejected. It tus appears tat te interval forecast ave te ardest time forecasting te tails of te spot rate distribution. 17
Tird, notice tat no regression slopes are significant in te JPY/EUR case. No dependence in te it sequence is apparent and te information in implied volatilities seems to be used optimally in tis case. Four wile te interval forecasts for te JPY/EUR are well specified, te intervals for te oter tree forecasts are typically rejected. Te slope on te 21-days lagged implied volatility is most often found to be significantly negative. Tis indicates tat te its tend to occur wen te implied volatility was relatively low on te day te forecast was made. If te intervals ad been using te implied volatility information optimally ten no dependence sould be found between te current implied volatility and te subsequent realization of te it sequence. Table 6 reports te interval forecast evaluation results using data from te euro sample only. Te results are now somewat different and can be summarized as follows. Firs te average it rate is typically not significantly different from te pre-specified p wit a couple of noteworty exceptions: Te average it rate is rejected across all te four FX rates for te widest intervals. Again, it appears tat te option implied densities ave trouble capturing te tails of te distribution. For all four FX rates it is te case tat te outside it frequency is lower tan it sould be, tus te wide-range option-implied intervals are too wide on average. Second, te average it rate is rejected in te two widest intervals for te pound, but in general te pound intervals are better calibrated in te euro sample tan before. Tird, te JPY/USD interval is now te most poorly calibrated interval. In summary we find tat te option-implied interval forecast for te euro cross rates perform well in te post January 1, 1999 sample. Te exception is te forecasts for te widest intervals, wic tend to be too wide on average. Te option-implied densities apparently ave trouble capturing te tail beaviour of te spot rate distributions. Te rejection of widest intervals and tus misspecification of te tails of te density forecasts sould peraps not come as a surprise. Te density tails are estimated on te basis of an extrapolation of te volatility smile from te values for wic option price information is available (tat is for deltas equal to.25,.5, and.75). It appears tat tis extrapolation could be improved. We will pursue te topic of density forecasting in more detail in te next section. 5. Density Forecast Evaluation Te option-implied interval forecasts analyzed above are constructed from te implied density, wic contains muc more information tan te intervals alone. We would terefore like 18
to evaluate te appropriateness of tese density forecasts in teir own rigt. Doing so is likely to yield some insigts into te poor performance of te widest interval forecasts, wic was noted above. We start off by outlining te general ideas beind density forecast evaluation developed by Diebold, Gunter and Tay (1998). Let S and f S F denote te cumulative and probability density function forecasts made on day t for te FX spot rate on day t+. We can ten define te so-called probability transform variable as U S. St f ( u) du Ft, t Te transform variable captures te probability of obtaining a spot rate lower tan te realization were te probability is calculated using te density forecast. Te probability will of course take on values in te interval [,1]. If te density forecast is correctly calibrated ten we sould not be able to predict te value of te probability transform variable U using information available at time t. Tat is we sould not be able to forecast te probability of getting a value smaller tan te realization. Moreover, if te density forecast is a good forecast of te true probability distribution ten te estimated probability will be uniformly distributed on te [,1] interval. 5.a Grapical Density Forecast Evaluation Figure 7 assesses te unconditional distribution of te probability transform variable U for eac spot rate troug a simple istogram. If te density forecast is correctly calibrated ten eac of te istograms sould be rougly flat and a random 1% of te 31 bars sould fall outside te two orizontal lines delimiting te 9% confidence band. It appears tat te istograms display certain systematic differences from te uniform distribution. Notice in particular tat te JPY/EUR istogram (top rigt panel) sows a systematically declining sape moving from left to rigt. Tis is indicative of te forecasted mean spot rate being wrong. Tere are too many observations were te realized spot rate lies in te left side of te forecasted distribution (and generates a U less tan.5) and vice versa. In te USD/EUR case (top left panel) it appears tat tere are not enoug observations in te two extremes, wic suggests tat te forecasted density as tails, wic are too fat. Tis finding matces Table 5 were we found tat te widest intervals were too wide for te USD/EUR. Finally, te JPY/USD distribution (bottom rigt panel) appears to be misspecified in te rigt tail. 19
For certain purposes, including statistical testing, it is more convenient to work wit normally distributed rater tan uniform variables for wic te bounded support may cause tecnical difficulties. As suggested by Berkowitz (21) 15 we can use te standard normal inverse cumulative density function to transform te uniform probability transform to a normal transform variable Z 1 U F S 1 If te implied density forecast is to be useful for forecasting te pysical density, it must be te case tat te distribution of U is uniformly distributed and independent of any variable X t observed at time t. Consequently te normal transform variable must be normally distributed and also independent of all variables observed at time t. Figure 8 assesses te unconditional normality of te normal transforms by plotting te istograms wit a normal distribution superimposed. 16 Te normal istograms typically confirm te findings in Figure 7 but also add new insigts. Wile it appeared in Figure 7 tat te GBP/EUR ad fairly random deviations from te uniform distribution, it now appears tat te normal transform is systematically skewed compared wit te superimposed normal distribution. Wile te grapical evidence in Figures 7 and 8 is quite informative of te potential deficiencies in te option implied density forecasts, it may be interesting to formally test te ypotesis of te normal transforms following te standard normal distribution. We do tis below. t 5.b Tests of te Unconditional Normal Distribution We first want to test te simple ypotesis tat te normal transform variables are unconditionally normally distributed. Basically, we want to test if te istograms in Figure 8 are significantly different from te superimposed normal distribution. Te unconditional normal ypotesis can be tested using te first four moment conditions E 2 3 4 Z, E Z 1, E Z, E Z 3 15 See also Diebold, Han and Tay (1999). 16 Te superimposed normal distribution functions ave different eigts due to te different number of observations available for eac currency. 2
We still need to allow for autocorrelation arising from te overlap in te data and so we estimate te following simply system of regressions Z Z Z Z 2 3 4 a 1 1 a a 3 2 3 a using GMM and test tat eac coefficient is zero individually as well as te joint test tat tey are all zero jointly. 17 In eac case we allow for 21 day overlap in te daily observations. Te results of tese tests are reported in Tables 7 and 8. Table 7 tests for unconditional normality on te entire sample and Table 8 restricts attention to te post 1999 period. Table 7 sows tat wile only a few of te individual moments are found to be significantly different from te normal counterpar te joint (Wald) test tat all moments matc te normal distribution is rejected strongly in tree cases and weakly in te case of te JPY/USD. Te post 1999 results are very similar. Now te Wald test strongly rejects all four density forecasts. We tus find fairly strong evidence overall to reject te option-implied density forecasts using simple unconditional tests. In order to focus attention on te performance of te density forecasts in te tails of te distribution, we report QQ-plots of te normal transform variables in Figure 9. QQ-plots display te empirical quantile of te observed normal transform variable against te teoretical quantile from te normal distribution. If te distribution of te normal transform is truly normal ten te QQ-plot sould be close to te 45-degree line. Figure 9 sows tat te left tail is fit poorly in te case of te dollar, and tat te rigt tail is fit poorly in te case of te pound and te JPY/USD. In te case of te dollar tere are too few small observations in te data, wic is evidence tat te option implied density as a left tail tat is too tick. Te pound as too many large observations indicating tat te rigt tail of te density forecast is too tin. In te JPY/USD case te rigt tail appears to be too tick. Tese findings are also evident from Figure 7. Rejecting te unconditional normality of te normal transform variables is of course importan but it does not offer muc constructive input into ow te option-implied density 4 (1) (3) (2) (4) 17 See Bontemps and Meddai (22) for related testing procedures. 21
forecasts can be improved upon. Te conditional normal distribution testing we turn to now is more useful in tis regard. 5.c Tests of te Conditional Normal Distribution We would like to know wy te densities are rejected, and specifically if te construction of te densities from te options data can be improved someow. To tis end we want to conduct tests of te conditional distribution of te normal transform variable. Is it possible to predict te realization of te time t+ normal transform variable using information available at time t? If so ten tis information is not used optimally in te construction of te density forecast. Te conditional ypotesis can be tested using te generic moment conditions E 2 Z f X, E Z f X 3 4 1, E Z f X, E Z f X 3 1 t 2 t 3 t 4 t Coosing particular moment functions and variables tese conditions can be implemented in a regression setup as follows Z Z Z Z 2 3 4 a b Z 1 1 a a 3 2 3 a 4 11 b b 31 Z t, 21 b 3 t, 41 Z Z b 2 t, 12 b 4 t, 32 IV t b 22 b 42 (1) IV 2 (2) t 3 (3) IV 4 (4) were we include te lagged power of te normal transform as well as te power of te current implied volatility as regressors. We can now test tat te regression coefficients are zero. Table 9 sows te estimation results of te regression systems for te four excange rates. In line wit previous results we find tat te information in te implied volatility is not used optimally in te construction of te option-implied density forecast for te GBP/EUR. Table 1 sows te regressions from Table 9 run only on te euro sample. Comparing te two tables, it is evident tat te clear rejection of te pound density forecasts in Table 9 is largely due to problems in te pre-euro sample. Restricting attention to te euro sample tere is more evidence on te implied volatility being misspecified in te JPY/USD rate. Looking across Tables 9 and 1 we see tat te Wald test of all coefficients being zero is strongly rejected for all four FX rates in bot samples. It would terefore seem possible in general to improve upon te option-implied density forecasts studied ere. IV t t 22
6. Conclusion and Directions for Future Work We ave presented evidence on te usefulness of te information in over-te-counter currency option for forecasting various aspects of te distribution of excange rate movements. We focused on tree aspects of spot rate forecasting, namely, volatility forecasting, interval forecasting, and distribution forecasting. Wile oter papers ave pursued volatility forecasting in manners similar to ours we believe to be te first to systematically investigate te properties of option-based interval and density forecasts. Furtermore, we are some of te first to investigate long time series of volatilities from over-te-counter options, wic we find to be muc more useful for volatility forecasting tan te market-traded options used in previous studies. Te reasons for tis important finding are likely to be 1) te so-called telescoping bias arising from rolling maturities in market-traded options is not an issue in te OTC options, 2) te timevarying moneyness in market-traded options, and 3) te volume of trades done over-te-counter is muc larger tan te excange trading volume for currency options. Our oter findings can be summarized as follows. Firs te implied volatilities from currency options typically offer predictions tat explain muc more of te variation in realized volatility tan do volatility forecasts based on istorical returns only. Tis ranking is owever sometimes reversed wen istorical volatility forecasts are constructed from intraday returns. Second, wen combining implied volatility forecasts wit return-based forecasts, te latter typically receive very little weigt. Tird, in terms of interval forecasting on te entire 1992-23 sample, te option-implied intervals are useful for te JPY/EUR but rejected for te oter tree currencies in te study. Four focusing on te euro sample, te option-implied interval forecasts are generally useful. Two notable exceptions are te widest-range intervals wit 9% coverage and te JPY/USD intervals in general. Te 9% intervals tend to be too wide due to te misspecification of te tails of te forecast distribution. Fif wen evaluating te entire implied density forecasts tese are generally rejected. Te grapical evidence again suggests tat te tails in te distribution are typically misspecified. We tus conclude tat te information implied in option pricing is useful for volatility forecasting and for interval forecasting as long as te interest is confined to intervals wit coverage in te 1-7% range. Te rejection of te widest intervals and te complete density forecast is of course interesting and warrants furter scrutiny. Te potential reasons are at least fourfold. Firs te option contracts used may not ave extreme enoug strike prices to be useful for constructing accurate distribution tails. Second, te information in options could be used sub-optimally in te 23
density estimates. Tird, we could be rejecting te densities because certain information available at te time of te forecasts is not incorporated in te option prices used to construct te densities, i.e. option market inefficiencies. Four te risk premium considerations, wic were abstracted from in tis paper could be important enoug to reject te risk-neutral density forecasts considered. Te misspecification of te mean in te case of te JPY/EUR rate suggests tat an omitted risk premium could be te culprit in tat case. For te oter tree currencies, owever, Figure 9 suggests tat te culprit is tail misspecification, wic is likely to arise from te lack of information on deep in-te-money and deep out-of-te-money options. We round off te paper by listing some promising directions for future researc. Firs policy makers may be interested in assessing speculative pressures on a given excange rate. Te option implied densities can be used in tis regard by constructing daily option-implied probabilities of say a 3% appreciation or depreciation during te next mont. Second, te accuracy of te left and rigt tail interval forecast could be analyzed separately in order to gain furter insigt on te probability of a sizable appreciation or depreciation. Tird, relying on te triangular arbitrage condition linking te JPY/EUR, te USD/EUR, and te JPY/USD, one can construct option implied covariances and correlations from te option implied volatilities. Tese implied covariances can ten be used to forecast realized covariances as done for volatilities in Tables 1-4. Four te misspecification found in te option-implied density forecasts may be rectified by assuming different tail-sapes in te density estimation or by incorporating returnbased information. Converting te risk-neutral densities to teir statistical counterparts may be useful as well but will require furter assumptions, wic may or may not be empirically valid. Bliss and Panigirtzoglou (24) present promising results in tis direction. Finally, one could consider making density forecasts combining option-implied and return based densities. We leave tese important issues for future work. 24
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