Testing Lower Boundary Conditions for Index Options Using Futures Prices: Evidences from the Indian Options Market

Transcription

1 R E S E A R C H includes research articles that focus on the analysis and resolution of managerial and academic issues based on analytical and empirical or case research Testing Lower Boundary Conditions for Index Options Using Futures Prices: Evidences from the Indian Options Market Alok Dixit, Surendra S Yadav and P K Jain Executive Summary Options are the contracts which serve as a tool for risk hedging, price discovery, and better allocation of capital. The efficiency of an options market, i.e., the correctness of option prices denotes that it is working well at its well-identified functions (Ackert and Tian, 2000). In view of this, the efficiency of options market has been of equal interest to the academics as well as practitioners and a number of studies on efficiency of options market have been carried out across the globe in different options markets. The present paper attempts to assess the pricing efficiency of the S&P CNX Nifty index options in India by testing the Lower Boundary Conditions (LBCs) using futures prices instead of spot values. The methodology adopted essentially tests a joint market efficiency hypothesis of index options and index futures. This has been done in view of the fact that the use of futures markets helps in doing away with the short-selling constraint as futures can easily be shorted. And, it becomes a natural choice for analysis as the short-selling has been banned in the Indian securities market during the period under reference. Moreover, the use of futures markets, to a marked extent, helps in ensuring the exploitability of arbitrage opportunities when underlying asset is an index. The study covers a period of six years from June 4, 2001 (starting date for index options in India) to June 30, The major findings of the study are: KEY WORDS Arbitrage Profits Ex-post Analysis Market Efficiency Options Market The put options market is more efficient than the call options market, given the existing market microstructure in India during the period under reference. Another equally important finding is that the put options market showed an improvement in the pricing efficiency over the years whereas the call options market demonstrated a counterintuitive and adverse development. However, the profit potential offered by highly traded opportunities both in the cases of call and put options seems to be unexploitable in the presence of transaction costs. Moreover, the dearth of liquidity in the case of otherwise exploitable opportunities which carry higher profit potentials has been the main inhibiting factor to arbitrageurs. Therefore, in short, it is reasonable to conclude that majority of violations in call as well as put options could not be exploited on account of the existing market-microstructure in India during the period under reference (especially short-selling constraint that caused underpricing in futures to persist) and the dearth of liquidity in the options market. In other words, the revealed state of options pricing can be designated to the short-selling constraints and dearth of liquidity. VIKALPA VOLUME 36 NO 1 JANUARY - MARCH

2 The options markets play a central role in an economy in view of the fact that they enhance better allocation of capital in securities market by virtue of their functions, namely, risk hedging and price discovery. In today s parlance where the demand for the structured financial products (which require excessive use of options contracts) is booming in India, the role of such markets has acquired greater significance. The open interests in the options segment of the Indian derivatives market has even surpassed that of the futures market for the last few months since April, This development has put the Indian derivatives market at an equal footing with the other international (developed) markets where the options are preferred to futures. There could be two major reasons for such a development. Firstly, increase in the portfolio management services (PMS) which provide structured financial products (using options market) to high profile investors. Secondly, there has taken place an increase in the variety of the products (in terms of maturity period) on account of the introduction of long-dated options on March 03, These options enable an investor to take a position up to five years. Considering the increasing importance of the options market in India, it is desired that the market should carry out its required functions in the best possible way. For the purpose, it is imperative that the market should be efficient. The reason is that well-functioning options markets are vital to a thriving economy as these markets facilitate price discovery, risk hedging, and allocation of capital to its most productive uses. Inefficiency of a financial market (e.g., options market in this paper) indicates that it is not performing the best possible job at the abovementioned important functions (Ackert and Tian, 2000). The present paper attempts to assess the pricing efficiency of the index options in India using the futures contracts on the same underlying asset, i.e., S&P CNX Nifty index. The use of futures markets helps, to a marked extent, in ensuring the exploitability of arbitrage opportunities when the underlying asset is an index. Moreover, the use of futures markets helps in doing away with the shortselling constraint as a futures can easily be shorted. Notably, the use of futures prices on the same underlying asset instead of spot prices essentially makes this approach a test of joint market efficiency, as opined by Fung, Cheng and Chan (1997). At the same time, the use of futures prices facilitates in assessing the degree of integration or pricing interrelationships between the different derivative instruments being traded in the market (Lee and Nayar, 1993). In other words, this approach helps in addressing the question whether market participants consider important pricing interrelationships while pricing the index options. The scope of the present study is confined to the pricing interrelationship between index options and index futures. The use of futures prices, however, puts one restriction on the otherwise model-free approach, i.e., it assumes costof-carry model to hold. Therefore, this approach cannot be designated as model-free unlike the test of the boundary condition using spot prices. However, the approach still remains less restrictive compared to those based on certain pricing models, e.g., those based on Black-Scholes model (1973) which assumes that the stock price and volatility are governed by some stochastic processes. The violations or mispricing signals observed from the test procedures have been classified as per liquidity with three specified levels and maturity with four specified levels. Also, the violations so classified as per the specified levels of maturity have further been sub-classified as per the three specified levels of liquidity. The classification facilitates a meaningful explanation to the exploitability of such violations and, therefore, is very crucial in assessing the efficiency of the market. This has been done in view of the fact that mere presence of violations does not indicate market inefficiency; it is the unexploitability and persistence of such violations which pose serious concerns/threats to the market efficiency. Moreover, the learning behaviour of the investors in options markets has also been examined. This has been done by analysing the number of violations vis-à-vis the number of observations analysed over the years under reference for both the call and put options. The learning hypothesis, which requires that the number of violations should go down over the years, has been proposed to gauge the developments related to the efficiency of the market. The analysis of violations over the years under reference is in line with Mittnik and Rieken (2000), a study done in German stock index options market. The findings of this research paper might be useful to brokerage houses, institutional investors --domestic as well as foreign, the National Stock Exchange (NSE), and the Securities and Exchange Board of India (SEBI); these findings are likely to be equally significant to the academics. 16 TESTING LOWER BOUNDARY CONDITIONS FOR THE S&P CNX NIFTY INDEX OPTIONS...

3 THE BOUNDARY CONDITION This section discusses lower boundary conditions using futures prices. The lower boundary conditions, first proposed by Merton (1973) and further extended by Galai (1978), play a crucial role in assessing the options market efficiency. A number of research studies have been carried out in different options markets using the lower boundary conditions to assess the efficiency of the markets including the first one by Galai (1978). The other studies which tried to diagnose the options market efficiency based on the violation of lower boundary conditions include Bhattacharya (1983), Halpern and Turnbull (1985), Shastri and Tandon (1985), Chance (1988), Puttonen (1993a), Berg, Brevik and Saettem (1996), Ackert and Tian (2001), Mittnik and Rieken (2000), Dixit, Yadav and Jain (2009). The lower boundary conditions of option prices denote the minimum price of an options contract at a given point of time during the life of an options contract. The violation of the condition indicates arbitrage opportunities. Therefore, the price for an options contract should necessarily be equal to or higher than that suggested by the lower boundary conditions. In order to ensure the correct pricing in an options market, this is a necessary condition which needs to be satisfied to uphold the well-known no-arbitrage argument of options pricing. In literature, the lower boundary conditions have been defined for the European options as well as American options. In this paper, as we are analysing the S&P CNX Nifty Index options which are European (that can be exercised only at maturity) in nature; the condition defined for European options constitutes the basis of the study. The method applied to test the efficiency of the options market is in line with other studies conducted in different markets for the same purpose with only one difference, i.e., use of the futures prices of the same underlying asset instead of its spot prices. The test of Lower Boundary Conditions using futures prices is in line with Puttonen (1993a), a study done in the Finnish index options market. Moreover, the test of options market efficiency using futures prices (on the same underlying asset) is in line with Lee and Nayar (1993), Fung and Chan (1994), Fung, Cheng and Chan (1997), Fung and Fung (1997), Fung and Mok (2001), etc., with the only difference that the condition tested in the paper is the lower boundary conditions for the option prices whereas all the above studies focus on the put-call-futures parity condition. The lower boundary conditions using corresponding futures prices (with the same maturity date) are given in the equations (1) and (2) for the call and put options respectively. These conditions are expected to hold in an efficient options market. Though the transaction costs have not been incorporated in the equations (1) and (2), the results have been interpreted considering the estimate of transaction costs discussed in the data section. In the above equations, C t = market price of a call option at time t, P t = market price of a put option at time t, F t = value of the S&P CNX Nifty futures (with same expiration date as of the options under consideration) at time t, K = strike price of the options contract, T = expiration time of the options contract at the time when it was floated, r = continuously compounded annual risk-free rate of return, (T-t) = time to maturity of the options contract at time t (measured in years). The dividends expected from the underlying asset during the life of the option have been ignored since the underlying asset used in the test is futures prices/value of the index instead of the spot prices/value. This has been done in view of the fact that the futures prices (in an efficient market) are expected to have impounded the effect of dividends on the prices of the underlying asset. In short, the use of futures prices essentially tests the joint hypothesis of the market efficiency of futures as well as options contracts in India. Testable Form of the Lower Boundary Conditions The equations (1) and (2) have been rearranged in order to make them testable to gauge the efficiency of the options market. The testable form, to gauge the Efficient Market Hypothesis (EMH) using lower boundary conditions, is given in the equations (3) and (4) for call and put options respectively. (1) (2) VIKALPA VOLUME 36 NO 1 JANUARY - MARCH

4 In the above equations, and denote the absolute amount of abnormal profits (ex-post) or mispricing signals from call and put options respectively, if the violation of lower boundary condition occurs. A violation of the lower boundary condition is recorded if >0 and >0 for call and put options respectively. Though the presence of such profits is only indicative of market inefficiency, it should not be treated as a conclusive remark on the efficiency of the market. The equations (3) and (4) have been specified assuming no transaction costs and zero or negligible bid-ask spread. It may be noted that there is always a chance that the arbitrage opportunities suggested by these equations may disappear in the presence of transaction costs and the bid-ask spread. Therefore, the violations have been interpreted considering the transaction costs and bid-ask spreads. In this regard, an attempt has been made to estimate the transaction costs. The details are summarized in the data section. On the contrary, given the fact that the bid-ask spread for options is not included in the transaction database provided by NSE and the difficulty to estimate such costs, it has been excluded in the above equations. In operational terms, our study is in line with that of Halpern and Turnbull (1985). In addition to this, commenting upon the exploitability of observed mispricing signals, Trippi (1977), and Chiras and Manaster (1978) concluded that the signals so observed were exploitable using a specified trading strategy to ensure ex-ante exploitation of such profit opportunities. However, in the present study, no strategy has been specified to ensure ex-ante exploitability of abnormal profits suggested by mispricing signals as the test procedure applied is ex-post in nature. Normalization of Abnormal Profits In order to facilitate comparison across different levels of liquidity, maturity, and time-period (years) under reference, the amount of absolute abnormal profits has been normalized to the strike price plus premium. That is, in the case of call options and for put options. The normalization criterion is in line with Nilsson (2008) since it has used strike price as a normalization parameter. Also, the normalization procedure is (3) (4) similar to the one used by Kamara and Miller Jr. (1995) since their results are more or less identical with those documented by Nilsson (2008) in this regard using strike prices as a normalization criterion. In the present study, the strike price plus premium has been used as a normalization criterion as it forms the basis of charging brokerage (which is a major constituent of the transaction costs in the Indian derivatives market). Therefore, this procedure facilitates a direct comparison of the violations (in percentage terms) vis-à-vis transaction costs and, thus, helps in assessing the exploitability of the mispricing signals in the presence of transaction costs. DATA Options, Futures and Interest Rates The data considered for the analysis can be broadly classified into three categories, namely, (i) data related to S&P CNX Nifty index options contracts, (ii) data related to the futures contracts, i.e., the S&P CNX Nifty index futures and (iii) data on the risk-free rate of return. The data on the options consist of daily closing prices of options, strike prices, deal dates, maturity dates, and number of contracts traded of call and put options respectively. In order to minimize the bias associated with nonsynchronous trading 1, only liquid option quotations 2 are being considered for the analysis. The second data set is regarding the futures contracts. It includes daily closing prices of S&P CNX Nifty index futures, deal dates, maturity dates and number of contracts traded. The third data set consists of monthly average yield on 91-days Treasury-bills. The yield on T-bills has been converted into continuously compounded annual rate of return using the relationship given in equation 5. Where, r = Proxy for continuously compounded annual risk-free rate of return, r = Average annual yield on 91- days T-bill of the maturity corresponding to the maturity 1 Nonsynchronous trading refers to the phenomenon of different timings of closing transactions in the two markets (i.e., the options market and the underlying s market in this case). 2 In the study, the liquid options quotations have been defined as those quotations which have at least one contract traded. Though the definition of liquid contracts is not useful in ensuring exploitability of arbitrage opportunities on account of the high bid-ask spread for such options, this has been done in order to gauge the total number of violations in Indian options market. Moreover, while ensuring the exploitability of the arbitrage opportunities, due consideration has been given to the liquidity. (5) 18 TESTING LOWER BOUNDARY CONDITIONS FOR THE S&P CNX NIFTY INDEX OPTIONS...

5 date of the options contract. Notably, the data on index futures has been matched with the corresponding options contracts on the basis of two criteria, viz., (a) deal date and (b) the expiration date of the options contract. The data for all the three mentioned categories have been collected from June 4, 2001 (starting date for index options in Indian securities market) to June 30, The first and second data sets have been collected from the website of NSE and the third category of data set has been collected from the website of the Reserve Bank of India (RBI). Transaction Costs Transaction costs typically include brokerage charged by the brokerage houses/trading members of the exchanges, service tax on the brokerage, stamp duty, opportunity cost of the margin deposits required in the case of futures contracts and short options positions, etc. In the Indian capital market, another charge, namely, Securities Transactions Tax (STT) was introduced and implemented with effect from October 1, Notably, such charges were to be levied only on the sell side of the transactions in the derivatives market unlike the equity market transactions where STT was proposed to be levied on both legs of the transactions. In short, the transaction costs being considered in the study typically include brokerage, service tax on the brokerage, and STT (October 1, 2004 onwards). Since the transaction costs constitute a major constraint to arbitrage (Ofek, Richardson and Whitelaw, 2004), an attempt has been made to have an estimate of such costs in Indian derivatives market. For the purpose, interviews were conducted with the senior employees of brokerage houses based at Delhi, India. This has been done in view of the fact that the trading member organizations (brokerage houses) are the best source of information in this regard as they themselves deal in the market and facilitate trading in F&O as well as cash market for different types of investors. Therefore, it would be reasonable to use the feedback of trading member organizations in this regard. Moreover, it may be noted that some of the studies on F&O segment (e.g., Vipul, 2008, a study in the Indian context) have estimated the transaction costs on the same lines. Based on the responses from the trading member firms, a consensus was arrived at an estimate of brokerage, viz., 0.05 per cent (including service taxes) of (strike price + premium)*lot size for options contracts and Futures price at the time of the transaction*lot size for futures contracts in the case of retail investors throughout the period under reference. However, such costs may go down up to 0.03 per cent (including service taxes) for the institutional investors. Notably, the brokerage houses bear the least cost of trading amongst all types of investors/players in the market as they are not required to pay any brokerage. However, it would be reasonable to consider the opportunity cost for the brokerage house and a logical estimate could be the cost incurred by the institutional investors, i.e., 0.03 per cent as pointed out by Vipul (2008). Moreover, the STT charge of 0.01 per cent (on the sales side of the transactions in derivatives market) has also been considered while interpreting the results over the years. Since the STT was introduced in October 1, 2004, it has been considered as part of the transaction costs for the arbitrage opportunities which occurred after October 2004, i.e., in years , , and in the present study. Therefore, for the analysis purpose, the major constituents of the transaction costs have been the brokerage and the service tax on it before October 1, 2004 and it additionally includes STT on the sales side of the transactions thereafter. The definition of the transaction costs has been confined to the brokerage and STT (wherever applicable) as these constitute, in general, more than 90 per cent of the transaction costs (excluding bid-ask spread). Though the analysis has been conducted ignoring the bid-ask spread and opportunity cost of the margin deposits, these have been given due consideration while ensuring the exploitability of mispricing signal. This has been done in view of the fact that bid-ask spread, in particular, plays a very important role in assessing the options market efficiency, as opined by Baesel, Shows and Thorpe (1983) and Phillips and Smith (1980). Before adding the transaction costs of options and futures contracts in order to arrive at the transaction costs for the arbitrage strategy, it becomes necessary to calculate them on the same basis, i.e., (K+C) for call options and (K+P) for put options. This is required in view of the fact that the mispricing signals have been normalized to these criteria. And, therefore, it would facilitate a fair and direct comparison of mispricing signals and transaction costs. Since the transaction costs in the case of options are based on (strike price + premium) and futures price forms the basis for the transaction costs in the case of VIKALPA VOLUME 36 NO 1 JANUARY - MARCH

6 futures contracts, the transactions costs of futures contracts has to be seen in relation to the (strike price + premium) to ensure the uniformity of basis for transaction costs and normalization of violations. In this regard, the transaction costs on the futures prices (0.05*Futures price) have been normalized to strike price+ premium. Empirically, it was found that the amount of transaction cost on futures when normalized to strike price+ premium remained intact at the level of 0.05 and 0.03 per cent on an average in the case of call options for the retail and institutional investors/trading members respectively; it turned out to be and per cent in the case of put options. Therefore, the transaction costs of futures (which should be added to that of options in order to arrive at the total transaction costs) are 0.05 per cent and per cent in the cases of call and put options respectively for retail investors; such costs are 0.03 per cent and per cent for institutional investors/trading member organizations. The transaction costs applicable in the case of a short hedged position, needed to exploit the mispricing indicated by a call option, was estimated at 0.10 per cent before October, 2004 and 0.11 per cent thereafter. In this case, the arbitrageur needs to buy the call option and sell a futures on the same underlying asset. In short, the transaction cost considered in the paper is 0.10 ( ) per cent prior to October 1, 2004 and 0.11(including STT of 0.01 per cent) thereafter for the retail investors. Likewise, in the case of a put option, there is a need to create a longhedged position, i.e., to buy the underpriced put option and take a long position in the futures contract on the same underlying asset. In this case, the applicable transaction costs would be ( ) for the whole period under reference and no STT needs to be included as it is levied on the sales side of a transaction. However, for the trading member firms and institutional investors, it would be 0.06 per cent before October 1, 2004 and 0.07 (including STT of 0.01 per cent) thereafter for call options, and ( ) per cent for put options throughout the period under reference. ANALYSIS AND EMPIRICAL RESULTS Magnitude of the Violations In order to have better insights about the behaviour of the mispricing signals obtained from the lower boundary conditions, the violations have been examined with respect to liquidity and maturity. In addition to this, both the parameters have further been classified into specified levels. Liquidity of the options has been decomposed into three levels based on the trading volume, namely, (i) thinly traded, options which have 1 to 100 contracts traded per day; (ii) moderately traded, options which have 101 to 500 contracts traded per day; and (iii) highly traded, options which have more than 500 contracts traded per day. Likewise, maturity of the options has been classified into four levels, namely, (i) 0-7 days to maturity; (ii) 8-30 days to maturity; (iii) days to maturity; and (iv) days to maturity. Also, the violations so classified as per the specified levels of maturity have further been sub-classified as per the three specified levels of liquidity in view of the fact that liquidity constitutes the basis for exploitation of arbitrage opportunities. The proposed classifications and their subclassifications enable us to draw some meaningful inferences about the exploitability of the observed mispricing signals which, in turn, help to assess the role of the existing market-microstructure in facilitating market efficiency in the Indian derivative market. The results related to violations, classified as per the specified levels of liquidity and maturity along with their sub-classifications, are summarized in Tables 1 and 2 respectively. Notably, the results across the specified levels of liquidity have direct implications for the exploitability of the observed mispricing signals as the higher the liquidity is, the lower would be the trading cost and bid-ask spread. Also, the higher liquidity ensures execution of the trading strategy required to tap such abnormal profit potentials. In contrast, the different specified levels of maturity have an indirect impact since these primarily influence the liquidity and, hence, the exploitability of such violations. And, therefore, the behaviour of violations with respect to maturity has been interpreted in light of the liquidity levels corresponding to their specified levels. The results summarized in Table 1 denote that the number of violations observed are 3,593 out of the total observations of 40,298 in the case of call options that amounts to 8.92 per cent of the total number of observations analysed. Likewise, 1,815 violations are found out of 35,171 observations for put options that accounts for 5.16 per cent of the total number of observations examined. In addition to this, the results regarding the different levels of liquidity, summarized in Table 1, clearly indicate that the mean percentage of the normalized violations (mag- 20 TESTING LOWER BOUNDARY CONDITIONS FOR THE S&P CNX NIFTY INDEX OPTIONS...

7 nitude) decreases as the liquidity increases. The majority of the violations in this category, i.e., about 96 per cent of the violations both in the case of call and put options, are confined to the thinly and moderately traded options which can be designated as unexploitable because of (i) higher bid-ask spread and (ii) difficulty in implementation of the strategy. Also, for the highly liquid contracts (which are approximately 4 per cent of the total violations both in the case of call and put options) where the possibility of exploitability is quite high as the bid-ask spread is expected to be considerably low, the means of abnormal profits are merely 0.08 and 0.13 per cent for call and put options respectively. In general, such meagre exploitable profit opportunities are clearly not attractive propositions for the retail arbitrageurs as the brokerage and securities transaction tax (STT) for exploiting such opportunities amounts to 0.10 and per cent for call and put options respectively. However, such opportunities might be exploitable on the part of the institutional investors and the trading member organizations (brokerage firms) as these firms enjoy relatively lower transaction costs of 0.06 and per cent in the cases of call and put options respectively. Moreover, amongst the violations relating to highly liquid contracts, only 25 per cent observations, i.e., the third quartile (as reported in the Table 1) seem to be exploitable as the profit for such violations is higher than 0.09 and 0.16 per cent for call and put options respectively. The remaining 75 per cent observations amongst highly liquid category yield the returns that are fairly below the exploitable level in the presence of transaction costs. Therefore, only 1 per cent (25 % of 4 %) of the violations for the call as well as put options seems to be exploitable by institutional investors and trading member organizations after factoring transaction costs. Besides, an attempt has been made to analyse the behaviour of liquidity (number of contracts traded) in S&P CNX Nifty index futures contracts vis-à-vis maturity over the period of analysis. This has been done in view of the fact that the liquidity of futures plays an important role in assessing the exploitability of mispricing signals identified using futures prices. To gauge the behaviour of liquidity across the different levels of maturity, the total and average number of contracts traded have been classified as per the three levels of maturity in the Indian derivatives market, namely, near-the-month contracts (NTM) having 0-30 days to maturity; next-the-month (NXTM) having days to maturity, and far-the-month (FTM) having days to maturity. The results in this regard are depicted in Figures 1 and 2. Figure 1: Total Number of Contracts Traded Classified as per the Three Levels of Maturity Maturity-wise Liquidity of S&P CNX Nifty Futures in India, June NTM 90.34% NXTM 9.43% FTM 0.22% Table 1: Liquidity-wise Magnitude of Violations of Lower Boundary Conditions for Call and Put Options in Indian Securities Market (June ) Liquidity Call Options Put Options Number of Magnitude of Violations Number of Magnitude of Violations Violations (in Percentage) Violations (in Percentage) Mean SD Q 1 Q 2 Q 3 Mean SD Q 1 Q 2 Q 3 Thinly traded 3,066 (85.3) ,557 (85.8) Moderately traded 372 (10.4) (10.0) Highly traded 155 (4.3) (4.2) Total 3, , Total number of observations analysed 40,298 35,171 Percentage of violations observed 8.92 % 5.16 % Note: 1. Figures in parentheses indicate percentage. 2. In the table, SD, Q 1, Q 2 and Q 3 denote standard deviation, first quartile, second quartile (i.e., median) and third quartile respectively. VIKALPA VOLUME 36 NO 1 JANUARY - MARCH

8 Figure 2: Average Number of Contracts Traded Classified as per the Three Levels of Maturity Maturity-wise Average Liquidity of S&P CNX Nifty Futures in India, June Average Liquidity Maturity Levels The results clearly indicate that the liquidity of futures is confined to the NTM contracts as per cent of the total number of contracts traded (along with the highest average) belong to this category. Therefore, the NTM futures contracts can be designated as the most traded contracts and are expected to have negligible bid-ask spread compared to NXTM and FTM. In another significant observation, the frequency of violations across different levels of maturity signifies a decreasing trend with an increase in the time to maturity. As reported in Table 2, majority of the mispricing signals are confined to the options having 0-7 and 8-30 days to maturity, approximately 94 per cent in call options and 90 per cent in put options. However, for the next two levels, i.e., and days to maturity, the combined percentage is merely 5.9 per cent and 9.8 per cent for call and put options respectively. The concentration of the violations in the 0-30 days to maturity category and especially in 0-7 days to maturity is quite similar to that reported by Bhattacharya (1983) for the US market where 42 per cent of the total violations had one week or less to maturity. The concentration of violations in 0-7 days to maturity category can be attributed to the fact that most of the arbitrageurs, in general, try to unwind their arbitrage positions when the options are nearing maturity. On account of this, the liquidity in such options is expected to be very thin as there are only a few or no buyers. This, in turn, causes the transaction costs especially the bid-ask Table 2: Magnitude of Violations of Lower Boundary Conditions in Indian Options Market as per the Specified Levels of Time-to-Maturity and their Sub-classification as per the Specified Levels of Liquidity for Call and Put Options (June ) Days to Liquidity Call Options Put Options Maturity No. of Magnitude of Violations No. of Magnitude of Violations Violations (in Percentage) Violations (in Percentage) Mean SD Q 1 Q 2 Q 3 Mean SD Q 1 Q 2 Q Days Thinly traded 1,327(77.29) (82.21) Moderately traded 249(14.50) (11.16) Highly traded 141(8.21) (6.63) Overall 1, Days Thinly traded 1,536(92.20) (88.51) Moderately traded 117(7.02) (9.30) Highly traded 13(0.78) (2.19) Overall 1, Days Thinly traded 186(96.37) (93.12) Moderately traded 6(3.11) (6.88) Highly traded 1(0.52) NA NA NA NA 0(0.00) NA NA NA NA NA Overall Days Thinly traded 17(100) (89.47) Moderately traded 0 (0.00) NA NA NA NA NA 2 (10.53) Highly traded 0(0.00) NA NA NA NA NA 0(0.00) NA NA NA NA NA Overall Note: 1. Figures in parentheses indicate percentage. 2. In the table, SD, Q 1, Q 2 and Q 3 denote standard deviation, first quartile, second quartile (i.e., median), and third quartile respectively. 22 TESTING LOWER BOUNDARY CONDITIONS FOR THE S&P CNX NIFTY INDEX OPTIONS...

9 spread to be considerably high. Therefore, lack of liquidity and less time to maturity might be cited as the major reasons why such observed mispricing signal remained unexploited. In addition to this, the reason why such options remained unexploited becomes clearer when they are seen in light of their corresponding levels of liquidity as reported in Table 2. The results indicate that only 8.21 per cent and 6.63 per cent of the total violations in 0-7 days to maturity category for call and put options respectively belong to the highly traded category which is apparently exploitable. However, their magnitude, i.e., the mean percentage turned out to be 0.08 per cent and 0.09 per cent respectively; even this insignificant number virtually ceases to be profitable when the transaction costs are considered for the retail investors. Moreover, from the viewpoint of institutional investors/trading member organizations, one-fourth of such opportunities seem to be profitable as the third quartile turns out 0.09 per cent and 0.11 per cent for call and put options respectively. And, obviously, less than 25 per cent of such opportunities would be profitable for retail investors. Precisely, 2 per cent (25 % of 8.21 %) and 1.6 per cent (25 % of 6.63 %) of the total observations having 0-7 days to maturity seem to be exploitable in call and put options respectively. And, the remaining violations in 0-7 days to maturity category (pertaining to the relatively illiquid category) amount to per cent and per cent in the cases of call and put options respectively. A plausible explanation for the existence of such options, despite the higher magnitude of profit they offer, could be the higher bid-ask spread. Notably, the behaviour of violations pertaining to the 8-30 days to maturity category is quite similar to that of 0-7 days to maturity category as the violations belonging to the highly liquid category are only 0.81 per cent in the case of call options and 2.19 per cent in the cases of put options of the total violations registered in this category. Equally revealing observation is that the majority of the violations belong to the relatively lower levels of liquidity, i.e., approximately 99 per cent and 98 per cent in the cases of call and put options respectively. Empirically, for the call options, the violations belonging to the highly liquid contracts in this category clearly seem to be unexploitable in the presence of transaction costs. In contrast, the violations in the case of put options, per-se, seem to be exploitable as the average magnitude of violations is 0.27 per cent. However, the number of exploitable violation further goes down as only the third quartile possesses profitable opportunities in the presence of transaction costs. In short, the percentage of such exploitable violations in put options becomes considerably low, i.e., 0.55 per cent (25 % of 2.19 %). In addition to the above, the last two levels of time to maturity, i.e., and days to maturity clearly depict a lack of the highly liquid contracts for call as well as put options. It may be noted that the majority of violations for both categories pertain to the thinly traded category, viz., per cent and 100 per cent in the cases of call options; per cent and per cent for put options. Also, the mean percentages of magnitude of these levels of maturity are significantly high compared to the first two levels and, prima-facie, seem to be exploitable. But the exploitability of such abnormal profit opportunities is questionable in view of the lack of liquidity in such options in the Indian options market. The unexploitability of such violations is reinforced by the fact that liquidity for the futures contracts having and days to maturity is relatively very low (Figure 1) which causes bid-ask spread for futures contracts to be high. Further Figure 2 depicts that the average number of contracts for such futures contracts are 13,138 and 310 vis-à-vis 1,14,494 contracts in futures having 0-30 days to maturity. In short, as far as the exploitability of the observed mispricing signals is concerned, the results regarding maturity are in line with those in the case of liquidity as the majority of violations pertain to the relatively illiquid categories for all the four levels of maturity for call as well as put options. Statistical Significance of the Differences in the Magnitude of Violations In view of the above findings, it can be observed that there seems to be a difference in the mean percentages of the magnitudes of violations among the specified levels of liquidity and maturity for call as well as put options. And, to validate the finding statistically, a well-known statistical test Analysis of Variance (ANOVA) was proposed initially. However, before applying the test statistics on the data, the main assumption of ANOVA, i.e., the samples have been drawn from a normally distributed population, has been validated using the one-sample Kolmogorov-Smirnov statistics. The results are summarized in Table 3. VIKALPA VOLUME 36 NO 1 JANUARY - MARCH

10 Table 3: Summary of One-sample Kolmogorov-Smirnov Statistics to Assess Normality Call Options Put Options Variable LBC_Normalized (in percentage) LBC_Normalized (in percentage) Number of Observations 3,593 1,815 Normal Parameters(a,b) Mean Std. Deviation Most Extreme Differences Absolute Positive Negative Kolmogorov-Smirnov Z Asymp. Sig. (2-tailed) a Test distribution is normal b Calculated from data. Since the results depict severe departure from the normality (revealed by the Kolmogorov-Smirnov (KS) statistics), ANOVA cannot be applied as it requires data to follow the normal distribution. Therefore, the differences have been analysed using a non-parametric statistics which does not require the data to follow any specified distribution. The test statistics applied in the present study is Kruskal-Wallis (H-statistics) test which is a non-parametric substitute for the one-way ANOVA. In addition to this, Dunn s multiple comparison test has been used for post-hoc analysis of all possible pairs in the analysis. The results of H statistics and Dunn s test for the differences across the specified levels of liquidity and maturity are summarized in Tables 4 (a) and (b), 5 (a) and (b) respectively. The significance values given in Tables 4 (a) and 5 (a) clearly indicate that the differences among the specified levels of liquidity as well as maturity are significant even at 1 per cent level of significance for both call and put options. Moreover, the post-hoc diagnosis, i.e., Dunn s multiple comparison test signifies that magnitude of violations (in percentage) for the thinly traded options is significantly different from that for the moderately traded as well as highly traded options both for call and put options. The magnitudes of violations across the specified levels of maturity are lower for exploitable options (i.e., 0-7 days to maturity and 8-30 days to maturity) compared to unexploiatble options (i.e., days to maturity and days to maturity) for call as well as put options; however, all the possible pairs are not significantly different at 5 per cent level of significance. Table 4(a): Kruskal-Wallis (H-statistics) Test for the Differences among the Violations across the Specified Levels of Liquidity for Call and Put Options (June ) Liquidity Call Options Put Options Rank Test Statistics (a,b) Rank Test Statistics (a,b) N Mean Rank Chi-Square df Sig. N Mean Rank Chi-Square df Sig. Thinly traded 3,066 1, , Moderately traded 372 1, Highly traded a. Kruskal Wallis Test b. Grouping Variable: Liquidity Table 4(b): Dunn s Test for Multiple Comparisons amongst the Specified Levels of Liquidity for Call and Put Options Dunn s Multiple Comparison Test Call Options Put Options Difference in Rank Sum Significant (P < 0.05) Difference in Rank Sum Significant (P < 0.05) Thinly traded vs.moderately traded 810 Yes Yes Thinly traded vs.highly traded 1,300 Yes Yes Moderately traded vs. Highly traded 450 Yes Yes 24 TESTING LOWER BOUNDARY CONDITIONS FOR THE S&P CNX NIFTY INDEX OPTIONS...

11 Table 5(a): Kruskal-Wallis (H-statistics) Test for the Differences among the Violations across the Specified Levels of Maturity for Call and Put Options (June ) Maturity Call Options Put Options Rank Test Statistics (a,b) Rank Test Statistics (a,b) N Mean Rank Chi-Square df Sig. N Mean Rank Chi-Square df Sig. 0-7 days to maturity , days to maturity , days to maturity 193 2, , days to maturity 17 2, , a. Kruskal Wallis Test b. Grouping Variable: Maturity Table 5(b): Dunn s Test for the Multiple Comparisons amongst the Specified Levels of Maturity for Call and Put Options Dunn s Multiple Comparison Test Call Options Put Options Difference in Significant Difference in Significant Rank Sum (P < 0.05) Rank Sum (P < 0.05) 0-7 days to maturity vs days to maturity -230 Yes -150 Yes 0-7 days to maturity vs days to maturity -380 Yes -250 Yes 0-7 days to maturity vs days to maturity -860 Yes -610 Yes 8-30 days to maturity vs days to maturity -140 No -95 No 8-30 days to maturity vs days to maturity -630 No -460 Yes days to maturity vs days to maturity -480 No -370 Yes In operational terms, the results imply that the average magnitude of violations for exploitable options contracts is significantly different from those for options contracts which can be designated as unexploitable. Empirically, the finding validates that the magnitude of exploitable violations is significantly less than that in the case of unexploitable options. It demonstrates a good sign for the market that the truly exploitable mispricing opportunities were meagre in magnitude and significantly noticeable violations existed only in the cases of unexploitable contracts due to lack of liquidity in such options. The Learning Curve In addition, an effort has been made to analyse the behaviour of the mispricing signals over the years under reference. This part of the paper attempts to test the learning hypothesis for call as well as put options markets in the Indian context. The hypothesis warrants improvement in the market efficiency over the years as investors are expected to be more familiar/experienced with the new market year after year and, thus, are expected to behave more rationally in pricing the options contracts. To put it explicitly, it has been hypothesized that the mispricing signals should depict a declining trend for call as well as put options. The examination of the violations over the years under reference is similar with the study done by Mittnik and Rieken (2000). In this regard, the results, summarized in Table 6 and Figure 3, indicate that the percentage of violations in call options has shown an increasing (counterintuitive) trend over the years except in the last year ( ) of the study. The percentage has gone up from 6.21 per cent in the year to per cent in the year with an alarming increase of per cent. The finding clearly indicates violation of the learning hypothesis for call options. In contrast, the results regarding the put options, conforming to the learning hypothesis, have shown a declining trend, i.e., percentage of violations has been decreasing over the years of analysis as presented in Table 6 and Figure 4. The percentage of violations has declined from 7.63 per cent in the year to 3.09 per cent in the year with a considerable decrease of per cent. This validates the warranted improvement in the efficiency of the put options market which, evidently, could not be observed in the case of call options market. The persistence of violations in the case of call options and observed decline in the case of put options are in line with those reported by Mittnik and Rieken (2000) in the context of German index options market. VIKALPA VOLUME 36 NO 1 JANUARY - MARCH

12 Table 6: Frequencies of the Violations of Lower Boundary Conditions for Call and Put Options (June ) Year Call Options Put Options No. of Observations No. of Violations Percentage No. of Observations No. of Violations Percentage Analysed Observed Analysed Observed , , , , , , , , , , , , Total 40, , Figure 3: Percentage of Violations in Relation to the Total Observations Analysed for Call Options The Learning Curve_Call Options Figure 4: Percentage of Violations in Relation to the Total Observations Analysed for Put Options The Learning Curve_Put Options Further, the results summarized in Table 7 reveal that the percentage of violations belonging to the thinly traded category has declined over the years for both call (except the last year, ) and put options. This finding, per se, indicates that the percentage of violations which could have been exploited has increased over the years. However, the exploitability of such arbitrage opportunities can be ascertained if we look at the magnitudes of abnormal profits they offered. In the year , there are no violations pertaining to the highly liquid category (which are the most exploitable as the bid-ask spread for such options is assumed to be very low) for call as well as put options. Moreover, the moderately traded options, where the bid-ask spread is assumed to be relatively high compared to the highly traded options, possess a mean percentage of 0.14 and 0.20 for call and put options respectively which, per se, seem to be exploitable. However, their unexploitability could be attributed to the high bid-ask spread. Apart from this, for the rest of the five years under reference (from to ), the percentage of violations belonging to the highly liquid category has offered considerably low abnormal profits as indicated by the third quartiles for both call and put options. And, therefore, such violations seem to be unexploitable in the presence of transaction cost except the year for call options (as the third quartile is 0.17 per cent); years and for put options(as the third quartiles are 1.07 and 0.16 per cent). In short, no exploitable violations (considering the bidask spread and transaction costs) were registered in the first four years of analysis for call as well as put options except the third year, i.e., in the case of call options where a negligible 0.39 per cent (25 % of 1.57 %) of all the violations observed seem to be exploitable. At the same time, the last two years ( and ) of analysis did not show any exploitable violations for call options. In contrast, the results regarding the put options depict that a meagre 1.86 per cent and 2.79 per cent of violations were exploitable in the years and TESTING LOWER BOUNDARY CONDITIONS FOR THE S&P CNX NIFTY INDEX OPTIONS...