Ex-Post Mispricing Analyses

The mispricing signals are identified by Equations (3) and (4) for SPDRs and Equations (5) and (6) for QQQs with various levels of transaction costs based on the theoretical futures prices. The total number of violations, the number of violations as percentage of number of observations, the average mispricing signal, the standard deviation of signals, and the statistics of violation within a 5-minute interval, are summarized in Table 2 (for SPDRs), and Table 3 (for QQQs), for both the pre- and post-decimalization periods. As the tables show, an increase in

transaction costs tends to lead to reductions in the number of violations and the percentage of violations.

For SPDRs, comparing Panel A and Panel B of Table 2, we can see that after decimalization there is a substantial decrease in number of violations and the percentage of violations. In contrast, from Table 3, the number of violations and the percentage of violations generally increase for QQQs. As a result, no general conclusions regarding pricing efficiency can be made from the number and frequency of pricing errors.

The ex-post profit (i.e., the average mispricing signal) is defined as the difference between the futures quoted price and the appropriate upper or lower boundary when an mispricing signal is observed. When comparing the mispricing signal sizes in the pre- and post-decimalization periods, we find that the average signal sizes are generally smaller after penny pricing than before for both ETFs. It appears that the smaller mispricing signals indicate an improvement in pricing efficiency, but it is also likely that the somewhat smaller average signal sizes in the post-decimalization period are partially attributable to a reduction in the bid-ask spread. More importantly, a decreased mispricing size also implies that the potential arbitrage profits are smaller after decimalization, which may weaken arbitragers’

incentive to participate in trading.

Table 2 and Table 3 also show that most of the boundary violations before and after penny pricing are clustered. From Panel B in Table 2, for the case with transaction cost of 0.5%, although the total number of violations is only 17, the maximum number of violations in a 5-minute interval is as high as 10.¹⁶ Similar results are also obtained with various levels of transaction costs for both ETFs. This result supports the argument proposed by Chung (1991) that mispricing signals tend to occur in clusters.

16 Note that the average number of violations in 5-minute intervals, the average percentage of violations in 5-minute intervals and the percentage of violations in clusters of 5 or more in 5-minute intervals are calculated based on those 5-minute intervals with mispricing signals observed.

Table 2 Ex-post boundary violations of SPDRs and S&P 500 E-mini futures

The ex-post tests refer to the frequency and persistence of boundary violations. No-arbitrage boundaries are constructed with the SPDR and S&P 500 E-mini (ES) prices, which are defined as:

where SPDR(t)bid is the SPDR bid price and SPDR(t)ask is the SPDR ask price at time t; ES(t)bid is the S&P 500 E-minis bid price and ES(t)ask is the S&P 500 E-minis ask price at time t; SDiv(t) refers to the present value of the SPDR dividend from time t to time T; and Cc is the trade commission. Since SPDR prices are 1/10th of the index level, an adjustment factor of 10 is applied.

Transaction costs are measured as a percentage of the theoretical futures value under the cost-of-carry model.

Number of Panel A: Pre-Decimalization (Jul 27, 2000 ~ Jan 28, 2001)

301,018 0.05% 97,094 32.26% 0.8677 0.7290 17.01 90 52.59% 74.55%

Panel B: Post-Decimalization (Jan 29, 2001 ~ Jul 30, 2001)

322,524 0.05% 81,276 25.20% 0.4357 0.3587 12.01 74 39.50% 69.95%

Table 3 Ex-post boundary violations of QQQs and NASDAQ 100 E-mini futures

The ex-post tests refer to the frequency and persistence of boundary violations. No-arbitrage boundaries are constructed with the QQQ and NASDAQ E-mini (NQ) prices, which are defined as:

where QQQ(t)bid is the QQQ bid price and QQQ(t)ask is the QQQ ask price at time t; NQ(t)bid is the NASDAQ 100 E-minis bid price and NQ(t)ask is the NASDAQ 100 E-minis ask price at time t; QDiv(t) refers to the present value of the QQQ dividend from time t to time T; and Cc is the trade commission. Since QQQ prices are 1/40th of the index level, an adjusting factor of 40 is applied. Transaction costs are measured as a percentage of the theoretical futures value under the cost-of-carry model.

Number of Panel A: Pre-Decimalization (Jul 27, 2000 ~ Jan 28, 2001)

387,404 0.05% 353,223 91.18% 6.4106 3.5484 35.83 91 90.22% 99.74%

Panel B: Post-Decimalization (Jan 29, 2001 ~ Jul 30, 2001)

463,823 0.05% 446,738 96.32% 4.6321 1.9584 45.35 94 95.79% 99.92%

Table 4 Ex-ante arbitrage analyses for SPDRs and S&P 500 E-mini futures

The ex-ante tests impose an execution lag for trading in SPDRs and S&P 500 E-mini futures, and thus the ex-ante mean profits are arbitrage profits after considering the transaction lag. A long arbitrage (ESAPL), triggered by futures overpricing, buys 500 SPDR shares and shorts an S&P 500 E-mini futures contract after observing an upper-boundary violation, whereas a short arbitrage (ESAPS), triggered by futures underpricing, executes the reverse transactions. Profits for long and short arbitrage are measured as:

( ) { [ ( ) ( ) ]

}

where t⁺ indicates the time of the first quote (trade) price of ETFs (E-minis) immediately after observation of the mispricing signal. SPDR(t⁺)bid is the SPDR bid price and SPDR(t⁺)ask is the SPDR ask price at time t⁺. ES(t⁺)bid is the S&P 500 E-mini futures bid price and ES(t⁺)ask is the S&P 500 E-mini futures ask price at time t⁺; SDiv(t⁺) refers to the present value of the SPDR dividend from time t⁺ to time T; and Cc is the trade commission. Since SPDR prices are 1/10th of the index level, an adjustment factor of 10 is applied.

Transaction

Panel A: Pre-Decimalization (Jul 27, 2000 ~ Jan 28, 2001)

0.05% 97,052 85,723 88.33% 0.8676 0.7289 0.8001 0.8058 0.8477 (<.0001) 0.10% 51,030 42,690 83.66% 0.6286 0.6972 0.5370 0.7749 0.8227 (<.0001) 0.15% 17,532 12,908 73.63% 0.4812 0.9153 0.3335 0.9883 0.8478 (<.0001) 0.20% 3,250 2,003 61.63% 0.6288 1.8628 0.3544 1.9405 0.8937 (<.0001) 0.25% 542 334 61.62% 1.9077 4.0091 1.3899 4.2122 0.9028 (<.0001) 0.30% 207 157 75.85% 3.9332 5.5026 3.2152 5.9267 0.8888 (<.0001) 0.35% 127 98 77.17% 5.5617 5.9675 4.5763 6.7278 0.8749 (<.0001) 0.40% 111 91 81.98% 5.6452 6.0114 4.6436 6.8690 0.8644 (<.0001) 0.45% 100 85 85.00% 5.5595 6.0471 4.4905 7.0124 0.8603 (<.0001) 0.50% 78 64 82.05% 6.3624 6.0732 5.2624 7.1859 0.8467 (<.0001) Panel B: Post-Decimalization (Jan 29, 2001 ~ Jul 30, 2001)

0.05% 81,250 64,508 79.39% 0.4357 0.3587 0.3302 0.4674 0.6229 (<.0001) 0.10% 20,182 12,674 62.80% 0.2900 0.3468 0.0971 0.4848 0.5271 (<.0001) 0.15% 1889 866 45.84% 0.3217 0.7478 -0.0283 0.8924 0.5750 (<.0001) 0.20% 191 89 46.60% 1.0461 1.8296 0.3226 2.2563 0.5388 (<.0001) 0.25% 63 37 58.73% 2.1201 2.4242 1.0092 3.4439 0.4122 (0.0008) 0.30% 36 23 63.89% 2.8677 2.4379 1.3992 4.1028 0.2765 (0.1026) 0.35% 28 16 57.14% 3.0061 2.2491 0.7249 4.4831 0.3790 (0.0467) 0.40% 25 16 64.00% 2.7175 2.1589 0.3362 4.6920 0.3537 (0.0828) 0.45% 21 13 61.90% 2.5454 2.0751 0.1274 4.8240 0.3142 (0.1654) 0.50% 17 10 58.82% 2.4601 1.9545 -0.3359 5.1468 0.3478 (0.1713)

Table 5 Ex-ante arbitrage analyses for QQQs and NASDAQ 100 E-mini futures

The ex-ante tests impose an execution lag for trading QQQs and NASDAQ 100 E-mini futures, and thus the ex-ante mean profits are arbitrage profits after considering the transaction lag. A long arbitrage (NQAPL), triggered by futures overpricing, buys 800 QQQ shares and shorts a NASDAQ 100 E-mini futures contract after the observation of an upper-boundary violation, whereas a short arbitrage (NQAPS), triggered by futures underpricing, executes the reverse transactions. Profits for the long and short arbitrage are measured as:

( ) { [ ( ) ( ) ]

}

where t⁺ indicates the time of the first quote (trade) price of ETFs (E-minis) immediately after observation of the mispricing signal. QQQ(t⁺)bid is the QQQ bid price and QQQ(t⁺)ask is the QQQ ask price at time t⁺. NQ(t⁺)bid is the NASDAQ 100 E-mini futures bid price and NQ(t⁺)ask is the NASDAQ 100 E-mini futures ask price at time t⁺; QDiv(t⁺) refers to the present value of the QQQ dividend from time t⁺ to time T; and Cc is the trade commission. Since QQQ prices are 1/40th of the index level, an adjustment factor of 40 is applied.

Transaction

Panel A: Pre-Decimalization (Jul 27, 2000 ~ Jan 28, 2001)

0.05% 353,120 341,812 96.80% 6.4107 3.5484 6.2775 3.7377 0.8204 (<.0001) 0.10% 324,440 306,998 94.62% 5.2944 3.3139 5.0962 3.5671 0.8046 (<.0001) 0.15% 279,138 255,802 91.64% 4.4244 3.0485 4.1420 3.3820 0.7783 (<.0001) 0.20% 223,104 197,362 88.46% 3.7663 2.7843 3.3872 3.2056 0.7413 (<.0001) 0.25% 167,616 142,501 85.02% 3.2216 2.5650 2.7361 3.0725 0.6998 (<.0001) 0.30% 116,932 94,684 80.97% 2.7937 2.4158 2.1832 3.0194 0.6566 (<.0001) 0.35% 75,093 57,474 76.54% 2.4989 2.3514 1.7402 3.0558 0.6183 (<.0001) 0.40% 44,967 32,573 72.44% 2.3261 2.3776 1.4054 3.1803 0.5855 (<.0001) 0.45% 26,012 18,078 69.50% 2.2221 2.4986 1.1553 3.3699 0.5556 (<.0001) 0.50% 14,839 9,875 66.55% 2.1576 2.7303 0.9230 3.6393 0.5355 (<.0001) Panel B: Post-Decimalization (Jan 29, 2001 ~ Jul 30, 2001)

0.05% 446,619 440,968 98.73% 4.6321 1.9583 4.5937 2.0357 0.8076 (<.0001) 0.10% 432,848 423,466 97.83% 3.8367 1.8439 3.7794 1.9445 0.7943 (<.0001) 0.15% 408,791 393,736 96.32% 3.1089 1.7088 3.0250 1.8400 0.7738 (<.0001) 0.20% 367,937 345,420 93.88% 2.4794 1.5547 2.3574 1.7246 0.7432 (<.0001) 0.25% 306,812 276,950 90.27% 1.9639 1.3997 1.7887 1.6120 0.7030 (<.0001) 0.30% 228,519 195,022 85.34% 1.5732 1.2649 1.3277 1.5373 0.6441 (<.0001) 0.35% 148,013 117,837 79.61% 1.2963 1.1748 0.9618 1.5139 0.5709 (<.0001) 0.40% 82,145 59,928 72.95% 1.1202 1.1585 0.6655 1.5792 0.4905 (<.0001) 0.45% 39,947 26,428 66.16% 1.0192 1.2529 0.4090 1.7637 0.4145 (<.0001) 0.50% 17,507 10,394 59.37% 0.9901 1.5165 0.1695 2.1472 0.3546 (<.0001)