High dimensional Markovian trading of a single stock

Robert Elliott; Dilip B. Madan; King Wang

doi:10.3934/fmf.2022001

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2022, Volume 1, Issue 3: 375-396. Doi: 10.3934/fmf.2022001 open access

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High dimensional Markovian trading of a single stock

1.
University of South Australia, City West Campus (WL3-44), Australia
2.
Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA
3.
Derivative Product Strats, Morgan Stanley, 1585 Broadway, 5th floor, New York, NY 10036, USA

Received: December 2021

Revised: April 2022

Early access: July 2022

Published: September 2022

This paper is the private opinion of the authors and does not necessarily reflect the policy and views of Morgan Stanley

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Abstract

OU processes with long term drifts that are Tempered Fractional Lévy Processes reduce to a $ d+1 $ dimensional Markovian system when the parameter $ d $ is an integer. Markovian optimization problems are formulated for the proportion of a dollar to be invested in a risky stock following the specified dynamics. The objective evaluates the cumulated discounted returns to a dollar being invested continuously through time. Risk sensitivity is accomplished by maximizing a conservative financial valuation seen as a nonlinear expectation. Trading policies are determined by solutions of nonlinear partial integro-differential equations. The policies are evaluated on a quantized set of representative Markovian states in the higher dimensions. Gaussian Process Regressions are then employed to deliver general functions of the state. The nonlinear policy functions deliver good trading outcomes on simulated data. The policy functions are then applied to trading $ SPY $ from $ 2008 $ through $ 2020 $ with good results. They are also employed to trade $ 874 $ stocks over a four year period with reasonable results. Only three policy functions trained on one year of $ SPY $ data for $ 2020 $ are reported on. It is conjectured that a variety of functions may be trained on other data sets over other periods and selections may then be made for the functions actually traded on a particular stock at a particular time from this collection. The underlying dynamics may also be further enriched by allowing for a Markov chain of states that code changes in the parameter values for the driving Lévy process.

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Mathematics Subject Classification: 60G18, 60G51, 91G10, 91G15, 91B55.

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Figure 1. HDMBG fit to SPY returns for 252 daily returns ending December 31, 2020. Observed tail probabilities are shown as circles and model probabilities are presented as dots

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Figure 2. Cumulated returns from following three policy functions over the years 2008 through 2020. The results for $ c = 1, $ $ 0.25, $ and $ 0.01 $ are presented in blue, red and black respectively

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Figure 3. Time series for the proportion of the dollar held in $ SPY $ over time by the three policies. The values for $ c = 1, $ $ c = .25, $ and $ c = .01 $ are represented by blue red and black dots

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Figure 4. Cumulated returns from three policies for four d values trading SPY over the years 2008 through 2020

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Figure 5. Cumulated returns on a dollar invested in $ 874 $ stocks daily generated by three policy functions trained to four values of $ d $ on $ SPY $ data for $ 2020. $ The trading period was $ 2016 $ through $ 2019 $

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Table 1. Marginal Percentiles of Variates

Percentile	Level	Drift	Speed	Accel.
1	-0.1021	-0.0573	-0.0673	-0.0896
5	-0.0623	-0.0332	-0.0387	-0.0501
10	-0.0433	-0.0225	-0.0267	-0.0313
25	-0.0193	-0.0085	-0.0101	-0.0099
50	-0.0012	0.0003	0.0005	0.0017
75	0.0090	0.0054	0.0059	0.0070
90	0.0157	0.0086	0.0097	0.0120
95	0.0199	0.0105	0.0120	0.0155
99	0.0284	0.0145	0.0168	0.0228

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Table 2. RSQR Results

		c=1	c=.25	c=.01
Policy	linear	0.4729	0.4179	0.2296
	nonlinear	0.9126	0.8981	0.8275
Value	linear	-0.3893	-0.2615	0.4087
	nonlinear	0.9848	0.9735	0.9968

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Table 3. Policy Function Regressions

	c=1		c=.25		c=.01
Variable	Coef	tstat	Coef	tstat	Coef	tstat
Constant	-0.0868	-30.17	-0.0109	-37.19	-0.0441	-10.82
Level	-8.8592	-19.50	-9.4211	-22.20	-2.6597	-4.13
Drift	10.3055	18.47	10.8714	20.86	5.7987	7.33
Speed	-2.3904	-7.47	-3.6848	-12.32	-0.8702	-1.92
Accel	0.6178	1.23	2.4821	5.30	-5.0021	-7.04

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Table 4. Gradient Percentiles c = .01

Percentile	Level	Drift	Speed	Accel.
1	-84.45	-67.82	-230.47	-75.49
5	-46.81	-37.07	-170.21	-44.61
10	-31.91	-24.63	-130.60	-31.60
25	-16.02	-6.54	-53.31	-13.52
50	-1.30	11.14	-8.27	0.36
75	10.26	52.03	10.87	15.40
90	23.10	103.67	36.96	33.61
95	33.37	141.53	60.29	48.37
99	54.40	212.27	116.18	99.08

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Table 5. Cumulated Return Percentiles on Simulated Paths

Percentile	c=1	c=.25	c=.01	benchmark
1	0.1881	0.1764	0.0514	-0.0106
5	0.2521	0.2459	0.1132	0.0114
10	0.2872	0.2787	0.1513	0.0171
25	0.3489	0.3436	0.2068	0.0258
50	0.4267	0.4252	0.2798	0.0353
75	0.5263	0.4998	0.3577	0.0452
90	0.6069	0.5809	0.4391	0.0560
95	0.6747	0.6330	0.4829	0.0650
99	0.8192	0.7515	0.5664	0.0853

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Table 6. Performance Measures

Measure	Quartiles	c=1	c=.25	c=.01	SPY
TR	25	-0.0035	-0.0029	0.0071	-0.0027
	50	0.0022	0.0021	0.0130	0.0392
	75	0.0092	0.0140	0.0204	0.0705
SH	25	-0.7304	-0.6990	1.0007	-0.0789
	50	0.5068	0.5397	2.1524	1.1415
	75	1.6990	1.6258	3.4839	2.1767
GL	25	0.8611	0.8745	1.2561	0.9876
	50	1.0976	1.0955	1.5333	1.2054
	75	1.3977	1.4129	1.9975	1.4584
PP	25	0.4844	0.4844	0.5469	0.5000
	50	0.5469	0.5156	0.5938	0.5391
	75	0.5781	0.5547	0.6406	0.5781
AI	25	-0.0263	-0.0259	0.0400	-0.0028
	50	0.0186	0.0196	0.0811	0.0414
	75	0.0643	0.0696	0.1386	0.0799
MD	25	0.0056	0.0059	0.0048	0.0383
	50	0.0085	0.0084	0.0090	0.0621
	75	0.0128	0.0150	0.0167	0.0993

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