# American Institute of Mathematical Sciences

doi: 10.3934/fmf.2021008
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## Implied price processes anchored in statistical realizations

 1 Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA 2 Morgan Stanley, 1585 Broadway, 5th floor, New York, NY 10036, USA

Received  August 2021 Revised  December 2021 Early access April 2022

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.

Citation: Dilip B. Madan, King Wang. Implied price processes anchored in statistical realizations. Frontiers of Mathematical Finance, doi: 10.3934/fmf.2021008
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##### References:
Fit of Bilateral Gamma based Sato Process across maturities between a month and a quarter on SPY options as at January 2, 2010
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
Fit of bilateral CGMY model to SPY options on April 4, 2017. The estimated number of days is 2.90
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
 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
 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
 Distance to Unity Regressions BG Upside Downside Coeff t-stat Coeff t-stat $Constant$ $-0.1251$ $-16,84$ $-0.3131$ $-8.53$ $b$ $119.7625$ $111.89$ $21.3256$ $18,59$ $c$ $0.0206$ $7.49$ $1.2206$ $51.61$ $a$ $0.0035$ $0.58$ $0.0072$ $0.22$ $g$ $-0.0018$ $-0.40$ $0.0419$ $1.27$
 Distance to Unity Regressions BG Upside Downside Coeff t-stat Coeff t-stat $Constant$ $-0.1251$ $-16,84$ $-0.3131$ $-8.53$ $b$ $119.7625$ $111.89$ $21.3256$ $18,59$ $c$ $0.0206$ $7.49$ $1.2206$ $51.61$ $a$ $0.0035$ $0.58$ $0.0072$ $0.22$ $g$ $-0.0018$ $-0.40$ $0.0419$ $1.27$
 Distance to Unity Regressions CGMY Upside Downside Coeff t-stat Coeff t-stat $Constant$ $-2.5279$ $-2.58$ $-5.8442$ $-4.39$ $C$ $152.77$ $206.41$ $128.29$ $133.18$ $M/G$ $0$ $-0.11$ $0$ $5.89$ $Y$ $1.3876$ $1.47$ $6.4309$ $4.77$ $a$ $0.5334$ $0.98$ $0.7$ $1.32$ $g$ $0.3370$ $0.99$ $-0.0985$ $-0.20$
 Distance to Unity Regressions CGMY Upside Downside Coeff t-stat Coeff t-stat $Constant$ $-2.5279$ $-2.58$ $-5.8442$ $-4.39$ $C$ $152.77$ $206.41$ $128.29$ $133.18$ $M/G$ $0$ $-0.11$ $0$ $5.89$ $Y$ $1.3876$ $1.47$ $6.4309$ $4.77$ $a$ $0.5334$ $0.98$ $0.7$ $1.32$ $g$ $0.3370$ $0.99$ $-0.0985$ $-0.20$
 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 $0.9827$ $0.9975$ $0.9960$ $0.9327$
 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 $0.9827$ $0.9975$ $0.9960$ $0.9327$
 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 $.9903$ $.9729$ $.7313$ $.7530$
 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 $.9903$ $.9729$ $.7313$ $.7530$
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