Empirical Evidence on Put-Call Parity in Australia: A Reconciliation and Further Evidence

Transcription

1 2 Empirical Evidence on Put-Call Parity in Australia: A Reconciliation and Further Evidence by R.L. Brown S.A. Easton Abstract: The results of the put-call parity studies by Loudon (1988) and Taylor (1990) are in direct conflict despite the authors reporting the use of virtually identical models and methods. Employing an improved version of Taylor s data collection procedures, we test the parity theorem in the period studied by Loudon. The results are similar to those of Loudon. As a result, we run separate checks of Taylor s data and analysis. The check of the data reveals that over sixty per cent of Taylor s observations are invalid. The check of the analysis reveals that the lower boundary of the put-call parity relation was incorrectly calculated by Taylor. Correcting this error results in fundamentally different conclusions. Keywords: PUT-CALL PARITY; DATA COLLECTION PROCEDURES. The Department of Accounting and Finance, Monash University, Clayton VIC We are grateful for the helpful comments made by Tim Brailsford, Barry Oliver, Graham Peirson, Alan Ramsay and Garry Twite and anonymous referees of this journal. We have also benefited from comments made at a seminar of the Department of Accounting and Finance, Monash Univerity, Clayton. We are also grateful for financial support provided by the Faculty of Economics Commerce and Management, Monash University. We are particularly indebted to Stephen Taylor who provided us with his data and made comments on earlier drafts. Australian Journal of Management, 17, 1, June 1992, The University of New South Wales

2 AUSTRALIAN JOURNAL OF MANAGEMENT June Introduction The Australian evidence on put-call parity consists of studies by Loudon (1988), Gray (1989) and Taylor (1990). For reasons discussed later, our main focus is on the studies of Loudon and Taylor. While both of these studies report the use of the same model and approach to testing, their results are in direct conflict. In this paper we summarise our reconciliation of their results and, in the process, provide some further evidence. 1 As a by-product, we are also able to offer some practical comments on the use of closing price data as against intra-daily data that is dated by input time. In Section 2 we review briefly the Australian evidence on put-call parity. In Section 3 we present our reconciliation and provide some further evidence. Our results accord very closely with those of Loudon, leading us to check Taylor s data and analysis. In Section 4, we report the results of these checks. We find that of Taylor s sample of 296, over half (150) consisted of put prices of zero and at least 41 of the remaining observations were also invalid. We then check Taylor s calculations and find a systematic error. We correct this error and find results that are fundamentally different from Taylor s. Some conclusions are offered in Section Australian Evidence on Put-Call Parity Following the format of Cox and Rubinstein (1985), put-call parity consists of two inequalities: 2 C S + PV e (K) P C S + K + PV p (D) (1) where: P = put price; C = call price; S = stock price; K = exercise price; PV e ( ) = present value, at the risk-free rate, for the period to the expiry of the options; D = amount of a dividend to be paid pursuant to an ex-dividend date falling within the life of the options; and, PV p ( ) = present value, at the risk-free rate, for the period to the dividend payment date. Our work is motivated by the fact that Loudon and Taylor report directly conflicting results on the empirical validity of this equation, despite reporting the 1. Further details are given in Brown and Easton (1991). 2. Although Equation (1) assumes there is only one ex-dividend date, it can be extended to cases where there is more than one dividend

3 Vol.17, No.1 Brown & Easton: PUT-CALL PARITY use of identical models and methodology. comparison of these two studies Data description Accordingly, we provide next a Loudon: Underlying share: BHP Time period: Options traded during the period January 1985 to December 1985 (inclusive). Taylor: Underlying shares: BHP and Woodside. 4 Time period (BHP options only): Options with expiry dates October 1982 to October 1985 (inclusive). Price observations begin in September 1982 and end in October Data collection Loudon: The primary data source is the Register of Sales Records. This source gives the time at which each price is entered into the exchange s recording system. This provides evidence on the latest time at which a given transaction could have occurred but does not provide the actual time of transaction. Taylor: 2.3 Results The primary data source is not identified. 5 The prices collected are monthly closing prices. But it is reported that prices outside the bid/ask spread are eliminated so as to exclude observations when it appears that the market has moved away from the closing price (Taylor 1990, p.208). Both studies provide details of their findings in terms of violation rates. That is, each put price in the sample is classified either as a non-violation if it falls between the boundaries specified in Equation (1), or as a lower (upper) boundary violation if it is less (greater) than the lower (upper) boundary. The central findings of the two papers are summarised in Table 1. The most noticeable characteristic of these results is, of course, the almost complete inconsistency between them. Whereas all of Taylor s violations are of the upper boundary, almost all of Loudon s violations are of the lower boundary. 3. The tests used by Loudon and Taylor are based squarely on the arbitrage framework of putcall parity. Gray uses tests that rely on expected profits and hence his work is difficult to compare with that of Loudon and Taylor. 4. But to enable a clean comparison with Loudon, we ignore Taylor s work on Woodside options. 5. But this information has been provided to us. The source is the Australian Financial Review

4 AUSTRALIAN JOURNAL OF MANAGEMENT June 1992 Table 1 Central Findings: Loudon and Taylor Loudon Taylor (BHP only) No. % No. % Non-violation Lower-violation Upper-violation Totals 1, Sources: Loudon (1988, Table 1) and Taylor (1990, Table 1). 3. Reconciliation and Further Evidence When confronted with conflicting empirical evidence, researchers will generally seek explanations in terms of different models, methodologies or data. In the present case it at first appeared that only data issues could be relevant. On a priori grounds it is not possible to decide which study has higher quality data. It is well recognised that non-synchronous prices can cause errors in tests of putcall parity. Loudon, however, provides no evidence that even the sequence of data input matches the sequence of transaction time. 6 Accordingly, non-synchronous data can be expected in Loudon s sample. Taylor s data consist of monthly closing prices and, accordingly, there is also a clear possibility that his prices will also be non-synchronous. There is also the problem that the two studies do not cover the same time period, although they overlap in the period January 1985 to October To provide independent evidence, we collected prices from Loudon s time period of January 1985 to December 1985 (inclusive), using procedures similar to those described by Taylor. But we took three steps to increase data quality. First, we used the Quote Sheets rather than the Australian Financial Review. Second, we collected daily closing prices instead of monthly closing prices. Third, following Brown and Shevlin (1983a and 1983b) and Gray (1989) we retained only those observations where the share, the put and the call had all traded on the same day. The last two steps were taken to reduce further the problem of temporal 6. Indeed, computer printouts of the Register of Sales carry the message, Time is when the trade was entered into the computer, it does not necessarily represent the time or sequence of the transaction

5 Vol.17, No.1 Brown & Easton: PUT-CALL PARITY mismatch between the security prices. In particular, we are able to avoid the gross form of non-synchronous prices in which puts are paired with calls that were in fact traded on different days. Specifically, our procedures for each Quote Sheet were as follows. 7 i. Record the closing stock price if it was within the bid/ask spread. If the closing stock price is outside the bid/ask spread, proceed to the next trading date. ii. iii. iv. Identify those puts that had the following three characteristics: a non-zero trading volume, a trading date that matched the date of the Quote Sheet and a closing price within the bid/ask spread. For each call that was a counterpart to a put identified in (ii), identify those that had the same three characteristics. Record prices for those matching options that satisfied both (ii) and (iii). Daily interest rate data were collected from the short-term interest rate information published in the Australian Financial Review. These data are derived from Treasury Notes traded by authorised dealers in the short-term money market. 8 Information on dividend amounts, ex-dividend dates and dividend payment dates was collected from stock exchange publications. 9 As a result of these procedures, we have 363 pairs of puts and calls with which to test Equation (1). A direct comparison between our results and Loudon s is provided in Table 2. The similarity between our results and those of Loudon is strong. It is also notable that the use of closing price data has produced a lower violation rate than the use of data which have been described (in our view, inaccurately) as transaction data (Loudon 1988, p.55). This suggests to us that prices classified according to input time do not necessarily provide a data base that is superior in quality to a data base consisting of daily closing prices extracted 7. Our procedures and those of Gray are very similar and may even be identical. For a description of his procedures, see Gray (1989, p.159). 8. Because of missing observations, many interest rates had to be estimated. To test the sensitivity of our results to alternative estimates we also collected data on 90-day and 180-day bank bill rates. There was little effect on the results. For details, see Brown and Easton (1991). 9. One slight complicating factor was a bonus share issue. Consistent with the model s assumption of perfect foresight of dividends, we assume that the correct dividend amount to use includes the extra amount payable pursuant to the bonus issue. In terms of implicit arbitrage transactions, this assumption means that to arbitrage an upper boundary violation, a trader would invest sufficient funds to make restitution for all dividends payable on shares sold short, including bonus shares issued subsequent to the short sale. But we also tested the sensitivity of our results to this assumption by using the simple declared dividend per share and there was no effect on the violation rate

6 AUSTRALIAN JOURNAL OF MANAGEMENT June 1992 Table 2 Parity Violations: January December 1985 Loudon This Paper Sample size: 1, Non-violations: Number Rate (%) Lower-violations: Number Rate (%) Mean size ($) Standard deviation of size ($) Upper-violations: Number Rate (%) Mean size ($) Standard deviation of size ($) Source (first column): Loudon (1988, Table 1). from the Quote Sheets using the methods we described above. 4. Data Investigation and Analysis Given the results reported in the previous section, we ran separate checks of Taylor s data and analysis. Our findings are summarised in Table 3. The data check revealed that of the 296 put prices used in the study, 150 were recorded at a price of zero. Since such prices cannot be market prices, we exclude these from further investigation. 10 When we checked the remaining 146 observations against Taylor s data source (the Australian Financial Review), we excluded a further 41 observations. Of these, one was a data transcription error and the remaining forty were cases where there was a breach of the bid/ask spread. 11 Furthermore, there 10. Put-call parity involves an arbitrage test. Were a zero-priced security to exist, it could be acquired for zero consideration. Under such conditions profits can be expected simply by acquiring all such securities in the maximum quantities available. 11. While some breaches were small, most (32) involved breaches of 1 or more. Of these, eleven involved breaches of 10 or more

7 Vol.17, No.1 Brown & Easton: PUT-CALL PARITY Table 3 Analysis of Taylor s Sample Upper- Percent of Violation Sample Non- Lower- Upper- and Number Remaining Violation Violation Violation Upper Boundary Positive less Full sample Put price of zero 150 equals less Data transcription error 1 equals less Outside bid/ask spread 40 equals Corrected Taylor sample less Bid/ask status doubtful 10 equals less Fails traded on day test 72 equals Sample after all data screens was doubt whether the closing price lay outside the bid/ask spread for an additional ten observations. For example, in one case the closing price was 15, the bid price was 5 and there was no recorded ask price. This left a sample of 95 observations, of which only 23 would have survived the traded on day test which we applied in our data collection. Consequently, considerations of both sample size and data quality cause us to prefer the evidence presented in Section 3. To check both Taylor s analysis and the effect of the data exclusions on the results, we used Taylor s data and tested for violations as each data screen was applied. Using the full sample, our calculations show that there were 47 upper boundary violations, which compares acceptably with Taylor s finding of 48 upper boundary violations. 12 But of the 47 upper boundary violations, there are only

8 AUSTRALIAN JOURNAL OF MANAGEMENT June 1992 seven cases where the upper boundary is positive. Because market prices must be positive, violations are inevitable in the other forty cases. Turning to violations of the lower boundary, we find 144 such violations using the full sample, whereas Taylor reports no violations of the lower boundary. Upon investigation, we found that for every parity observation, Taylor had incorrectly calculated the lower boundary as C S + rk (where r is the risk-free rate) instead of making the correct calculation of C S + PV e (K). The effect of this error is to understate dramatically the lower boundary, 13 resulting in a correspondingly dramatic understatement of the number of lower boundary violations. Removal of the put prices of zero reduces the number of lower boundary violations from 144 to 28 and the number of upper boundary violations from 47 to 35. Using the corrected Taylor sample, there are 24 lower boundary violations, as against eighteen upper boundary violations. The sample after all data screens consists of only 23 observations and hence inferences based on so small a sample would be unreliable. 5. Conclusions We draw three conclusions. First, our evidence is consistent with that of Loudon. Second, the quality of closing price data is not necessarily inferior to that of so-called transaction data where prices are classified by the time of data input. Third, we believe there are strong grounds to disregard Taylor s results using BHP options. (Date of receipt of final typescript: April 1992.) References Brown, R.L. and S.A. Easton, 1991, Put-call parity in Australia: a review and some further evidence, Discussion Paper 8/91, Department of Accounting and Finance, Monash University, Clayton. Brown, R. L. and T. J. Shevlin, 1983a, Modelling option prices in Australia using the Black- Scholes model, Australian Journal of Management, 8, Brown, R. L. and T. J. Shevlin, 1983b, Stock market efficiency and price predictions implicit in option trading, Australian Journal of Management, 8, The difference of one upper boundary violation is due to a calculation error by Taylor. 13. The maximum value found for the incorrect lower boundary is $ Therefore, no lower boundary violations could have been found

9 Vol.17, No.1 Brown & Easton: PUT-CALL PARITY Cox, J.C. and M. Rubinstein, 1985, Options Markets (Englewood Cliffs, Prentice-Hall). Gray, S.F., 1989, Put-call parity: an extension of boundary conditions, Australian Journal of Management, 14, Loudon, G.F., 1988, Put-call parity: evidence from the Big Australian, Australian Journal of Management, 13, Taylor, S.L., 1990, Put-call parity: evidence from the Australian Options Market, Australian Journal of Management, 15,

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