Put-Call Parity in Equity Options Markets: Recent Evidence

There have been various studies of potential violations of put-call parity in US equity options markets, and the purpose of this study is to examine one potential explanation of these anomalous results. Cremers and Weinbaum [1] indicate a potential trading strategy that can obtain excess returns of up to 50 basis points per week, which is quite remarkable. However, none of these studies consider the fact that options markets have historically maintained different trading hours than those of their underlying security markets. While the US stock market has traditionally closed at 3:00 PM CST, options markets have variously closed between 3:10 and 3:02 PM CST over the past two dec-ades. Using over ten million individual options implied volatility estimations since 1996, it is documented that these anomalies have all but disappeared since stock and option markets synchronized their trading hours. Beginning in the late 1990’s, stock prices often move slightly or to a larger degree in “af-ter-hours” trading, enabled by the advent of electronic trading platforms. Options markets that are still open may adjust to subsequent stock market movements, although closing stock prices are reported as of 3:00 PM CST. Prior studies may have ignored these effects, and this is the first study to indicate that apparent deviations from put-call parity have decreased markedly over recent years, if they were ever economically significant at all.

where S equals the current stock price, P equals a particular strike price and maturity of a put option, C equals the equivalent strike price and maturity of a call option, and Ke −rT equals the present value of the strike price using the risk-free rate and time to maturity in years. If this relation is violated, potential arbitrage opportunities exist in the absence of transactions costs and/or a bid/ask spread.
In their study, Cremers and Weinbaum [1], they find that differences in call and put implied volatilities create potential opportunities for excess profits. However, they do not consider the effects of "after hours" trading, which became prevalent during the late 1990's, via online the earliest electronic trading networks such as Instinet, Island, and NYSE Arca. This phenomenon may have resulted in potentially spurious put-call parity results based on non-synchronous data observations, since options markets historically have closed later than their underlying security markets. Over the past three decades, options markets have alternatively closed at 3:10, 3:02, and 3:00 CST, which may have created the appearance of arbitrage profits that did not really exist.
Regarding the implications of put-call parity, An, Ang, Bali, & Cakici [2] finds a positive relation among increases in call implied volatilities and future stock returns, which may have been artificially supported by the differences in options and stock trading hours, although they report multi-month return differences.
Therefore, their results may have been given a "head start" due to nonsynchronous trading issues, although the overall results may not be perfectly valid.
Nishiotis and Rompolis [3] demonstrate that "the 2008 short-sale ban significantly enhanced the return predictability of put-call parity violations and attribute the significant increase in violations to stock over-valuation". Klemkosky and Resnick [4] [5] provide early evidence of potential arbitrage profits due to violations of put-call parity. In a study of the Israeli stock options market, Nissim and Tchahi [6] find evidence that violations of put-call parity are frequent and may result in potential arbitrage opportunities.
Additional evidence on this topic is provided by Haug and Taleb [7], who indicate that options traders do not price options based on traditional options pricing formulas, but rather based on very sophisticated heuristics, lending credence to the theory that option pricing models are not utilized in a rigorous fashion, and that traders adjust option pricing theory to reflect market realities.
Kamara and Miller [8]  profits. The contribution of the present study, however, is that it demonstrates that these apparent opportunities may have been an illusion. Prior to February 2006, when options markets closed at 3:02 PM CST, market price movements in electronic trading networks distorted the appearance of put-call parity relationships, because option market-makers adjusted their bid and ask prices even though stock prices are reported as of 3:00 PM CST. Overall, a statistically significant decline in the differences between put and call implied volatilities occurs over time. This effect is clear even in a subset of the sample that only includes the two years prior to and after the change in option market closing time from 3:02 PM to 3:00 PM CST. Some part of this decline may be attributed to general overall improvements in market liquidity, but the previously documented deviations from put-call parity were also likely affected by the non-synchronicity of stock and option closing market hours.

Data and Methodology
The data for the present study is obtained from the Option Metrics® database, and includes 30-day standardized at-the-money call and put option implied volatilities (IV) for the period from January 4, 1996 to August 29, 2014, representing almost nineteen years of options trading activity for approximately 2500 individual stock options series. Implied volatilities for these stock options are calculated using the same methodology that the CBOE® employs to calculate the VIX index for S & P 500 index options, which is robust to dividends for American equity options.
After trimming the data for obviously erroneous entries (negative or missing IV estimates are removed from the database), there are 10,366,256 individual observations. The missing or erroneous observations most likely occur during expiration weeks for US equity options, when implied volatility can be notoriously volatile and difficult to measure. These observations number just 366,769, or 2.78 percent of the initial sample of 13,177,147. Additionally, this phenomenon is random and affects all US equities equally, therefore, the removal of these observations does not create an issue related to selection bias. Finally, the empirical analysis below examines deviations from put-call parity before and after periods of stock and option market trading time synchronicity. Therefore, only stocks with options that traded during the full sample period are included in the analysis of the final sample database.
Summary statistics for this data are presented in Table 1. As is evident from these statistics, the average of closing bid-ask put volatilities are significantly higher than call volatilities, which indicate potential violations of put-call parity, as noted by previous studies (e.g. Cremers and Weinbaum [1], Klemkosky and Resnick [4] [5], and Nishiotis and Rompolis [3]). The average call implied volatility is 44.72%, while that for puts is slightly higher at 44.89%. In both cases, the medians of these values are significantly lower than the means, indicating the presence of some outlying observations at high volatilities. The average IV difference in percentage points indicates that, on average, at-the-money put volatilities are approximately 1.41 percent higher than call volatilities, indicating a potential opportunity for arbitrage profits. However, the economic significance of these differences is the most important aspect of this study. Therefore, the question that needs to be answered is whether this difference is economically significant and whether it can be exploited in an economically significant way.

Empirical Results
In order to examine the nature of these differences and how they have evolved over time, two pairwise tests of means are conducted for the periods prior to The results of two pairwise comparisons of means are contained in Table 2 below and indicate that the deviations from put-call paritymaynot be as signifi-  "official" 3:00 CST stock exchange closing price is observed. The statistical difference between these results is highlighted in Panel C of Table 2 where the t-statistic of 46.12 is extremely highly significant at the one-percent level. While this result may be inferred to be the result of the general increase in market efficiency over recent years, further analysis may indicate otherwise.
As noted, this result may be due to increased market efficiency over the past several decades, and a more specific approach may supply more direct results. This is because, although these results may be strongly statistically significant, some observers may point out that they may just be related to increased efficiency in markets created as electronic systems and increased market liquidity have affected options markets. In order to examine this issue more directly, the two-year periods immediately preceding and following the change in trading hours are examined in Panel B of Table 2. The mean level of IV differences increases during this period which presages and occurs during the global financial crisis (GFC), but there is still a marked difference in the "before and after" periods. Mean implied volatility differences between call and put options decline from 2.20% to 2.10% surrounding this event. While the difference in these elevated spread levels (due to the GFC) may seem small and the decline may not seem to be that great, the t-statistic for the difference remains strong at 3.50, which is once again significant at the one-percent level. Therefore, there may have been an effect on put-call parity relationships from the changing market closing hours.
As a test of robustness, one other unique US stock and option market pheno-

Conclusions
The results of the present exploratory study do not include the extensive asset-pricing tests that are conducted by Cremers and Weinbaum [1] and other studies referenced in this article. However, the analysis indicates that these pre- Overall, the paper documents increased efficiency in equity options markets, despite the disruptions that occurred during the global financial crisis. These efficiencies may have been responsible for eliminating potential arbitrage opportunities that have been documented in previous research, especially when considering the effects of changing option market closing times.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.