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Implied price processes anchored in statistical realizations

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  • It is observed that statistical and risk neutral densities of compound Poisson processes are unconstrained relative to each other. Continuous processes are too constrained and generally not consistent with market data. Pure jump limit laws deliver operational models simultaneously consistent with both data sets with the additional imposition of no measure change on the arbitrarily small moves. The measure change density must have a finite Hellinger distance from unity linking the two worlds. Models are constructed using the bilateral gamma and the CGMY models for the risk neutral specification. They are linked to the physical process by measure change models. The resulting models simultaneously calibrate statistical tail probabilities and option prices. The resulting models have up to eight or ten parameters permitting the study of risk reward relations at a finer level. Rewards measured by power variations of the up and down moves are observed to value negatively(positively) the even(odd) variations of their own side with the converse holding for the opposite side.

    Mathematics Subject Classification: Primary: 60G18, 60G51, 91G20.

    Citation:

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  • Figure 1.  Fit of Bilateral Gamma based Sato Process across maturities between a month and a quarter on SPY options as at January 2, 2010

    Figure 2.  Fit of bilateral gamma based model to the observed tail probabilities as estimated from bootstrapped data for a thousand immdeiately prior returns over a gamma distributed number of days with a mean of five days and a volatility of 3 days. The estimated number of days is 8.24

    Figure 3.  Fit of bilateral CGMY model to SPY options on April 4, 2017. The estimated number of days is 2.90

    Figure 4.  Fit of bilateral CGMY based model to tail probabilities estimated from bootstrapped return data for a gamma distributed random number of days with a mean of five days and a volatility of three days. The estimated number of days is 2.90

    Table 1.   

    Distance to Unity
    BG Based CGMY Based
    Percentile Up Moves Down Moves Up Moves Down Moves
    1 0 0.9432 0 0
    5 0 1.0081 0 0
    10 0 1.0840 0 0
    25 0.0006 1.2174 0.9459 2.0824
    50 0.0248 1.4629 1.4314 2.8091
    75 0.1287 1.8979 2.2129 3.7624
    90 0.2744 2.4683 2.9719 5.5502
    95 0.3880 2.9356 3.8536 6.7162
    99 0.8490 3.8934 40.3766 41.2440
     | Show Table
    DownLoad: CSV

    Table 2.   

    Distance to Unity Regressions BG
    Upside Downside
    Coeff t-stat Coeff t-stat
    $ Constant $ $ -0.1251 $ $ -16,84 $ $ -0.3131 $ $ -8.53 $
    $ b $ $ 119.7625 $ $ 111.89 $ $ 21.3256 $ $ 18,59 $
    $ c $ $ 0.0206 $ $ 7.49 $ $ 1.2206 $ $ 51.61 $
    $ a $ $ 0.0035 $ $ 0.58 $ $ 0.0072 $ $ 0.22 $
    $ g $ $ -0.0018 $ $ -0.40 $ $ 0.0419 $ $ 1.27 $
     | Show Table
    DownLoad: CSV

    Table 3.   

    Distance to Unity Regressions CGMY
    Upside Downside
    Coeff t-stat Coeff t-stat
    $ Constant $ $ -2.5279 $ $ -2.58 $ $ -5.8442 $ $ -4.39 $
    $ C $ $ 152.77 $ $ 206.41 $ $ 128.29 $ $ 133.18 $
    $ M/G $ $ 0 $ $ -0.11 $ $ 0 $ $ 5.89 $
    $ Y $ $ 1.3876 $ $ 1.47 $ $ 6.4309 $ $ 4.77 $
    $ a $ $ 0.5334 $ $ 0.98 $ $ 0.7 $ $ 1.32 $
    $ g $ $ 0.3370 $ $ 0.99 $ $ -0.0985 $ $ -0.20 $
     | Show Table
    DownLoad: CSV

    Table 4.   

    Risk Reward Relations BG
    Dependent Variable
    Explanatory Variable mp mn mp-mn mp-mn
    Constant $ -0.8953 $ $ -4.8001 $ $ 3.5352 $ $ 25.6063 $
    $ -4.69 $ $ -13.62 $ $ 8.06 $ $ 13.81 $
    $ s_{p}/s $ $ 2.6610 $ $ -0.0049 $ $ 2.8800 $ $ -0.7228 $
    $ 120.02 $ $ 0.12 $ $ 56.38 $ $ -12.72 $
    $ c_{p}/c $ $ -3.6327 $ $ -0.0206 $ $ -3.9597 $ $ 3.7913 $
    $ -90.60 $ $ 0.28 $ $ -42.87 $ $ 33.29 $
    $ q_{p}/q $ $ 1.8544 $ $ 0.0187 $ $ 2.0101 $ $ 2.9772 $
    $ 73.54 $ $ 0.40 $ $ 34.60 $ $ 46.13 $
    $ s_{n} $ $ -0.0403 $ $ 4.9494 $ $ -5.0418 $
    $ -2.23 $ $ 147.94 $ $ -121.01 $
    $ c_{n} $ $ 0.1079 $ $ -5.5852 $ $ 5.8038 $
    $ 2.76 $ $ -77.38 $ $ 64.57 $
    $ q_{n} $ $ -0.0639 $ $ 2.0908 $ $ -2.2128 $
    $ -3.05 $ $ 53.98 $ $ -45.88 $
    RSQ $ 0.9827 $ $ 0.9975 $ $ 0.9960 $ $ 0.9327 $
     | Show Table
    DownLoad: CSV

    Table 5.   

    Risk Reward Relations CGMY
    Dependent Variable
    Explanatory Variable mp mn mp-mn mp-mn
    Constant $ 30.6410 $ $ 10.3237 $ $ 5.2518 $ $ 26.5127 $
    $ 5.56 $ $ 1.35 $ $ 0.69 $ $ 4.08 $
    $ s_{p}/s $ $ 49.0703 $ $ 23.2615 $ $ 15.0069 $ $ 1.7536 $
    $ 108.85 $ $ 37.14 $ $ 24.08 $ $ 26.56 $
    $ c_{p}/c $ $ -224.1851 $ $ -128.8385 $ $ -28.0084 $ $ 10.8180 $
    $ -72.01 $ $ -29.79 $ $ -6.51 $ $ 175.89 $
    $ q_{p}/q $ $ 302.6475 $ $ 181.1018 $ $ 20.0436 $ $ 3.4408 $
    $ 63.64 $ $ 27.41 $ $ 3.05 $ $ 19.16 $
    $ f_{p}/f $ $ -126.1078 $ $ -76.1065 $ $ -5.5934 $ $ -2.1609 $
    $ -60.63 $ $ -26.34 $ $ -1.95 $ $ -16.55 $
    $ s_{n} $ $ -2.0138 $ $ 22.3100 $ $ -22.9098 $
    $ -7.37 $ $ 58.79 $ $ -60.68 $
    $ c_{n} $ $ 8.4408 $ $ -61.4901 $ $ 61.8770 $
    $ 6.23 $ $ -32.67 $ $ 33.05 $
    $ q_{n} $ $ -8.4234 $ $ 61.0450 $ $ -58.6393 $
    $ -4.30 $ $ 22.45 $ $ -21.67 $
    $ f_{n} $ $ 2.6993 $ $ -20.8879 $ $ 19.2105 $
    $ 3.19 $ $ -17.75 $ $ 16.41 $
    RSQ $ .9903 $ $ .9729 $ $ .7313 $ $ .7530 $
     | Show Table
    DownLoad: CSV
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