Correlated squared returns

Dilip B. Madan; King Wang

doi:10.3934/puqr.2021007

Article Contents

2021, Volume 6, Issue 2: 139-158. Doi: 10.3934/puqr.2021007

This issue Previous Article An infinite-dimensional model of liquidity in financial markets Next Article

Correlated squared returns

Dilip B. Madan^1, , and
King Wang^2,

1.
Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA
2.
Derivative Product Strats, Morgan Stanley, New York, NY 10036, USA

Author Bio: Email: King.Wang@morganstanley.com
Email: dbm@umd.edu
Email: dbm@umd.edu

Received: April 17, 2020

Accepted: June 03, 2021

Published: June 2021

Abstract / Introduction Full Text(HTML) Figure(5) / Table(12) Related Papers Cited by

Abstract

Joint densities for a sequential pair of returns with weak autocorrelation and strong correlation in squared returns are formulated. The marginal return densities are either variance gamma or bilateral gamma. Two-dimensional matching of empirical characteristic functions to its theoretical counterpart is employed for dependency parameter estimation. Estimations are reported for 3920 daily return sequences of one thousand days. Path simulation is done using conditional distribution functions. The paths display levels of squared return correlation and decay rates for the squared return autocorrelation function that are comparable to these magnitudes in daily return data. Regressions of log characteristic functions at different time points are used to estimate time scaling coefficients. Regressions of these time scaling coefficients on squared return correlations support the view that autocorrelation in squared returns slows the rate of passage of economic time. An analysis of financial markets for 2020 in comparison with 2019 displays a post-COVID slowdown in financial markets.

Keywords:

Mathematics Subject Classification: G11, G13, G17.

Citation:

FullText(HTML)

Figure 1. Fit of Variance Gamma (VG) and Bilateral Gamma (BG) to the stationary equilibrium return distribution of a GARCH(1,1) process. The observed tail probabilities are represented by circles. The VG and BG model tail probabilities are shown by red and black dots.

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Figure 2. Sample paths for 63 days of returns from VGCTC and BGCTC. The VGCTC paths employ Cases 7 and 2 from the respective Tables of representative parameter values.

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Figure 3. Bilateral Gamma parameter quantiles for the return at the five-, ten-, fifteen-, and twenty-day horizons. The parameters were fit to simulated returns for the period using a 512 point quantization of the full set of 3920 estimated parameter values. Presented are the quantiles for the 512 sets of long horizon return densities.

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Figure 4. VGCTC second-period variance conditional on the first-period squared return. The eight cases refer to Table 2.

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Figure 5. BGCTC second-period variance conditional on the first-period squared return. The eight cases refer to Table 4.

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Table 1. Parameter quantiles for VGCTC. Displayed are the quantiles for VGCTC model parameters estimated for data on a thousand returns for 196 stocks taken at 20 randomly selected end dates between January 2008 and December 2019.

Quantile	$\sigma$	$\nu$	$\theta$	$\alpha$
1	0.0215	0.2098	2.1693	0.02
5	0.1321	0.3169	6.2207	0.02
10	0.1481	0.3785	8.5184	0.02
25	0.1819	0.4843	11.867	0.3091
50	0.2316	0.6068	15.575	0.5431
75	0.3148	0.7485	20.244	0.7738
90	0.4367	0.9314	26.874	0.9795
95	0.5278	1.1636	36.265	0.98
99	0.6602	6.5334	255.21	0.98

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Table 2. Representative VGCTC parameters. The 3920 parameter sets for 196 stocks taken at 20 randomly selected end dates for data on a thousand returns were quantized into 8 representative centroids. The final column shows the proportion represented by each centroid.

Case	$\sigma$	$\nu$	$\theta$	$\alpha$	Prop.
1	0.0153	0.5785	0.0015	0.9038	15.43
2	0.0144	0.5605	0.0016	0.6572	11.12
3	0.0175	0.7606	0.0017	0.6935	14.65
4	0.0194	0.7224	0.0017	0.3558	17.51
5	0.0126	0.3350	0.0014	0.9397	10.94
6	0.0285	1.1169	0.0040	0.4664	8.62
7	0.0146	0.4685	0.0015	0.5550	10.13
8	0.0156	0.4822	0.0016	0.3177	11.60

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Table 3. Parameter quantiles for BGCTC. Displayed are quantiles of estimated parameters for the BGCTC model. Estimates were obtained for 196 stocks taken at 20 random end dates with a thousand prior returns between January 2008 and December 2019.

Quantile	$b_{p}$	$c_{p}$	$b_{n}$	$c_{n}$	$\alpha$
1	0	0.1782	0.0033	0.1615	0.02
5	0.0043	0.9129	0.0042	0.8294	0.02
10	0.0050	1.0638	0.0048	0.9918	0.02
25	0.0064	1.3026	0.0062	1.2678	0.2526
50	0.0084	1.5988	0.0082	1.5789	0.5010
75	0.0118	2.0148	0.0120	2.0002	0.7316
90	0.0171	2.5118	0.0186	2.5029	0.9339
95	0.0215	2.9618	0.0256	2.9599	0.98
99	0.0327	4.4274	0.5241	4.3772	0.98

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Table 4. Representative parameter sets for BGCTC. The 3920 parameter estimates for 196 stocks taken at 20 random end dates between January 2008 and December 2019 were quantized into eight centroids. The last column shows the proportions of points represented by each centroid.

Case	$b_{p}$	$c_{p}$	$b_{n}$	$c_{n}$	$\alpha$	Prop.
1	0.0091	1.5746	0.0102	1.3424	0.9284	10.67
2	0.0076	1.9910	0.0080	1.7468	0.9532	18.29
3	0.0050	3.5498	0.0058	2.8806	0.8854	8.20
4	0.0087	1.7630	0.0102	1.3803	0.2893	9.52
5	0.0091	1.7250	0.0098	1.5000	0.9541	10.58
6	0.0115	1.3579	0.0125	1.1849	0.7729	17.00
7	0.0064	2.4888	0.0073	2.0861	0.9140	13.14
8	0.0158	1.0939	0.0178	0.9166	0.3755	12.60

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Table 5. Squared return correlations. For 196 stocks taken at 20 random end dates squared return correlations were estimated from paths of 252 days. For the data column, they were prior returns. For the models they were from a single simulated path using estimated parameter values based on a thousand days of data.

Quantile	Data	VGCTC	BGCTC
1	0.0265	0.1034	0.1457
5	0.0605	0.1659	0.1928
10	0.0913	0.1906	0.2279
25	0.1639	0.2352	0.2983
50	0.2459	0.2850	0.3474
75	0.3187	0.3281	0.4076
90	0.3820	0.3649	0.4646
95	0.4241	0.3936	0.4912
99	0.4972	0.4786	0.5449

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Table 6. Decay rate quantiles. For 196 stocks and 20 randomly selected end dates between January 2008 and December 2019 from a sequence of 252 squared returns an autocorrelation function was computed. The log autocorrelation function was regressed on the logarithm of the time lag. Displayed are quantiles for slope coefficients. The data column employed return sequences. The models used paths simulated from parameter estimates.

Quantile	Data	VGCTC	BGCTC
1	−0.0718	−0.0580	−0.0210
5	−0.0175	−0.0249	0.0436
10	0.0055	−0.0075	0.0726
25	0.0482	0.0238	0.1158
50	0.1100	0.0610	0.1943
75	0.1888	0.1105	0.3393
90	0.2843	0.1714	0.6415
95	0.3520	0.2206	0.9243
99	0.5271	0.2760	1.6103

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Table 7. VGCTC scaling coefficient quantiles. Displayed are quantiles of scaling coefficients. The coefficients are obtained by regressing the empirical cumulant at the longer horizon on the same for the shorter horizon. The empirical cumulant is based on 10000 simulated VGCTC paths.

Quantile	$c_{5,10}$	$c_{5,15}$	$c_{5,20}$
1	0.6178	0.6160	0.6271
5	0.6774	0.6808	0.6821
10	0.7203	0.7177	0.7187
25	0.7776	0.7768	0.7695
50	0.8545	0.8548	0.8453
75	0.9601	0.9625	0.9423
90	1.0890	1.0886	1.0803
95	1.1828	1.2213	1.2029
99	2.0039	3.0077	4.0080

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Table 8. BGCTC scaling coefficient quantiles. Displayed are quantiles of scaling coefficients. The coefficients are obtained by regressing the empirical cumulant at the longer horizon on the same for the shorter horizon. The empirical cumulant is based on 10000 simulated BGCTC paths.

Quantile	$c_{5,10}$	$c_{5,15} $	$c_{5,20}$
1	0.6673	0.6851	0.6513
5	0.7673	0.7741	0.7568
10	0.8164	0.8346	0.8125
25	0.9364	0.9462	0.9453
50	1.2345	1.2530	1.2425
75	1.9509	2.3806	2.4516
90	2.0126	3.0054	4.0017
95	2.0346	3.0318	4.0375
99	2.1364	3.1371	4.1237

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Table 9. Regression of time scaling coefficients on squared return correlations (SQRC) dependent. Presented are the results of regressing time scaling coefficients at the ten-, fifteen-, and twenty-day horizon on the associated squared return correlations. t-statistics in parenthesis.

Variable	Constant	SQRC
	VGCTC
$c_{5,10}$	0.9391	−0.2335
$c_{5,10}$	(62.48)	−(4.66)
$c_{5,15}$	0.9715	−0.3410
$c_{5,15}$	(66.91)	(−6.71)
$c_{5,20}$	0.9398	−0.2758
$c_{5,20}$	(62.38)	−(5.49)
	BGCTC
$c_{5,10}$	1.9408	−2.6867
$c_{5,10}$	(87.57)	(−30.06)
$c_{5,15}$	2.6457	−4.8397
$c_{5,15}$	(66.91)	(−30.35)
$c_{5,20}$	3.2558	−6.8205
$c_{5,20}$	(58.23)	(−30.25)

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Table 10. p-value quantiles. Presented are selected quantiles of p-values for distributional equality between 2019 and 2020 for 824 stocks.

Quantile	p-value
1	2.92e−9
5	2.8e−7
10	5.92e−6
25	0.00022
50	0.0061
75	0.0466
90	0.1862
95	0.3316
99	0.6267

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Table 11. Bilateral gamma parameter quantiles.

Quantile	2019				2020
Quantile	$b_{p}$	$c_{p}$	$b_{n}$	$c_{n}$	$b_{p}$	$c_{p}$	$b_{n}$	$c_{n}$
1	0.0001	1.1635	0.0007	0.7347	0.0069	0.5897	0.0084	0.4720
5	0.0003	1.5256	0.0012	1.0004	0.0095	0.7328	0.0117	0.6220
10	0.0006	1.7956	0.0021	1.1641	0.0109	0.8460	0.0136	0.6921
25	0.0019	2.3483	0.0042	1.5314	0.0138	0.9958	0.0169	0.8474
50	0.0048	3.4532	0.0068	2.3362	0.0182	1.2260	0.0215	1.1009
75	0.0067	16.343	0.0095	6.7286	0.0250	1.5685	0.0272	1.4138
90	0.0092	113.26	0.0129	18.250	0.0347	2.1524	0.0331	1.8616
95	0.0110	184.27	0.0148	93.635	0.0413	2.6990	0.0370	2.2755
99	0.0171	312.31	0.0198	123.74	0.0583	5.4866	0.0503	4.6835

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Table 12. Squared return correlation quantiles.

Quantile	2019	2020
1	0.0095	0.0650
5	0.0341	0.1212
10	0.0553	0.1593
25	0.1155	0.2390
50	0.2095	0.3463
75	0.2854	0.4370
90	0.3500	0.5212
95	0.3766	0.5743
99	0.4454	0.6511

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