The procedures for Gibbs-sampling described in the previous sections are applied here to the PCP deviations of S&P 500 and the DAX. Gibbs-sampling is run such that the first 2,000 draws are discarded and the next 10,000 are recorded. We employ almost non-informative priors for all the models’ parameters. Table 1 presents the marginal posterior distributions of the parameters that result from Gibbs-sampling for the PCP deviations of S&P 500 and the DAX, respectively. At the end of each run of Gibbs-sampling, we have a simulated set of
{ S
t,t
=1,2...T }
and thus, of{ S
jt,t
=1,2,...T
,j
=1,2,3}
, and . Figures 2(a), 2(b), 2(c) and Figures 3(a), 3(b), and 3(c) depict probabilities of low-, medium-, and high-variance states for the PCP deviations of S&P 500 and the DAX, respectively, that result from the Gibbs-sampling simulation.Using the particular realizations of the states and the parameters for each run of Gibbs-sampling, we can calculate
σ
t2 fort
=1,2,LT
using equation (8). Thus, when all the iterations are over, we have 10,000 sets of realized variances{
tt T }
T , 1,2,L
~2 =
σ
2 =σ
of PCP deviations. Figures 2(d) and 3(d) plot the average of 10,000 sets of ~2σ
T, which are our estimates of the variance of the S&P 500’s and the DAX’s PCP deviations. Tables 2 (a) and (b) present variance ratios for original daily deviations from PCP for S&P 500 and DAX, respectively. Only the DAX displays mean reversions at long horizons. The smallest p-value is 0.028 at 45 days lag.Table 3 (a) and (b), in which variance ratios for standardized daily deviations from PCP for S&P 500 and DAX are presented, respectively. The DAX also displays mean reversion at long horizons and its smallest p-value is 0.028 at a lag of 40 days.
The evidence is weak that the standardized returns approach to estimating the VRs suggests that mean reversion, if it is present, occurs at shorter lags.
8. Summary
Previous studies indicate that when market frictions are taken into account, the deviations from PCP can fluctuate within a bounded interval without giving rise to any arbitrage profit. This study presents a model of the option price mean reverting to a function form of PCP. The variance ratio test is employed to examine whether the deviations of PCP exhibit mean reversion. We make appropriate allowance for heteroskedasticity when basing inference on the VR statistic by using the Gibbs-sampling approach in the context of a three-state Markov switching model.
The empirical result shows that PCP deviations from the electronic screen-traded DAX index options, which are calculated as if the dividends are reinvested in the index, display mean reversion at long horizons. On the other hand, those deviations from floor-traded S&P 500 index options, which do not correct for dividend payments, vary randomly.
Table 1：The estimated parameters from the Bayesian Gibbs-sampling approach to a three-state Markov-switching model of heteroskedasticity for S&P 500’s and DAX’s daily PCP deviations
Posterior distribution
S&P 500 DAX Parameter Mean Std. Mean Std.
P11 0.8912 0.0344 0.9244 0.0168 P12 0.0561 0.0402 0.0713 0.0165 P21 0.0197 0.0230 0.0677 0.0511 P22 0.8926 0.0489 0.6355 0.0975
P
31 0.2575 0.0689 0.6257 0.1234P
32 0.0882 0.0519 0.0390 0.0379σ
12 0.0216 0.0064 0.0280 0.0026σ
22 0.0559 0.0174 0.0843 0.0274 σ32 1.1807 0.2525 2.6624 0.7211Table 2(a)：Variance ratios for original daily deviations from PCP in S&P500
Lag(days) VR sampling distribution
( VR
r( ) k )
K VR(k) Mean Std Median p-value
2 1.4616 1.0343 0.0621 1.0340 1.0000
5 2.1525 1.0796 0.1266 1.0738 1.0000
10 1.1489 0.9957 0.0578 0.9950 0.9950
15 1.1816 0.9699 0.0922 0.9671 0.9854
20 1.2202 0.9536 0.1177 0.9491 0.9824
25 1.2907 0.9561 0.1385 0.9490 0.9864
30 1.3059 0.9564 0.1602 0.9467 0.9790
35 1.2527 0.9491 0.1812 0.9376 0.9440
40 1.1663 0.9425 0.2004 0.9288 0.8667
45 1.1181 0.9464 0.2172 0.9284 0.7957
50 1.0792 0.9496 0.2341 0.9290 0.7294
Note：1. The sampling distribution is based on Gibbs-sampling-augmented randomization.
2. The p-value is the frequency with which the simulated VR is smaller than the historical sample value, which is observed in the Gibbs-sampling-augmented randomization under the null hypothesis.
Table 2(b)：Variance ratios for original daily deviations from PCP in DAX
Lag(days) VR sampling distribution
( VR
r( ) k )
K VR(k) Mean Std Median p-value
2 1.2283 1.0245 0.0648 1.0245 0.9991
5 1.5501 1.0438 0.1184 1.0401 0.9999
10 1.0528 0.9789 0.0471 0.9794 0.9456
15 1.0141 0.9520 0.0808 0.9529 0.7782
20 0.9131 0.9313 0.1043 0.9315 0.4293
25 0.8369 0.9294 0.1237 0.9257 0.2279
30 0.7666 0.9291 0.1450 0.9230 0.1305
35 0.6626 0.9216 0.1671 0.9131 0.0513
40 0.5893 0.9153 0.1859 0.9049 0.0282
45 0.5635 0.9174 0.2007 0.9050 0.0237
50 0.5484 0.9189 0.2158 0.9029 0.0254
Note：1. The sampling distribution is based on Gibbs-sampling-augmented randomization.
2. The p-value is the frequency with which the simulated VR is smaller than the historical sample value, which is observed in the Gibbs-sampling-augmented randomization under the null hypothesis.
25
Table 3(a)：Variance ratios for standardized daily deviations from PCP in S&P500
Lag(days) VR posterior distribution
( VR
*( ) k )
VR sampling distribution(
VRr*( )
k)
k Mean Std Median Mean Std Median p-value
2 1.5519 0.0209 1.5503 1.0001 0.0346 0.9998 1.0000
5 2.6193 0.0887 2.6043 0.9998 0.0758 0.9981 1.0000
10 1.3209 0.0387 1.3120 0.9982 0.0645 0.9984 1.0000
15 1.4444 0.0788 1.4244 0.9971 0.1047 0.9938 0.9996
20 1.5463 0.0975 1.5243 0.9963 0.1358 0.9911 0.9997
25 1.6176 0.0995 1.6000 0.9959 0.1619 0.9890 0.9995
30 1.6019 0.0992 1.5897 0.9958 0.1850 0.9862 0.9959
35 1.5043 0.0997 1.4955 0.9961 0.2063 0.9827 0.9811
40 1.3640 0.1008 1.3590 0.9966 0.2263 0.9803 0.9228
45 1.2431 0.1001 1.2410 0.9971 0.2449 0.9760 0.8256
50 1.1392 0.1031 1.1396 0.9975 0.2622 0.9726 0.7141
Note：1. The sampling distribution is based on Gibbs-sampling-augmented randomization 2. The p-value is the frequency with which the realizations of the Gibbs sampling of the
posterior distribution are smaller than the corresponding realization under the null hypothesis.
26
Table 3(b)：Variance ratios for standardized daily deviations from PCP in DAX
Lag(days) VR posterior distribution
( VR
*( ) k )
VR sampling distribution(
VRr*( )
k)
k Mean Std Median Mean Std Median p-value
2 1.3804 0.0299 1.3852 1.0001 0.0343 1.0002 1.0000
5 2.0731 0.0830 2.0853 1.0001 0.0752 0.9979 1.0000
10 1.1008 0.0201 1.1017 0.9970 0.0642 0.9966 0.9388
15 0.9341 0.0389 0.9332 0.9963 0.1050 0.9931 0.2978
20 0.8265 0.0461 0.8252 0.9960 0.1367 0.9906 0.1166
25 0.8670 0.0449 0.8664 0.9963 0.1632 0.9876 0.2313
30 0.8349 0.0455 0.8347 0.9970 0.1868 0.9845 0.2062
35 0.7005 0.0483 0.7003 0.9976 0.2081 0.9801 0.0641
40 0.6078 0.0504 0.6068 0.9983 0.2281 0.9769 0.0277
45 0.6136 0.0526 0.6121 0.9991 0.2467 0.9728 0.0411
50 0.6213 0.0542 0.6196 1.0000 0.2645 0.9683 0.0580
Note：1. The sampling distribution is based on Gibbs-sampling-augmented randomization.
2. The p-value is the frequency with which the realizations of the Gibbs sampling of the
posterior distribution are smaller than the corresponding realization under the null hypothesis.
27
Figure 1(a)：Daily PCP Deviations from S&P500
Figure 1(b)：Daily PCP Deviations from DAX
28
Figure 2(a)：Probability of a Low-variance State for S&P500’s PCP Deviations (Gibbs Sampling)
Figure 2(b)：Probability of a Medium-variance State for S&P500’s PCP Deviations (Gibbs Sampling)
29
Figure 2(c)：Probability of High-variance State for S&P500’s PCP Deviations (Gibbs Sampling)
Figure 2(d)：Estimated Variance of S&P500’s PCP Deviations (Gibbs Sampling)
30
Figure 3(a)：Probability of a Low-variance State for DAX’s PCP Deviations (Gibbs Sampling)
Figure 3(b)：Probability of a Medium-variance State for DAX’s PCP Deviations (Gibbs Sampling)
31
Figure 3(c)：Probability of a High-variance State for DAX’s PCP Deviations (Gibbs Sampling)
Figure 3(d)：Estimated Variance of DAX’s PCP Deviations (Gibbs Sampling)
32
33
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