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High dimensional Markovian trading of a single stock

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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  • 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.

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

    Citation:

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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

    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

    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

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

    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 $

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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