**1. Introduction**

Options are a conduit of carrying information into the market, which subsequently leads to stock price changes Grossman (1988). Because informed traders prefer to trade in options market for leverage and low transaction cost Black (1975) and Easley et al. (1998) 1, trading activities of options market measured in terms of volume and open interest are informative to predict the future price of their respective underlying assets. Both options volume and open interest have been used in addition to other factors in modeling early warning system for market crisis Li et al. (2015). Further, as per Jena and Dash (2014), trading volume and open interest represent the strength and potential of price change of the underlying asset, respectively. In addition, traders and technical analysts use open interest data to study the behavior of the underlying asset and design appropriate options strategies. Fodor et al. (2011) found individual call and put open interest have the power to predict future stock return. Most often PCR remains in the news as one of the important and parsimonious information variables used by traders to predict the market return2. This ratio is a contrarian indicator of the market by looking at build up options. That means, if there is excessive fall or rise in the market, PCR will move

<sup>1</sup> Informational role of derivative markets was discussed by Back (1993), Biais and Hillion (1994), Brennan and Cao (1996) and John et al. (2000) and others who further enriched the linkage among trade, price and private information in derivative market. In addition, few empirical studies support the informational role of derivative markets (e.g., Du et al. (2018), Ryu (2015), Cao and Ye (2016) and Chordia et al. (2018)).

<sup>2</sup> https://economictimes.indiatimes.com/markets/stocks/news/rising-nifty-put-call-ratio-bringssolace-for-bullsno-big-fall-likely/articleshow/60456238.cmshttps://economictimes.indiatimes.com/markets/stocks/news/ spike-in-put-call-ratio-shows-niftymay-correct-1-or-more-in-a-single-session/articleshow/59484386.cmshttps: //economictimes.indiatimes.com/markets/stocks/news/rising-put-call-ratio-falling-volatilitysupporting-the-bulls/ articleshow/60727912.cms

towards an extreme value based on which the traders can take a contrarian call. Thus, the direction of the market can be determined from the options market by using this most popular indicator, i.e., PCR, which is estimated as follows on a given day for both the measures of trading activity such trading volume and open interest.

PCR (OI) = open interest of put options on a given day/open interest of call options on the same given day

PCR (VOL) = volume of put options on a given day/volume of call options on the same given day

The objective of our study was to discern the efficacy of PCR (OI) and PCR VOL) in predicting the market return. However, is the predictability power of PCR stable across different time scales? Therefore, to answer this question, we investigated the strength and direction of causality at different frequencies using the novel frequency domain causality methodology of Breitung and Candelon (2006) in a rolling framework.

However, few academic studies are found in the literature related to this ratio. Billingsley and Chance (1988) found volume PCR as an effective forecasting tool in predicting the direction of the market. Blau and Brough (2015) in the US market found that current daily PCR of stock options is negatively related to next day's return, thus, as a contrarian trading strategy, PCR has the power of return predictability. Pan and Poteshman (2006) stated that the PCR constructed from buyer initiated volume (signed volume) contains information about future stock prices. Economically, stocks with low PCR are outperforming their higher counterpart stocks by 40 basis points and 1% on the next day and one week, respectively. However, this relative predictability of the PCR is short-lived Pan and Poteshman (2006). Therefore, in our study, we investigated whether the predictability of this ratio is consistent at a different frequencies over a period of time. Unlike Pan and Poteshman (2006), Blau et al. (2014) used unsigned trading volume in their study and investigated the relative information content of PCR and Option to Stock (O/S) ratio. They found that the nature of the information content of PCRs is fleeting at different frequencies. In our study, we tested this fleeting property of PCR at different frequencies in a time-varying framework.

Although information content of option ratios was studied by Roll et al. (2010) and Johnson and So (2012), they both used Option to Stock (O/S) volume ratio3. Further, in the literature, only PCR based on volume is studied, ignoring open interest, which is an important trading activity variable. Thus, in our study, in addition to volume PCR, we studied the efficacy of PCR open interest ratio in predicting the future market return. Thus far, existing literature provides one-shot statistic in the time domain in predicting the market return, thereby ignoring the causality dynamics at different frequencies. Thus, we applied Breitung and Candelon's (2006) frequency domain causality for this comparative study of predictability of PCR in both the short and long run. Since in sample frequency domain causality is not robust to structural changes<sup>4</sup> Batten et al. (2017) and Bouri et al. (2017), we estimated out of sample rolling frequency domain causality using a fixed window size of 250 days of observations.

Our contribution to the literature of the derivative market in general and options market, in particular, is threefold.

First, horizon heterogeneity requires information regarding the market at different time periods for trading and investment at different time horizon. We investigated the predictability of option ratios at different frequencies, thereby providing a robust measure for trading and investment at different time horizon for the investors. Second, in addition to volume PCR, we took PCR based on open interest, it being one of the important measures of investors' activity in the derivative market that is currently missing in the literature. Finally, we studied the robustness of our results at the different time periods as well as in the presence of the futures market.

<sup>3</sup> Other studies on markets include those by Roll et al. (2009) and Chang et al. (2009).

<sup>4</sup> We estimated the Bai and Perron (2003) test and the results show five breakpoints in both the cases, i.e., volume PCR and market return, and open interest PCR and market return. The results are available on request.

We found that open interest PCR is an efficient predictor of market return in the long period of 12 days and volume PCR in the short period of 2.5 days. The results are robust after controlling for the information generated in the futures market.

The rest this paper is outlined as follows. Section 2 describes the data and methodology used in the study. The empirical results are presented and discussed in Section 3. Section 4 concludes the paper.

## **2. Data and Methodology**

Daily volume and open interest data were collected for the Nifty Index<sup>5</sup> call and put option from the official website of National Stock Exchange of India (NSE)<sup>6</sup> from 1 June 2001 to 16 May 2013. The daily volume and open interest were aggregated across expiry and moneyness for both call and put options and taken for further calculation of daily PCR, the information variable for our study, by following Blau et al. (2014) and Bandopadhyaya and Jones (2008). Put–Call volume (open interest) ratio is the total volume (open interest) of put divided by total volume of call for the day. Log (P*t*/P*<sup>t</sup>*−1) was taken as market return, where P*t* and P*t*−<sup>1</sup> are closing price of the Nifty index at *t* and *t* − 1, respectively. To control for the information originating from futures trading, we took the trading volume of the NIFTY index futures as a control variable. The descriptive statistics of the variables are presented in Table 1.

**Table 1.** Descriptive statistics of volume put–call ratio (PCRTO), open interest put–call ratio (PCROI), market return (RET) and log Nifty index futures volume (LFTO).


\*\*\* and \*\* represent significance at 1% and 5% levels, respectively.

An average PCROI (PCRTO) greater than one (less than one) indicates a positive (negative) market sentiment. This justifies the PCROI taken in this study in addition to PCRTO. All the series were stationary. Moreover, since all the series were non-normal and fat-tailed, it further justified our methodology.

For the purpose of estimation, we used the frequency domain Granger causality (GC) test by following Bouri et al. (2017) as the widely utilized GC test (Granger 1969) is the one-shot measure of GC, which is assumed to be constant over time and frequency. Hosoya (1991) suggested that the causal influence may change across frequencies; nonetheless, they pointed out estimation difficulties owing to nonlinearities of the data to measure GC, which was made possible by Breitung and Candelon (2006)<sup>7</sup> by imposing linear restrictions on the autoregressive parameters in a VAR model and thus allowing

<sup>5</sup> The bellwether index of National Stock Exchange of India (NSE) represents 65% of the total market capitalization and 12 sectors of the economy.

<sup>6</sup> www.nseindia.com.

<sup>7</sup> Yamada and Yanfeng (2014) through theoretical evaluation tested the usefulness of the methodology even at a frequency close to zero.

for the estimation of the frequency domain approach to causality at different frequency bands. Several studies have used this approach (for example, Tiwari et al. 2014, 2015 and references therein), therefore we provide a small introduction to the approach.

Let us present an equation of a stationary VAR framework of two series *xt* and *yt* as follows:

$$\mathbf{x}\_{t} = a\_{1}\mathbf{x}\_{t-1} + \dots + a\_{p}\mathbf{x}\_{t-p} + \beta\_{1}y\_{t-1} + \dots + \beta\_{p}y\_{t-p} + \varepsilon\_{t} \tag{1}$$

The null hypothesis that *yt* does not Granger-cause *xt* at frequency (*ω*) in Equation (1) is tested by,

$$H\_0: \mathcal{R}(\omega)\mathcal{B} = 0\tag{2}$$

where *β* is the vector of the coefficients of *yt* i.e., *β* = [*β*1, *β*1,... *βp*] and

$$R(\omega) = \begin{bmatrix} \cos(\omega)\cos(2\omega) \dots \cos(p\omega) \\ \sin(\omega)\sin(2\omega) \dots \sin(p\omega) \end{bmatrix} \tag{3}$$

According to the Breitung and Candelon (2006), an ordinary *F* statistic for Equation (2) can be used to test the hull hypothesis at any frequency interval (i.e., *ω* ∈ (0, *π*)) as it is approximately distributed as *F*(2, *T* − <sup>2</sup>*p*). Further, for the purpose of interpretations in time framework, the frequency parameter *ω* (omega) can be used to obtain the time period of the causality in days (T) by using formula *T* = 2*π*/*<sup>ω</sup>*.
