Horizon Effect of Options in Predicting Index Returns

Fear of Future: Horizon Effect of Options in Predicting Index Returns

David Sun* Kainan University

Shuai Qiao† Huazhong University of Science and Technology

ABSTRACT

Trading in options, as well as futures, has been shown to help forecasting movements in underlying securities. While put-call ratios have been used extensively in gauging the ‘fear’ of market about its further development, the option moneyness used in our study measures that fear more directly. Based on the idea of Fodor, Krieger and Doran (2011), we extend the formula of Bergsma, Csapi, Diavatopoulos and Fodor (2019) to show in this study that option moneyness exhibits a horizon effect, which says that in a longer horizon options forecast equity returns better than futures. Studying the Taiwan’s TXO index option, 5th in volume globally, we find option moneyness provides good information about index returns further in the future. Futures contracts, even as as a collector of information about the immediate future, perform not as good as option moneyness. Information on the longer horizons is only provided by option contracts of various expiration dates. With our analysis, information for longer horizons can be priced better, which elevates market thickness in general. Our results would benefit literature in understanding the price discovery function of options markets. The long horizon forecasting capability of option moneyness would also help investors in constructing effective hedges with options.

Keywords: Price discovery, moneyness, forecasting, Kalman filter. JEL Classification: C11, C33, E22, E44, G2, G31

* Correspond author at: Department of Banking and Finance, Kainan University, Taoyuan, Taiwan 33857. † Correspond author at: Department of Economics and Finance, Huazhong University of Science and Technology, China 430074. 1. Introduction

Futures, as well as options, markets have been argued to have provided useful information about their underlying markets. Those who have valuable information about future levels of underlying cash markets would prefer to engage trading in the futures markets (Stoll and Whaley, 1990; Antoniou, Koutmos, and Periclic, 2005) or options markets (Manaster and Renleman, 1982; Mihir, 1987; Easley, O’Hara, and

Srinivas, 1998). Particularly on the movements of general equity markets, many works were on the relationship between equity indices and their futures (Kawaller, Koch and

Koch, 1987; Stoll and Whaley, 1990; Hasbrouck, 2003), as well as the relationship between cash indices and index options (Joseph, 1988, Jens and Robert, 1990;

Matthew, 1999), have been studied extensively, what simultaneous relationship lies among futures, options and underlying markets has also been of empirical interests in literature (Fung and Chan, 1994; Frank and Monique, 1998; Booth, So and Tse, 1999).

Chakravarty, Gulen, and Mayhew (2004) found that on average, the information share of out-of-the-money (OTM) options is higher than that of at-the-money (ATM) options, suggesting that both leverage and liquidity are important in promoting price discovery. Motivated by the observation above we intend to examine in this study how forecasting future equity index can be improved with the help of an index option’s average moneyness (AM) across all contracts at various exercise prices of a given month. The improvement could be achieved, going through or together with a related index futures contract, either in a nearby or a distant month for the options and futures contract of interest.

Observations from options market are also related to the motivation of this study.

In general, open-interest based put-call ratio (PCR) is around 30% to 50% higher than the volume based one. Fodor, Krieger and Doran (2011) argued that individual call and

2 put open interest have predicting power about future stock return. Jena, Tiwari and

Mitra (2019) found, based on data from the Nifty Index options, most active contract globally and accounting for two thirds of world volume, in India, open interest PCR is an efficient predictor of index return for a horizon of 12 days and volume PCR is only good for 2.5 days. Their results are robust after controlling for information from the futures contract on the same underlying. The structure of the options market in Taiwan supports the argument above in further details. Since the Taiwan Futures Exchange

(TAIFEX), whose TXO index option is 5th in volume globally, launched in 2012 weekly expiring options on the side of existing monthly options, the volume for the former has been around 8 to 11 times that for the latter. But the daily open interest for the single-series weekly options is only about 20% higher than the over all open interest for the multi-series monthly options. So the volum-based PCR measure calculated daily is only one primarily gauging a short-horizon measure of ‘fear’.

Market-wide open-interest-based PCR would still contain about 45% the kind of long-horizon ‘fear’ represented by figures from the monthly options. For the weekly options the open-interest-based PCR is about 30% higher than the volume-based one, and that difference for the monthly options is about 35%. But the composition of open-interest-based PCR obviously makes it more of a long-horizon ‘fear’ measure.

The AM measure captures this exact spirit much better than PCR. One crucial distinction is that while PCR, whichever base is used, for the weekly contracts is higher than that for the monthly contracts, the weekly AM measure is on the contrary much lower than the monthly one. For the spot month put contract of the TAIFEX weekly index option, the open-interest-based AM is only 24.8% that of the corresponding monthly measure in terms of its distance above 1, whereas for the volume-based AM, the distance of weekly measure is 17.8% that of the monthly one.

In this comparison, the AM measure seems to have revealed the long-horizon ‘fear’

3 about market future better than PCR. It is in this regard that we investigate how index option AM can provide long-horizon information about the underlying cash index returns.

On the definition of AM, we adopt a modified version from Bergsma, Csapi,

Diavatopoulos and Fodor (2020). Our results in this study show that in a longer horizon, index options forecast underlying equity index better than index futures. AM of put options is better than that of call options in forecasting future index returns, and the pattern is more significant for contracts expiring in more distant months.

Particularly, option moneyness exhibits a horizon effect, which says that in a longer horizon options forecast equity returns better than futures. we find option moneyness provides good information about index returns further in the future. Futures contracts, even as as a collector of information about the immediate future, perform not as good as option moneyness. Information on the longer horizons is only provided by option contracts of various expiration dates. Along a separate dimension, we place the currently dominant weekly options in the same analysis framework as that for their horizon-rich monthly counterpart. In addition to differences documented by general market statistics, we find that the short-horizon nature of the market, discouraged by the difficulty of taking advantage of leverage benefits, contains little information about long-horizon cash index. Compared with Pan and Poteshman (2006), our study, using

AM instead, considers directly the issue of leverage benefits, which rewards information collection, by integrating trading volume or open interest with moneyness of individual strike prices. This construction allows us to compare information revelation across expiration dates and identify the effect of time horizon as suggested by Jena, et al. (2019).

Our results would benefit literature in understanding how to price long-horizon information better in the absence of perfect foresight. The market for derivatives

4 expiring more distantly can become more active as it attracts more participants attempting to take advantage of their horizon-dependent information. This improvement elevates eventually market thickness in general and helps investors in constructing effective hedges with derivatives. A brief review of existing literature is given in Section 2, followed by the introduction of our econometric models in Section

3. Empirical results are reported in Section 4, with robustness discussion given in

Section 5. Concluding remarks are given in Section 6.

2. Related literature

Related to our study is the three-way relationship, among cash, futures and options investigated in literature primarily along a short-horizon approach with one period forecast. PCR is considered primarily as an informative measure, which obviously stresses positions in nearby, at-the-money options contracts. When PCR is used to extract information in the options markets, attention is placed more on short horizon expectations. As Christoffersen and Diebold (1998) particularly indicates that long-horizon forecasts need to be handle differently from near-horizon ones, we attempt to address the three-way relationship along a longer horizon with the help of option moneyness which, available currently, summarizes information on market prospects further into the future. Option moneyness, reflecting the extent of leverage assumed by option traders, was argued by Ge, Lin, and Pearson (2016) as a reason why options could predict stock returns.

A number of works find that futures prices lead spot prices and contribute more to the price discovery process (Chan, 1992; Frino et al., 2000). It is often argued that price discovery occurs first in a market where informed traders profit most from their information. Futures markets offer higher leverages, lower trading costs, and fewer short-sale restrictions than spot markets (Black, 1975; Kawaller et al., 1987; Easley et

5 al., 1998). As informed traders prefer futures market for the reasons above, futures market responds to new information ahead of, and therefore can help predicting, spot markets (Finnerty and Park, 1987; Kawaller et al., 1987; Harris, 1989; Stoll et al., 1990;

Chan, 1992). Price movements in futures lead stock indices in mature markets such as

S&P 500 (Kawaller et al., 1987; Ghosh, 1993; Stoll et al., 1990), Nasdaq 100

(Hasbrouck, 2003), DAX (Booth et al., 1999) and FTSE 100 (Stoll et al., 1990;

Abbyankar, 1995; Kavusssanos, Visvikis and Alexakis, 2008).

On the other hand, there is literature presenting the informative role of options on its underlying market. A study by Easley, et al. (1998) indicates that option volume by itself can be informative and information-based trading prefers options markets.

Billingsley and Chance (1988) study the predictive power of PCR and how it gives direction of market. This strand of literature also shows both positive and negative option volumes have predictive power for stock price movements (Chakravarty, et al.,

2004; Blau and Wad, 2013). Blau, Nguyen, and Whitby (2014) compare PCR and

Option-to-Stock Volume Ratio (OSR) and find that the former has better predictability about future stock returns at a daily level while the OSR works better at weekly and monthly levels. Similarly, Bandopadhyaya and Jones (2008) suggest PCR has better explanatory power about stock returns than the Volatility Index (VIX). Pan and

Poteshman (2006) indicate that open-buy PCR for equity options are informative about the next-day stock return, and that the informational content is significantly stronger for OTM options. Using the the same model, Chang, Hsieh, and Lai (2009) investigate the information content of options trading in Taiwan and show that option volume

(PCR), as a whole, carries no valid information on spot index return.

3. Econometric models

6 Extending the work of Bergsma, et al. (2020), we adopt a modified measure of weighted average moneyness of option. We intend to compare the informativeness of this measure, against that of futures, on future stock returns. Our modified average moneyness (AM) measure for calls is different from that for puts to stress the leverage implications of the measure. Our measures, for a given contract month, are defined as

K OICall AveCallMoney   i i (1) i F TotalCallOI And

F OIPut j AvePutMoney   j K j TotalPutOI (2) where

TotalCallOI  OICallm m

TotalPutOI  OIPutn n and Ki and Kj denotes respectively call and put exercise prices, F stands for price of a futures contract on the same underlying as the options of interest. OICalli, OIPutj are open interest of respective calls and puts, whereas TotalCallOI and TotalPutOI are total market interest figures for calls and puts of the contract month of interest.

Our measures differ from that of Bergsma, et al. (2020) in several ways. First, in calculating moneyness we use the ratio between the level of exercise prices and the price of a related futures contract. Prices, and their related patterns, of options contracts, when they share a common underlying security or commodity with an active futures contract, often correlate heavily with the prices of that futures contract prior to expiration. For the purpose of identifying how options moneyness reflects the information about leverage or volume of the options contract of interest, we decide to use the concurrent futures price to measure relatively the distance of a certain exercise

7 price from the price level where ‘money’ is at. Another reason for using the futures price instead of the cash price is that rules to halt trading in volatile market sessions could cause the cash price to deviate from its equilibrium level.

The second difference is that the average moneyness in our formula is not weighted in calculation by a ‘midpoint option price’ (MP) variable as in the definition of Bergsma, et al. (2020). This modification is there to give more wight to the OTM options, as adding MP would naturally add weights to ATM and in-the-money options due to their high MP’s. Various works in literature (Chakravarty, et al., 2004; Xing,

Zhang, and Zhao, 2010; Kehrle and Puhan, 2015; Kang, Kim, and Lee, 2018) has suggested that OTM options convey relatively more information about future cash returns. Our measure would obviously produce values of AM different from those of

Bergsma, et al. (2020), but we argue that, in identifying the price discovery role of options prices at various exercise prices, our modification does not lose much of its generality.

The third difference is that in calculating weights for moneyness of each exercise price we use open interest instead of trading volume to focus on the horizon effect of

AM. This modification addresses implications from the comparison, which is indicated earlier, between volume- and open-interest-based PCR. Using open interest to weight the moneyness of individual strike prices makes the calculated measure more relevant to long-horizon related information. The existence and improvement of measure like this contribute to increasing thickness of markets.

In estimating how one market instrument predicts another, price discovery analysis is widely used. One approach is the common factor model with its measure coming primarily from either the permanent-transitory (PT) model of Gonzalo and

Granger (1995) or the information share (IS) model of Hasbrouck (1995). The former measure provides an unique and all-inclusive gauge of price discovery which the latter

8 lacks. But the latter measure incorporates both the error-correction coefficients as well as the innovation covariances, whereas the former considers error-correction. Lien and

Shrestha (2009) propose a modified information share (MIS) measure to resolve the uniqueness issue of IS. However, both of the measures above are still in themselves static and regressive. Also static are the approaches of error-correction (Ghosh, 1993;

Tse, 1995; Booth, et al., 1999; Lehmann, 2002) and volatility-spillover (Carchano and

Pardo, 2009; Zhong, Darrat and Otero, 2004; Salm and Schuppli, 2010; Arnade and

Hoffman, 2015; Miao, Ramchander, Wang and Yang, 2017).

In order to evaluate performances of predicting capability, we adopt the view of

Hamilton (1989) to address possible discrete shifts in regime in handling potentially nonstationary time series of futures, options as well as equity indices. Related issues of time-varying forecasts on financial time series are thoroughly investigated by Kalman filter (Barrasi, Caporale and Hall, 2005; Lettau and Nieuwerburgh, 2008; Ling and

Stone, 2016; Monach, Petrella and Vendetti, 2020). Lagos-Alvarez, Padilla, Mateu and

Ferreira (2019) conclude that Kalman filter is capable of capturing temporal dependence as well as spatial correlation structure through its state-space equations, and it gives statistical inferences in both parameter estimation and prediction. Caporale,

Ciferri and Giradi (2014) estimated price discovery under the PT specification for oil spot and futures with a standard Kalman filter model. A special state-space model other than the standard Kalman filter was employed to deal with prediction of long-horizon stock returns by Conrad and Kaul (1988) where nonstationary and unoberservable expected returns are properly estimated and extracted expected returns predict realized returns well. Modified alternatives of the original Kalman filter are therefore considered for the purpose of economics as well as econometrics.

For our study, we follow the state-space model for time-varying expected returns of stocks proposed by Conrad and Kaul (1988). For the comparison of informativeness

9 of futures against that of options, we model two separate measurement equations, one for futures price and the other for AM of options, where each contains a permanent and a transitory component according to the specification of Conrad and Kaul (1988). For the futures , we have the following specification

S t  F t  Z f ,t (3)

F t  F t-1   f ,t (4)

   (5) Z f ,t  f Z f ,t-1 x f ,t  f ,t

where 푆 t is logarithm of detrended stock price, with Ft as the logarithm of detrended futures price and permanent component, whereas Zf.t , equivalent to futures basis, is the transitory component in the measurement equation for futures. In (5), xf,t is an unobservable state variable governing the transitory component Zf.t. εf.t and ηf.t are assumed to be uncorrelated with each other and both have zero means and finite variances. Also, it is assumed that |Φf|≤1. Similarly, for options the specification would be

S t   AM t  Z o,t (6)

AM t  AM t-1   o,t (7)

    (8) Z o,t o Z o,t-1 xo,t o,t

where AMt is the permanent component, with Zo,t as the transitory component. xo,t in (8) is also an unobservable state variable, governing the transitory component Zo.t. εo.t and

ηo.t are assumed to be mutually independent, with zero means and finite variances. It is also assumed that |Φo|≤1.

Rewriting rt as the differenced St, we would have measurement equations for futures in the following form,

10 r f , t  Z f ,t  Z f ,t1   f ,t . (9)

A transition equation for the state variable xf,t can take on the form of

x f , t   f x f , t  1   f ,t . (10)

Thus, given in addition that |λf|<1, the state-space system consisting (8)~(10) is one for estimating index returns from futures returns and can be be estimated with Kalman filter. Similarly, (6) can be expressed with respect to rt and carry the form of

r o, t  Z o,t  Z o,t1   o,t , (11) along with (7) and the transition equation of

xo, t   o xo, t  1   o,t (12)

, assuming in addition that |λo|<1, to constitute a state-space system for estimating index returns from changes of options average moneyness.

4. Estimations

Our data cover the period from January 2007 to December 2017. Included are daily index options closing prices and trading volume, for all strike price levels of the monthly-expiring contracts listed on TAIFEX, from each of the three consecutive near-term delivery months. Index futures data are also from TAIFEX, including daily closing prices, up to 6 trading days from expiration, from the contract with the nearest expiration. The futures price series is constructed such that the price from the contract with the next expiration is used in place of the nearest-month price for the last 5 trading days prior to current expiration. In addition, daily closing levels of the underlying index, Taiwan Stock Exchange Capitalization Weighted Stock Index

(TAIEX), are obtained from the Taiwan Stock Exchange (TWSE).

11 【Table 1】

Table 1 gives the descriptive statistics of AM constructed from volume as well as open interest, separately for calls and puts from the monthly TXO contracts. On average AM measures for puts, compared with those for calls, are 33.7% more away from the exact at-the-money mark of 1, which suggests that participants in this option market are more inclined to hold OTM puts than OTM calls, reflecting how leverage benefits intensify the ‘fear’ motive. Against the spot month contract, the average volume-based call moneyness for the first near-term contract is 99% further away from

1, whereas for the next near-term contract the measure is another 112% further away.

This phenomenon suggests that participants in the options markets with further expiration are also more aggressive in betting on the OTM options due to their leveraging advantage. In contrast, the differences in the aggressiveness for puts are not as drastic as those for calls. In terms of standard errors on AM, those for puts are around 5.5% higher than those for calls, while those based on open interest are 34% higher than those based on volume. These statistics summary is also consistent with the horizon effect of options open interest as indicated in Jena, et al. (2019).

【Table 2】

Table 2 reports the performance evaluation for both the futures and the options state-space systems. Root mean squared errors RMSEf,r on estimating the daily index returns, based on the futures related system (8)~(10), are given for comparisons is defined as

T 1 ~ 2 RMSE f ,r   r f , t  rt  (13) T t  1 whereas RMSEo,r,i,

12 T 1 ~ 2 RMSEo,r,i   r o, t, i  rt  , i=1,2,3 (14) T t  1 denote respectively for the options related system consisting (7), (11) and (12) the root mean squared errors out of estimations on the three near-term option contract months.

The RMSE, according to (13), for estimated TAIEX returns with index futures is shown in Table 2 to be around 30% lower than those for options. Although the measures for more distant option contracts are higher and closer to futures’ RMSE, the filtered results still suggest that index futures provides an informative role uniformly better than all index options. Even in terms of the filtering process itself, using index futures gives lower estimated standard errors for the transitory component in index returns, σ(η), as well as lower σ(ξ), the standard error for the unobservable state variable xf,t. However, across the three near-term options contracts these indicators suggest that less distant contracts perform better, due possibly to relatively more variation in moneyness across time in options markets with further expiration. As open-interest-based AM varies more than the volume-based ones, the filtering performance is accordingly better for the volume-based AM. For a similar reason, results for AM on puts are worse than those for calls, due apparently to the higher variation of AM. Compared with the filtering performance of futures, the level of imprecision on state variable for open-interest-based AM on puts is twice as large.

To evaluate the relative performance in forecasting cash index returns rt, we compare the RMSE according to

1 N 2 RMSE   r N ,  r N  , 10,20,30, (15) N N  1

as well as mean absolute percentage errors (MAPE) following

13 N  1  r N r N, MAPE   , 10,20,30, (16) N N  1 r N on the differences between realized and fitted rt 10, 20 and 30 days ahead. While the first measure of two above reports the raw levels of fitting errors, the second one gives a characterization of how close the fitted returns are to the realized ones proportionately. The process is carried out by first fitting a window of 500 days and followed by rolling over 15 days each time. The averages across all the fitting windows are reported in Table 3.

【Table 3】

The out-of-sample forecasts suggest that AM measures compiled from more distant option contracts are more relevant to cash index returns in the future. Although the forecasting errors, with index futures, for index returns 10, 20 and 30 days beyond the estimation period are lower than those with spot month option moneyness, errors from the more extended forecasts tend to be lower with moneyness out of more distant options contracts. The volume-based AM for calls, at a 10-day forecast window, produces a level of RMSE and MAPE around 10% to 20% above those forecasted with futures. But when we lengthen the window to 30 days and move from calls to puts, the forecasting error from the most distant option is already lower than that from the index futures by about 10%. This pattern is more pronounced for open-interest-based AM.

Although forecasting errors range from 17% to 26% above those for futures for the shortest horizon, they fall to 12% below futures for the nearest calls and 17% for the furthest puts at a forecasting horizon of 30 days. This indicates that option AM measures from the more distant months forecast index returns further in the future better index futures, especially for AM on more distant puts, suggesting again a horizon effect reflected by AM on the fear for future.

To further clarify the relationship between option moneyness and index futures,

14 we construct another state-space system, recognizing that trading of index options is primarily connected to trading of the immediately expiring index futures contract. So the system includes

AM t   F t  Z A,t (17)

    (18) Z A,t A Z A,t-1 xA,t A,t

x A, t   A x A, t  1  A,t (19) and

Δ      (20) AM f ,t a f , t Z A,t Z A,t1  A,t .

Although AMt contains Ft according to (1) and (2), we would argue that in a daily setting the positions at various moneyness are all established prior to the realization of

Ft, the closing price, and hence AMt would only be a function of the expectation of Ft instead of being affected by it directly.With (18) and (20) as the measurement equation and (19) as the transition equation, we can extract parameters about how moneyness prices transitory information which index futures does not. Results of this system is given in Table 4.

【Table 4】

Filtering results reported in Table 4 shows that futures price is more informative on AM than on cash index, which is consistent with Fleming, Ostdiek and Whaley

(1996) and Nam, Oh, Kim and Kim (2006). In evaluating how explanatory futures price is about cash index, RMSE of estimating index returns from futures returns in

Table 2 is about 2.3 times the standard error of index returns. While in Table 4, RMSE of estimating AM changes from futures returns is only about 54% of the standard error for AM from the nearest call option contract. Both RMSE and its ratio against standard errors of AM increase with the expiration date, suggesting that futures price works

15 better in providing current information. Similar pattern follows for puts, as well as for open-interest-based AM, with all measures being larger. Information not contained in futures price, as captured by ZA,t in (17), reflects leverage-induced trading and cannot be properly extracted by from futures prices when we attempt to measure AM with Ft.

The argument above can also be reinforced by the results on σ(η), which gives the variability of the part in (17) unexplained by Ft. From the state-space system (3)~(5), where index returns are estimated with futures returns, the residuals in (5) have a standard error 58% that of the index returns. In this system where ΔAM is estimated with futures returns, σ(η) is about 32% above the standard error of AM, indicating Ft has left much more of AM unexplained, relatively to what Ft is able to explain about St.

The proportion of unexplained residuals in Table 4 grows to be 46% above the standard error of AM when open-interest-based moneyness is estimated. This unexplained variability also goes up with distant to expiration suggesting a possibility that Ft cannot influence the leverage incentives on the trading of OTM options, especially in the case of more distant contracts.

【Table 5】

To determine whether futures or option predicts index return better, we carry out first regressions of index returns, one to sixty days ahead, on current futures returns and control variables, as in

          VIX  ,i 1,...60 (21) .r s, t  i 0 1 r s, t 2 r f , t 3lnVol s, t 4 t  t with rs,t being the current index return, rf,t the current futures return, VIXt the current volatility index calculated daily from options and lnVols,t the logarithm of daily trading volume on all common equities listed on the Taiwan Stock Exchange. Then we conduct a similar set of regressions by replacing the futures returns with AM, separately for calls and puts from all near-term contract months, as follows,

16       AM     VIX  ,i 1,...60 . (22) .r s, t  i 0 1r s, t 2 t 3lnVol s, t 4 t  t

The results are reported in Table 5 and it is shown that while rf,t is significantly informative about rs,t about a week or a month ahead, volume-based AM for calls predicts index returns in various days, twice as many as those for futures returns, in the future. Although the spot call contract would expire on average in about two to three weeks from now, an increase of average moneyness for call contracts today is related to a drop in index returns many days ahead, even after the spot call contract expires.

One possibility is that some informed investors are trading on the private information they possess. Specifically, information of those actively trading call contracts is bearish as a larger AM predicts a negative index return. The coefficients are significantly negative for days between one to two weeks before maturity, and again in about a month later than that, of the contracts considered, indicating the informed are trading against their bearish information about the cash index. One could argue that the increase in AM could be a result of drop in futures today, but the second column of

Table 5 shows that futures returns are not predicting negative index returns for the days when index returns are forecasted with significance. Another factor to cause AM to increase could be spread trading, in that situation we would observe synchronous significance across calls and puts or across expiration, which is not the case in Table 5.

For the call contracts, most of the significant incidences suggest the informed assumes a seller’s position, taking advantage of the strong optimism from buyers. The concentration of incidences for the first near-term contract appear about four to five weeks after the spot contract. Low trading volume is obviously responsible for the lack of significant incidences for the second near-term contract, and they do appear in days much later than the spot contract.

For the spot put contract, the incidences happen about 10 days after the

17 corresponding call contract. The positive coefficients indicate also that the informed are taking a seller’s position, covering the over-pessimistic orders from buyers. But there are several strongly significant incidences of negative coefficients clustered on about a month from the current trading day, suggesting the informed are actively buying puts based on bearish private information. One difference between put and calls is that the average moneyness of distant put contracts are much less informative.

Another difference is that trading of puts seems to exhibit a longer horizon than that of calls.

According to Jena, et al. (2019), futures contracts could serve as a collector of information about the immediate future, our findings in Table 5 show that option trading provides more information for the very near future in addition to information at least a month further into the future.

【Table 6】

We also examine AM based on weights according to open interest of various strike prices. Similar regressions based on (22) are carried out and results are given in

Table 6. The number of significant incidences is much smaller across the board, possibly due to the reality that open interest in general is low relative to trading volume, which is also reflected by the high standard deviation observed for open-interest-based

AM as reported in Table 1.

Although results in Table 6 seem to suggest that open-interest-based AM is not as informative as the volume-based one, the horizon effect is obviously stronger.

Compared with Table 5, the first cluster of significant incidences for spot calls appears about 10 days later, coinciding with the second cluster when AM is volume-based. For the first near-term put contract, there is a cluster appearing at around the contract’s expiration, which is absent in corresponding volume-based results. It’s also worth noticing that all of the significant incidences in Table 6 indicate the informed are taking

18 a seller’s position. The fact that the informed utilize their information like a contrarian over longer horizons reflects that option trading is unique in dealing with the fear of future.

5. Robustness discussions

To validate our results in separating short- and long-horizon effects of moneyness, we proceed to change the horizons of the options contracts used in calculating AM. We use weekly expiring options contracts listed on TAIFEX, which were introduced in

November 2012, to replace the monthly expiring ones. We compiled closing prices for

255 weekly options between 2013 and 2017 and calculated AM measures. Due to the nature of weekly option, there is only one day, namely Wednesday, every week that there are concurrently two option contract series, one each for calls and puts, traded together in the same session. One series expires at the close of that day and the other is only listed after the open of the day. Although there is only one overlapping day, we can still compare AM measures between the two. Therefore we still construct a space-system over the 255 weeks, and except for Wednesday there is always one weekly spot contract estimated. The weekly near-term contract exists only on

Wednesdays, when the then weekly spot contract is to expire at the end of the day. So each weekly spot contract starts on Thursdays and ends on Wednesdays. The results for filtering performances are presented in Table 7.

【Table 7】

For weekly options, results given by Table 7, fitting errors are substantially higher, possibly due to the facts that the number of states is reduced by about 57%, and also to the AM effect is greatly limited by the short expiration period. RMSE measures for spot contracts are about 25% above those for the monthly contracts and also 88% higher than that for index futures. Moving from the weekly spot to the weekly

19 near-term contract, the number of states drops by 75% and filtering error level increases by 30%. However, compared with the first near-term monthly contract

RMSE is already 105% higher. The informativeness of distant options seen in the case of monthly contracts disappears, suggesting that OTM trading contains less information-based trading as benefits of leverage may not realize before the weekly option expires.

【Table 8】

To validate the conjecture given above, we repeat the calculation of the performance for weekly options in forecasting index returns, but limited to 10 and 20 days beyond the estimation period. Results are reported in Table 7. For the weekly spot contracts, forecasts are carried out for the non-Wednesday weekdays, whereas forecasts are only run on Wednesdays for weekly near-term contracts. The horizon effect, which is exemplified by distant options’ providing precision superior to index futures in long-horizon forecasts of index returns, disappears as we find AM measures from more distant weekly options produce much higher forecasting errors. As we conjecture above, the short expiration seems to be the most important difference between weekly and monthly options. The lack of leverage benefits to be anticipated in investing in

OTM options is on the one hand the reason for the absence of the horizon effect, and supports on the other hand why AM of OTM options exhibit a horizon effect found in

Table 3 and Table 4.

【Table 8】

For comparisons, we also repeat the forecasting regressions as given in Table 5 by replacing monthly AM measures with weekly ones, limiting the forecasting horizon to

21 days. Almost no significant incidences can be found, suggesting that the market processes only same-day information. All incidences are seller-induced, with two for calls and one for puts. The directions and timing for these incidences coincide roughly

20 with those in Table 6. Except for the three incidences, weekly-based AM in Table 9 exhibits no horizon effect, as argued previously. The informed seem to have chosen only the monthly, whose volume is only half that of the weekly contract, contract to profit from their information.

6. Conclusions

Trading in futures, as well as options, has been shown to help forecasting movements in underlying securities. Based on the idea of Chakravarty, et al. (2004), we extend the formula of Bergsma, et al. (2019) to show in this study that option moneyness exhibits a horizon effect, which says that in a longer horizon options forecast equity returns better than futures. Option moneyness provides information about index returns further in the future. Futures contracts, even as as a collector of information about the immediate future, perform not as good as option moneyness.

Information on the longer horizons is only provided by option contracts of various expiration dates.

The kind of horizon effect mentioned above depends on analysis of trading in markets for options, or other derivatives, expiring in different times. Although more distant contracts are traded less actively, the information contained there essentially follows different distributions. So studying the horizon effect expands the dimension of our information vector. The advantage brought by the comparisons of short- and long-horizon trading is a good alternative to overcoming the limit of dimension by continued modification. For this reason, we need to investigate the distinctions between the dominant weekly options and their horizon-rich monthly counterpart.

Many derivative markets do not have active trading for instruments with distant expiration, partly due to the lack of framework where risks and returns can be properly

21 priced according to informative market measures1. In this regard, our results would benefit literature in understanding how to price long-horizon information better in the absence of perfect foresight. The market for derivatives expiring more distantly can become more active as it attracts more participants attempting to take advantage of their horizon-dependent information. This improvement elevates eventually market thickness in general. and helps investors in constructing effective hedges with derivatives.

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25 Table 1 Descriptive Statistics of Options Moneyness, 2007-2017 Daily index options closing prices and trading volume, for all strike price levels of the monthly-expiring contracts listed on TAIFEX, from each of the three consecutive near-term delivery months. AveCallMoney and AvePutMoney, daily average moneymess measures are calculated according to (1) and (2). The underlying index is Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) compiled by the Taiwan Stock Exchange (TWSE). Mean Median Std.Err. Max Min

Volume-based AveCallMoney Spot 1.0271 1.0205 0.0282 1.3343 0.9769 1st Near-Term 1.0540 1.0470 0.0299 1.2545 0.9754 2nd Near-Term 1.1145 1.0998 0.0745 1.6692 0.0000 AvePutMoney Spot 1.0375 1.0335 0.0272 1.3361 0.9381 1st Near-Term 1.0720 1.0644 0.0465 1.7827 0.8564 2nd Near-Term 1.1480 1.1457 0.0804 1.6370 1.6459

Open Interest-based AveCallMoney Spot 1.0435 1.0343 0.0443 1.4263 0.8921 1st Near-Term 1.0663 1.0517 0.0747 1.9278 1.4685 2nd Near-Term 1.1145 1.1008 0.1056 1.9070 0.8182 AvePutMoney Spot 1.0657 1.0674 0.0458 1.2526 0.8493 1st Near-Term 1.0888 1.0832 0.0680 1.4686 0.6158 2nd Near-Term 1.1332 1.1559 0.1090 1.6012 0.6809

26 Table 2 Performances of Kalman Filter Estimations Measuring index returns with futures price and option AM

Root mean squared errors on estimating the daily index returns, based on the futures related system (8)~(10), are given for comparisons is defined as T 1 ~ 2 RMSE f ,r   r f , t  rt  T t  1 whereas T 1 ~ 2 RMSE f ,r,i   r f , t, i  rt  T t  1 , i=1,2,3, denote respectively for the options related system consisting (7), (11) and (12) the root mean squared errors out of estimations on the three near-term option contract months. RMSE σ(η) σ(ξ) Futures 0.0272 0.0108 0.0136

Volume-based Moneyness

AveCallMoney Spot 0.0298 0.0116 0.0153 1st Near-Term 0.0325 0.0120 0.0191 2nd Near-Term 0.0385 0.0127 0.0226 AvePutMoney Spot 0.0327 0.0121 0.0187 1st Near-Term 0.0366 0.0125 0.0223 2nd Near-Term 0.0413 0.0133 0.0258 Open-interest--based Moneyness

AveCallMoney Spot 0.0317 0.0120 0.0159 1st Near-Term 0.0329 0.0131 0.0222 2nd Near-Term 0.0368 0.0144 0.0269 AvePutMoney Spot 0.0349 0.0136 0.0214 1st Near-Term 0.0397 0.0159 0.0265 2nd Near-Term 0.0426 0.0173 0.0287

27 Table 3 Forecasting Performances of Futures and Options RMSE according to

1 N 2 RMSE   r N ,  r N  , 10,20,30, N N  1

as well as mean absolute percentage errors (MAPE) following

N  1  r N r N , MAPE   , 10,20,30, N N  1 r N on the differences between realized and fitted rt 10, 20 and 30 days ahead. The process is carried out by first fitting a window of 500 days and followed by rolling over 15 days each time. The averages across all the fitting windows are reported.

τ=10 τ=20 τ=30 RMSE MAPE RMSE MAPE RMSE MAPE Futures 0.0354 183.28 0.0439 235.84 0.0622 367.51

Volume-based Moneyness

AveCallMoney Spot 0.0407 210.96 0.0450 237.89 0.0563 335.05 1st Near-Term 0.0411 216.97 0.0443 227.46 0.0545 328.45 2nd Near-Term 0.0426 224.85 0.0459 239.86 0.0570 343.26 AvePutMoney Spot 0.0419 222.11 0.0455 238.07 0.0547 321.31 1st Near-Term 0.0423 228.29 0.0449 230.29 0.0535 315.16 2nd Near-Term 0.0435 237.62 0.0451 244.56 0.0531 318.77 Open-interest-based Moneyness

AveCallMoney Spot 0.0415 222.53 0.0468 246.64 0.0552 327.63 1st Near-Term 0.0421 224.11 0.0465 243.91 0.0536 314.02 2nd Near-Term 0.0437 235.68 0.0477 250.54 0.0540 309.72 AvePutMoney Spot 0.0428 224.26 0.0475 248.27 0.0539 311.31 1st Near-Term 0.0446 247.13 0.0459 232.45 0.0527 295.45 2nd Near-Term 0.0449 259.34 0.0461 230.88 0.0521 288.30

28 Table 4 Performances of Kalman Filter Estimations, Measuring AM Root mean squared errors on estimating the change of AM, based on the futures related system (18)~(20), are given for comparisons is defined as T 1 ~ 2 RMSE f ,a   a f , t  at  T t  1 RMSE σ(η) σ(ξ)

Volume-based Moneyness

AveCallMoney Spot 0.0152 0.0372 0.0239 1st Near-Term 0.0275 0.0543 0.0352 2nd Near-Term 0.0747 0.0784 0.0454 AvePutMoney Spot 0.0177 0.0384 0.0255 1st Near-Term 0.0314 0.0558 0.0369 2nd Near-Term 0.0771 0.0807 0.0480 Open-interest-based Moneyness

AveCallMoney Spot 0.0231 0.0646 0.0349 1st Near-Term 0.0476 0.0988 0.0509 2nd Near-Term 0.0918 0.1376 0.0654 AvePutMoney Spot 0.0273 0.0669 0.0358 1st Near-Term 0.0431 0.0961 0.0522 2nd Near-Term 0.0989 0.1418 0.0693

29 Table 5 Horizon Predicting Power of Option Moneyness, Volume-based, 1- to 60-day ahead We carry out OLS regressions, for calls and puts from all near-term contract months, as follows, r s, t  i   0   1 r s, t   2 AMt   3 lnVol s, t   4VIXt   t ,i 1,...60 with rs,t being the current index return, rf,t the current futures return, VIXt the current volatility index calculated daily from options and lnVols,t the logarithm of daily trading volume on all common equities. Only β2 are reported. For comparison, we also run a similar regression by replacing AMt with rf,t, the futures return. Calls Puts Futures i Spot 1st Near-Term 2nd Near-Term Spot 1st Near-Term 2nd Near-Term 1 0.1528(0.0594)** 1.8750(1.1697) 0.1241(1.2021) -0.5518(0.3568) 1.0145(1.0848) 0.1583(0.5311) 0.0626(0.2884) 2 0.0441(0.0596) 2.0515(1.1718) 0.2041(1.2043) -0.3077(0.3576) 1.4400(1.0867) -0.4327(0.5320) 0.0925(0.2892) 3 0.1513(0.0593)** 0.7870(1.1725) 1.3568(1.2041) -0.1553(0.3577) 0.6684(1.0870) -0.1798(0.5321) 0.0790(0.2892) 4 0.0148(0.0596) 0.9928(1.1721) 0.9499(1.2039) -0.2521(0.3576) 1.3513(1.0865) -0.6636(0.5318) 0.4428(0.2890) 5 -0.0358(0.0595) -0.6960(1.1713) -0.5593(1.2031) -0.2753(0.3573) 1.7545(1.0855) -0.4648(0.5314) 0.3938(0.2879) 6 -0.0397(0.0595) -3.5781(1.1695)** -3.3438(1.2015)** 0.0696(0.3575) 1.8185(1.0857) 0.0461(0.5315) 0.6787(0.2886)** 7 -0.1638(0.0594)** -3.1434(1.1710)** -2.6187(1.2032)* 0.1207(0.3578) 1.9103(1.0869) 0.4749(0.5319) 0.8097(0.2887)** 8 0.1076(0.0596) -1.1963(1.1727) -2.6096(1.2036)* -0.8670(0.3575)** 2.5451(1.0869)** 0.1015(0.5321) 0.2405(0.2892) 9 0.0835(0.0596) -1.2521(1.1729) -2.7498(1.2038)* -0.6896(0.3578) 1.0663(1.0882) 0.6289(0.5321) 0.2075(0.2893) 10 -0.0569(0.0596) -3.8049(1.1710)** -2.7015(1.2040)* -1.1954(0.3573)** 2.3250(1.0877)* 0.4547(0.5322) -0.1065(0.2894) 11 0.0568(0.0596) -3.8304(1.1703)** -1.7926(1.2040) -0.5915(0.3577) -0.8470(1.0879) 0.2833(0.5320) 0.3699(0.2891) 12 -0.1173(0.0596)* -4.5339(1.1702)** -1.9278(1.2047) -0.4708(0.3580) -0.7867(1.0887) 0.2225(0.5323) 0.5403(0.2892) 13 0.0560(0.0595) -3.8216(1.1694)** -2.3361(1.2027) -0.8479(0.3572)** -0.1520(1.0872) 0.0252(0.5316) 0.5176(0.2888) 14 0.0050(0.0595) -4.1537(1.1688)** -1.8892(1.2027) -0.3820(0.3574) 0.2269(1.0870) -0.2474(0.5314) 0.2054(0.2890) 15 -0.0976(0.0596) -3.6833(1.1716)** -2.2324(1.2048) -1.0702(0.3576)** 0.0321(1.0891) -0.1106(0.5325) 0.2013(0.2896) 16 0.1165(0.0596) -2.2915(1.1737) -2.2909(1.2054) -0.0736(0.3584) 0.8067(1.0896) 0.1688(0.5327) 0.2869(0.2897) 17 0.0536(0.0597) -1.9227(1.1738) -0.0944(1.2061) -0.3182(0.3583) 1.5446(1.0895) 0.0507(0.5327) 0.3748(0.2897) 18 0.0393(0.0596) -0.7691(1.1730) -0.4444(1.2046) -0.8978(0.3575)** 2.0686(1.0879) -0.1192(0.5320) 0.3245(0.2894) 19 -0.0123(0.0597) 0.2175(1.1744) -0.3839(1.2061) -0.6593(0.3582) 1.9665(1.0893) 0.6278(0.5325) 0.5307(0.2896) 20 0.0799(0.0595) -0.9721(1.1719) 0.4225(1.2036) -0.0215(0.3577) 3.4026(1.0859)** -0.0123(0.5316) 0.5422(0.2891) 21 0.0082(0.0597) -0.2059(1.1752) 1.1534(1.2066) -0.2256(0.3586) 2.6511(1.0896)** -0.2236(0.5330) 0.1489(0.2901) 22 -0.1031(0.0596) -0.0980(1.1742) 0.0039(1.2058) -0.8228(0.3579)* 1.4343(1.0896) 1.3999(0.5318)** 0.2586(0.2898) 23 0.0156(0.0597) -1.0534(1.1757) -1.3365(1.2071) -0.7122(0.3585)* 2.4551(1.0903)* 0.9522(0.5329) 0.2580(0.2902) 24 -0.0770(0.0597) -0.6609(1.1759) -0.3874(1.2075) -0.8059(0.3585) 2.3610(1.0905)* 1.5762(0.5324)** 0.6184(0.2900)* 25 -0.0667(0.0597) -1.1483(1.1754) -3.0542(1.2058)** -0.6427(0.3585) 3.4870(1.0892)** 1.3996(0.5325)** 0.5002(0.2901) 26 0.0354(0.0596) 0.1279(1.1737) -2.2369(1.2043) -0.2573(0.3581) 2.4048(1.0885)* 0.7908(0.5320) 0.5178(0.2896) 27 0.1486(0.0596)** -1.6251(1.1744) -1.3887(1.2058) -0.4061(0.3583) 1.8570(1.0900) 0.5123(0.5329) 0.6415(0.2897)* 28 -0.0718(0.0597) 1.2844(1.1756) -0.1895(1.2071) -0.7189(0.3584)* 1.6812(1.0910) 0.5389(0.5330) 0.2396(0.2902) 29 0.0180(0.0597) -0.3473(1.1756) -2.5413(1.2059)* -0.0478(0.3586) 0.3021(1.0913) 0.6463(0.5328) 0.4024(0.2901) 30 -0.0265(0.0597) -2.4686(1.1757)* -1.7076(1.2074) -0.0107(0.3589) -2.0349(1.0916) 0.5182(0.5333) -0.0518(0.2905) 31 0.1655(0.0597)** -3.3149(1.1760) -3.4787(1.2070)** -0.7584(0.3588)* -3.8627(1.0907)** -0.0468(0.5338) 0.2318(0.2906) 32 -0.0219(0.0597) -4.1564(1.1748) -3.3109(1.2031)** 0.0508(0.3590) -4.0008(1.0904)** 0.2631(0.5336) 0.3355(0.2905) 33 -0.2015(0.0598)** -1.4086(1.1777) -2.4239(1.2080)* -0.1423(0.3592) -3.5526(1.0915)** 0.4862(0.5338) 0.4461(0.2906) 34 0.1548(0.0597) -0.7866(1.1776) -3.3934(1.2067)** -0.2776(0.3590) -1.3536(1.0931) 0.1380(0.5337) 0.4624(0.2905) 35 0.0266(0.0597) -0.5589(1.1762) -1.7262(1.2067) -0.0365(0.3586) -0.3704(1.0922) 0.4085(0.5330) 0.1602(0.2903) 36 0.0001(0.0598) 0.1526(1.1773) -1.0559(1.2080) -0.0468(0.3589) -1.4590(1.0929) 0.0949(0.5335) -0.1511(0.2905) 37 0.0249(0.0597) -3.2032(1.1746)** -1.3293(1.2068) -0.2263(0.3586) -0.3362(1.0922) -0.1033(0.5331) 0.3848(0.2902) 38 -0.0940(0.0597) -3.9260(1.1733)** -0.8987(1.2065) -0.4910(0.3583) 0.3664(1.0917) 0.1458(0.5329) 0.2068(0.2902) 39 -0.0813(0.0597) -2.4293(1.1746)* -0.1025(1.2064) 0.1409(0.3584) 0.9975(1.0915) 0.3201(0.5328) 0.0849(0.2902) 40 0.0443(0.0596) -0.9144(1.1750) -1.4288(1.2057) 0.0395(0.3583) 0.9655(1.0911) 0.7164(0.5325) 0.6064(0.2898)* 41 -0.0742(0.0597) -4.2558(1.1726) -2.6710(1.2053)* -0.4462(0.3583) -0.3733(1.0916) 0.4077(0.5330) 0.3123(0.2901) 42 0.0580(0.0596) -1.1564(1.1747) -3.0019(1.2044)** -0.3219(0.3582) 1.9576(1.0904) 1.0730(0.5324)* 0.0418(0.2900) 43 0.0526(0.0596) 0.6189(1.1747) -2.9509(1.2043)** -0.4378(0.3581) 2.7825(1.0896)** 1.0786(0.5323)* 0.2600(0.2900) 44 0.0249(0.0597) -2.1999(1.1753) 0.2012(1.2069) -0.4196(0.3585) 1.8314(1.0915) 0.9350(0.5331) 0.2994(0.2903) 45 0.0100(0.0596) -3.5710(1.1728)** -0.8001(1.2056) -0.6619(0.3580) 0.6262(1.0909) 0.5179(0.5328) 0.1759(0.2900) 46 0.0155(0.0597) -3.8220(1.1735)** -1.4523(1.2064) 0.2903(0.3585) -0.3865(1.0919) 0.4588(0.5333) 0.2323(0.2902) 47 0.0874(0.0597) -2.7989(1.1744)** -1.3858(1.2061) 0.7791(0.3581)* -0.1679(1.0917) 0.2745(0.5333) 0.0917(0.2902) 48 -0.0475(0.0597) 0.9716(1.1763) -1.7554(1.2068) 0.4486(0.3586) -0.5976(1.0925) 0.6466(0.5336) 0.2597(0.2904) 49 -0.0045(0.0597) -0.3605(1.1762) -1.1630(1.2068) 0.0853(0.3586) -0.6365(1.0923) 0.0850(0.5337) 0.1532(0.2903) 50 -0.1706(0.0596) -1.6427(1.1750) -2.5738(1.2052)* 0.7479(0.3581)* -0.5379(1.0916) 0.7568(0.5333) 0.4830(0.2900) 51 0.0471(0.0597) -1.3934(1.1769) -2.8755(1.2068)** 0.0748(0.3589) 0.4797(1.0933) 0.2187(0.5344) 0.3114(0.2905) 52 0.0443(0.0597) -2.4440(1.1758)* -1.4118(1.2072) 0.9296(0.3583)** -0.0265(1.0930) 0.3294(0.5342) 0.2642(0.2904) 53 0.0772(0.0598) -1.2330(1.1774) -2.9296(1.2072)** -0.1740(0.3590) 0.2429(1.0940) 0.1028(0.5346) 0.4692(0.2905) 54 -0.0905(0.0597) -2.4559(1.1763) -3.2275(1.2066)** -0.0064(0.3589) -0.7764(1.0935) -0.2761(0.5344) 0.5511(0.2904) 55 0.0317(0.0598) -5.4688(1.1727) -3.1588(1.2079) -0.7351(0.3587)* 0.3901(1.0938) 0.9153(0.5342) 0.3516(0.2905) 56 -0.1013(0.0597) -1.4164(1.1770) -1.4032(1.2083) -0.0610(0.3589) 0.6302(1.0936) 0.6473(0.5343) 0.3507(0.2905) 57 0.0504(0.0598) -2.6548(1.1774)* -0.8295(1.2098) -0.5704(0.3591) -0.0037(1.0949) 1.4281(0.5343)** 0.4680(0.2907) 58 0.0025(0.0598) -2.1035(1.1778) -1.5645(1.2095) -0.6291(0.3591) 0.3384(1.0949) 0.3137(0.5349) 0.2404(0.2908) 59 0.0009(0.0598) -0.9429(1.1783) -0.6939(1.2098) -0.1667(0.3593) 1.1358(1.0947) 0.2574(0.5350) 0.3894(0.2908) 60 -0.0747(0.0598) -0.5596(1.1786) -0.8615(1.2100) 0.1937(0.3593) 0.5257(1.1009) 0.1738(0.5351) 0.4805(0.2908) **: Significant at 1%. *: Significant at 5%. 3 30 Table 6 Horizon Predicting Power of Option Moneyness, Open-interest-based, 1- to 60-day ahead We carry out OLS regressions, for calls and puts from all near-term contract months, as follows, r s, t  i   0   1 r s, t   2 AMt   3 lnVol s, t   4VIXt   t ,i 1,...60 with rs,t being the current index return, rf,t the current futures return, VIXt the current volatility index calculated daily from options and lnVols,t the logarithm of daily trading volume on all common equities. Only β2 are reported. For comparison, we also run a similar regression by replacing AMt with rf,t, the futures return. Calls Puts i Spot 1st Near-Term 2nd Near-Term Spot 1st Near-Term 2nd Near-Term 1 0.4687(0.7466) -0.4779(0.3542) -0.1918(0.2340) 0.2848(0.5341) -0.0086(0.3433) 0.3235(0.2239) 2 0.5413(0.7449) -0.4548(0.3549) -0.1860(0.2345) 0.1132(0.5351) -0.1578(0.3440) 0.2306(0.2244) 3 1.3115(0.7466) -0.3666(0.3550) -0.1300(0.2345) 0.0702(0.5351) -0.0780(0.3440) 0.2020(0.2244) 4 0.3647(0.7456) -0.3024(0.3549) -0.2025(0.2344) 0.3880(0.5349) -0.0709(0.3439) 0.1938(0.2244) 5 -0.1248(0.7448) -0.5251(0.3545) -0.3164(0.2343) 0.8050(0.5343) 0.1591(0.3436) 0.2820(0.2241) 6 -0.9919(0.7466) -0.5834(0.3545) -0.4412(0.2342) 1.2169(0.5340)* 0.4304(0.3436) 0.3641(0.2241) 7 -0.9572(0.7472) -0.3382(0.3550) -0.4128(0.2344) 1.1401(0.5346)* 0.4642(0.3438) 0.3139(0.2243) 8 -1.4131(0.7464) -0.3875(0.3550) -0.5206(0.2344)* 1.2250(0.5346)* 0.3309(0.3440) 0.3182(0.2244) 9 -1.2342(0.7468) -0.4179(0.3551) -0.4532(0.2345) 1.0313(0.5349) 0.2821(0.3441) 0.3623(0.2244) 10 -1.5407(0.7461) -0.6381(0.3550) -0.4580(0.2345) 1.0050(0.5350) 0.4384(0.3441) 0.3977(0.2244) 11 -1.0055(0.7454) -0.7868(0.3547) -0.3676(0.2345) 0.6794(0.5350) 0.4814(0.3438) 0.3932(0.2243) 12 -1.3798(0.7469) -0.9112(0.3548) -0.3942(0.2346) 0.9610(0.5352) 0.6170(0.3440) 0.4648(0.2244)* 13 -0.7877(0.7459) -0.6993(0.3545) -0.2647(0.2343) 0.6610(0.5346) 0.4738(0.3436) 0.3386(0.2242) 14 -0.3424(0.7477) -0.5748(0.3545) -0.2834(0.2343) 0.3746(0.5346) 0.3451(0.3436) 0.2821(0.2242) 15 -1.3843(0.7469) -0.6090(0.3552) -0.3855(0.2347) 0.7607(0.5355) 0.4661(0.3442) 0.3658(0.2245) 16 -1.3791(0.7483) -0.4037(0.3555) -0.4741(0.2347)* 1.0600(0.5356)* 0.4028(0.3445) 0.4059(0.2246) 17 -0.6242(0.7486) -0.3443(0.3554) -0.3954(0.2348) 0.8705(0.5357) 0.4654(0.3444) 0.3676(0.2246) 18 0.5035(0.7471) -0.3183(0.3550) -0.3198(0.2345) 0.7013(0.5352) 0.2626(0.3441) 0.2305(0.2246) 19 -0.9964(0.7479) -0.1777(0.3555) -0.3213(0.2348) 1.1735(0.5357)* 0.3376(0.3445) 0.2966(0.2247) 20 0.5394(0.7473) 0.1699(0.3548) -0.2799(0.2343) 0.7535(0.5349) 0.1403(0.3438) 0.1698(0.2243) 21 -0.7053(0.7491) 0.3577(0.3556) -0.2683(0.2350) 0.7588(0.5364) 0.1378(0.3448) 0.2230(0.2248) 22 -1.6378(0.7473)* 0.1244(0.3554) -0.3116(0.2347) 0.9556(0.5359) 0.4021(0.3444) 0.2867(0.2246) 23 -1.8697(0.7480)** 0.0329(0.3559) -0.3382(0.2350) 1.0379(0.5367) 0.4672(0.3448) 0.3138(0.2249) 24 -2.2399(0.7486)** -0.0982(0.3559) -0.4521(0.2350) 0.2233(0.5367)* 0.4199(0.3449) 0.3187(0.2249) 25 -2.4167(0.7476)** -0.1319(0.3558) -0.4755(0.2349)* 1.3263(0.5367)** 0.4108(0.3448) 0.3190(0.2249) 26 -1.7033(0.7472)* 0.2539(0.3552) -0.3500(0.2346) 0.8444(0.5363) 0.0926(0.3443) 0.2479(0.2245) 27 -0.5766(0.7473) 0.3222(0.3554) -0.2273(0.2348) 0.4835(0.5371) 0.0523(0.3446) 0.2479(0.2247) 28 0.2063(0.7492) 0.3782(0.3557) -0.1513(0.2350) -0.0148(0.5378) -0.0300(0.3449) 0.1928(0.2249) 29 -0.5847(0.7492) -0.0191(0.3557) -0.1951(0.2350) 0.2178(0.5378) 0.2164(0.3447) 0.2610(0.2248) 30 -0.2220(0.7497) 0.0632(0.3560) -0.1089(0.2352) -0.0088(0.5384) 0.2350(0.3450) 0.1796(0.2250) 31 -0.2527(0.7506) -0.0882(0.3563) -0.0730(0.2354) -0.1448(0.5390) 0.2863(0.3452) 0.1486(0.2252) 32 -0.1796(0.7508) 0.0611(0.3562) -0.0123(0.2353) -0.3958(0.5389) 0.1285(0.3452) 0.0999(0.2251) 33 -0.6162(0.7506) 0.0542(0.3563) -0.0408(0.2354) -0.1578(0.5391) 0.0834(0.3453) 0.0907(0.2252) 34 -0.1822(0.7496) 0.0672(0.3562) -0.1047(0.2353) -0.1686(0.5390) 0.0558(0.3452) 0.0954(0.2251) 35 -0.3025(0.7499) 0.1647(0.3558) -0.0930(0.2350) -0.0157(0.5384) 0.0171(0.3448) 0.1168(0.2249) 36 -0.0979(0.7504) 0.1864(0.3561) -0.0115(0.2352) -0.2608(0.5388) 0.0436(0.3452) 0.0743(0.2251) 37 -0.1705(0.7496) 0.0846(0.3557) -0.0559(0.2350) -0.0527(0.5383) 0.1158(0.3449) 0.1264(0.2249) 38 -0.2647(0.7500) -0.1362(0.3556) -0.0933(0.2349) 0.0480(0.5381) 0.3971(0.3447) 0.0787(0.2248) 39 -0.6259(0.7499) -0.2574(0.3552) -0.1428(0.2349) 0.5014(0.5380) 0.6053(0.3446) 0.1412(0.2248) 40 -0.3824(0.7496) -0.6991(0.3552) -0.2104(0.2348) 0.2869(0.5378) 0.8010(0.3443)** 0.1524(0.2247) 41 -0.6715(0.7499) -0.7027(0.3553) -0.2338(0.2348) 0.0076(0.5380) 0.8179(0.3448)** 0.1462(0.2248) 42 -0.9166(0.7496) -0.7157(0.3551) -0.3462(0.2347) 0.5791(0.5376) 0.8378(0.3443)** 0.2144(0.2246) 43 -0.6476(0.7496) -0.6351(0.3551) -0.2118(0.2347) 0.2352(0.5377) 0.7416(0.3444)* 0.0885(0.2246) 44 -0.2716(0.7504) -0.6209(0.3554) -0.2096(0.2349) 0.2025(0.5383) 0.7363(0.3449)* 0.0858(0.2249) 45 0.6121(0.7496) -0.4881(0.3552) -0.0984(0.2347) -0.0300(0.5377) 0.6041(0.3446) -0.0321(0.2247) 46 1.0852(0.7480) -0.3761(0.3556) -0.0198(0.2349) -0.2679(0.5382) 0.5444(0.3450) -0.0361(0.2249) 47 0.9247(0.7501) -0.4028(0.3555) -0.0761(0.2349) -0.2177(0.5381) 0.7109(0.3449) 0.0216(0.2248) 48 0.5075(0.7500) -0.1488(0.3558) -0.0486(0.2350) -0.1960(0.5385) 0.5277(0.3453) 0.0090(0.2250) 49 0.4052(0.7502) -0.2770(0.3557) -0.0853(0.2350) 0.1548(0.5383) 0.6409(0.3453) 0.0007(0.2249) 50 -0.5468(0.7490) -0.4169(0.3554) -0.1787(0.2348) 0.5216(0.5379) 0.8044(0.3449) 0.0642(0.2248) 51 -0.9467(0.7500) -0.4767(0.3559) -0.2126(0.2351) 0.6937(0.5386) 0.7776(0.3456) 0.0668(0.2250) 52 -1.2154(0.7497) -0.2651(0.3558) -0.1844(0.2350) 0.5508(0.5385) 0.5714(0.3457) 0.0338(0.2251) 53 -1.7748(0.7504)** -0.2616(0.3561) -0.2458(0.2352) 0.7964(0.5387) 0.4147(0.3460) 0.0954(0.2252) 54 -1.5780(0.7505)* -0.3709(0.3559) -0.1719(0.2351) 0.3353(0.5387) 0.3387(0.3459) 0.0726(0.2251) 55 -1.7482(0.7501)** -0.2683(0.3560) -0.2239(0.2351) 0.8349(0.5386) 0.4850(0.3459) 0.1175(0.2251) 56 -0.6610(0.7509) -0.0437(0.3560) -0.0775(0.2351) 0.3815(0.5387) 0.2884(0.3459) 0.0017(0.2251) 57 -0.5581(0.7513) -0.2986(0.3563) -0.0804(0.2354) 0.3311(0.5393) 0.3929(0.3462) 0.0515(0.2254) 58 -0.3631(0.7516) -0.3624(0.3563) -0.1108(0.2354) 0.4329(0.5392) 0.2946(0.3462) 0.0979(0.2253) 59 0.2081(0.7519) -0.0524(0.3563) -0.0383(0.2354) 0.2838(0.5392) 0.0584(0.3463) 0.0575(0.2253) 60 -0.2600(0.7517) -0.1275(0.3569) -0.1144(0.2354) 0.6135(0.5652) 0.0362(0.3464) 0.0768(0.2254) **: Significant at 1%. *: Significant at 5%. 31 Table 7 Filtering Performances for Weekly Options We use weekly expiring options contracts listed on TAIFEX, which were introduced in November 2012, to replace the monthly expiring ones. We compiled closing prices for 255 weekly options between 2013 and 2017 and calculated AM measures. Due to the nature of weekly option, there is only one day, namely Wednesday, every week that there are concurrently two option contract series, one each for calls and puts, traded together in the same session. One series expires at the close of that day and the other is only listed after the open of the day. Although there is only one overlapping day, we can still compare AM measures between the two. Therefore we still construct a space-system over the 255 weeks, and except for Wednesday there is always one weekly spot contract estimated. The weekly near-term contract exists only on Wednesdays, when the then weekly spot contract is to expire at the end of the day. So each weekly spot contract starts on Thursdays and ends on Wednesdays. RMSE σ(η) σ(ξ) AveCallMoney Weekly Spot 0.0513 0.0347 0.0681 Weekly Near-Term 0.0668 0.0392 0.0736 AvePutMoney Weekly Spot 0.0521 0.0325 0.0702 Weekly Near-Term 0.0659 0.0414 0.0788

32 Table 8 Forecasting Performances Based on Weekly Options We apply the a procedure similar to what is done for Table 3 to forecast index returns only for 10 and 20 days beyond the estimation period. For the weekly spot contracts, forecasts are carried out for the non-Wednesday weekdays, whereas forecasts are only run on Wednesdays for weekly near-term contracts. τ=10 τ=20 RMSE MAPE RMSE MAPE

AveCallMoney Weekly Spot 0.0589 271.22 0.0725 328.01 Weekly Near-Term 0.0635 293.56 0.0803 375.33 AvePutMoney Weekly Spot 0.0604 285.47 0.0747 349.67 Weekly Near-Term 0.0666 318.80 0.0886 423.78

33 Table 9 Predicting Power of Weekly Option Moneyness, 1- to 21-day ahead We carry out a OLS regression, for calls and puts from weekly contracts, as follows, r s, t  i   0   1r s, t   2 AMt   3lnVol s, t   4VIX t   t ,i 1,...21 , where lnVols,t denotes the logarithm of daily trading volume on all common equities listed on the Taiwan Stock Exchange. Only β2 are reported for comparisons. Calls Puts i Volume-based OI-based Volume-based OI-based 1 2.5443(4.9000) -0.3559(4.1030) -0.3284(3.8834) -1.8024(3.6856) 2 -2.6085(4.9204) 1.2141(4.1191) 1.6282(3.8985) 1.2684(3.7003) 3 -0.9336(4.9224) 3.0954(4.1200) 2.8903(3.8987) 0.2867(3.7016) 4 0.9650(4.9148) 2.1251(4.1139) 2.9885(3.8928) 2.4008(3.6951) 5 -6.4327(4.9067) -6.3620(4.1062) 6.2957(3.8857) 10.3902(3.6785)** 6 1.2787(4.9230) -7.0954(4.1200) 5.8082(3.8979) 13.9930(3.6765) 7 2.0468(4.9081) -2.8250(4.1088) 0.0349(3.8925) 3.1689(3.6926) 8 -6.3009(4.9281) -4.7385(4.1244) 5.3089(3.9136) 3.4621(3.7105) 9 -0.3842(4.9287) -1.3611(4.1240) 3.8597(3.9124) 5.0898(3.7052) 10 -7.3264(4.9234) -1.8895(4.1233) 4.0765(3.9116) 0.4250(3.7115) 11 -6.2056(4.9278) -0.2567(4.1277) 1.3797(3.9153) 0.4550(3.7156) 12 -15.3991(4.8919)** -7.0607(4.1141) 5.9004(3.8996) 2.9462(3.7040) 13 -8.4781(4.9349) -8.0231(4.1373) 2.2662(3.9240) 4.2181(3.7225) 14 -1.9716(4.9334) -3.7194(4.1360) 1.0995(3.9184) 3.1229(3.7168) 15 -0.1946(4.9364) -6.4278(4.1334) 2.8436(3.9182) 6.2764(3.7149) 16 -3.7399(4.9399) -4.9900(4.1399) 2.8032(3.9218) 4.2237(3.7221) 17 -8.3233(4.9355) -8.2483(4.1389)* 0.9472(3.9239) 3.9630(3.7235) 18 -8.7876(4.9411) -1.9455(4.1518) -0.2819(3.9307) 2.4767(3.7326) 19 -1.5841(4.9522) 1.3117(4.1560) 4.6201(3.9308) 1.2846(3.7361) 20 -0.6693(4.9510) -2.2061(4.1557) 6.5553(3.9272) 2.1734(3.7351) 21 1.0149(4.9546) 1.6663(4.1591) 2.4510(3.9349) -1.6380(3.7387) **: Significant at 1%. *: Significant at 5%.