# American Institute of Mathematical Sciences

March  2022, 7(1): 45-66. doi: 10.3934/puqr.2022004

## Lower and upper pricing of financial assets

 1 School of Business, University of South Australia, Adelaide, SA 5001, Australia 2 Department of Finance, Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA 3 Department of Actuarial Studies and Business Analytics, Macquarie Business School, Macquarie University, Sydney, NSW 2109, Australia

dbm@rhsmith.umd.edu

Received  October 15, 2021 Accepted  March 16, 2022 Published  March 2022 Early access  March 2022

Modeling of uncertainty by probability errs by ignoring the uncertainty in probability. When financial valuation recognizes the uncertainty of probability, the best the market may offer is a two price framework of a lower and upper valuation. The martingale theory of asset prices is then replaced by the theory of nonlinear martingales. When dealing with pure jump compensators describing probability, the uncertainty in probability is captured by introducing parametric measure distortions. The two price framework then alters asset pricing theory by requiring two required return equations, one each for the lower upper valuation. Proxying lower and upper valuations by daily lows and highs, the paper delivers the first empirical study of nonlinear martingales via the modeling and simultaneous estimation of the two required return equations.

Citation: Robert Elliott, Dilip B. Madan, Tak Kuen Siu. Lower and upper pricing of financial assets. Probability, Uncertainty and Quantitative Risk, 2022, 7 (1) : 45-66. doi: 10.3934/puqr.2022004
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Moment transforms of bilateral gamma parameter estimates
 Quantile d v s k 1 −0.0038 0.0070 −0.9097 3.0281 5 −0.0016 0.0084 −0.6409 3.2262 10 −0.0007 0.0093 −0.4789 3.4830 25 0.0004 0.0115 −0.1229 3.9540 50 0.0012 0.0150 0.0418 4.5340 75 0.0017 0.0209 0.1493 5.1621 90 0.0023 0.0305 0.2635 5.8350 95 0.0027 0.0408 0.3535 6.3046 99 0.0038 0.0707 0.5982 7.4617
 Quantile d v s k 1 −0.0038 0.0070 −0.9097 3.0281 5 −0.0016 0.0084 −0.6409 3.2262 10 −0.0007 0.0093 −0.4789 3.4830 25 0.0004 0.0115 −0.1229 3.9540 50 0.0012 0.0150 0.0418 4.5340 75 0.0017 0.0209 0.1493 5.1621 90 0.0023 0.0305 0.2635 5.8350 95 0.0027 0.0408 0.3535 6.3046 99 0.0038 0.0707 0.5982 7.4617
Lower and upper return quantiles
 Quantile Lower Upper $1$ $-0.0540$ $-0.0479$ $5$ $-0.0267$ $-0.0227$ $10$ $-0.0167$ $-0.0146$ $25$ $-0.0053$ $-0.0056$ $50$ $0.0020$ $0.0010$ $75$ $0.0083$ $0.0070$ $90$ $0.0167$ $0.0151$ $95$ $0.0237$ $0.0223$ $99$ $0.0458$ $0.0480$
 Quantile Lower Upper $1$ $-0.0540$ $-0.0479$ $5$ $-0.0267$ $-0.0227$ $10$ $-0.0167$ $-0.0146$ $25$ $-0.0053$ $-0.0056$ $50$ $0.0020$ $0.0010$ $75$ $0.0083$ $0.0070$ $90$ $0.0167$ $0.0151$ $95$ $0.0237$ $0.0223$ $99$ $0.0458$ $0.0480$
SPY states
 State $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $1$ $0.0054$ $1.2937$ $0.0097$ $0.6195$ $2$ $0.0057$ $1.2603$ $0.0059$ $1.0112$ $3$ $0.0053$ $1.3110$ $0.0077$ $0.7423$ $4$ $0.0034$ $2.3575$ $0.0049$ $1.3701$ $5$ $0.0061$ $1.1068$ $0.0085$ $0.6450$ $6$ $0.0042$ $1.7842$ $0.0059$ $1.0625$ $7$ $0.0058$ $1.2971$ $0.0077$ $0.8604$ $8$ $0.0011$ $12.049$ $0.0028$ $4.4096$
 State $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $1$ $0.0054$ $1.2937$ $0.0097$ $0.6195$ $2$ $0.0057$ $1.2603$ $0.0059$ $1.0112$ $3$ $0.0053$ $1.3110$ $0.0077$ $0.7423$ $4$ $0.0034$ $2.3575$ $0.0049$ $1.3701$ $5$ $0.0061$ $1.1068$ $0.0085$ $0.6450$ $6$ $0.0042$ $1.7842$ $0.0059$ $1.0625$ $7$ $0.0058$ $1.2971$ $0.0077$ $0.8604$ $8$ $0.0011$ $12.049$ $0.0028$ $4.4096$
Measure distortion parameters for the myopic case
 $b$ $c$ $2017$ $8.1711$ $863.3857$ $t{\text{-}}stat$ $(3.62)$ $(3.12)$ $2018$ $1.6311$ $78.1643$ $t{\text{-}}stat$ $(16.31)$ $(13.13)$ $2019$ $1.8594$ $85.7347$ $t{\text{-}}stat$ $(16.36)$ $(13.25)$
 $b$ $c$ $2017$ $8.1711$ $863.3857$ $t{\text{-}}stat$ $(3.62)$ $(3.12)$ $2018$ $1.6311$ $78.1643$ $t{\text{-}}stat$ $(16.31)$ $(13.13)$ $2019$ $1.8594$ $85.7347$ $t{\text{-}}stat$ $(16.36)$ $(13.25)$
Measure distortion parameters for the Markov modulated case
 $b$ $c$ $2017$ $0.008962$ $0.475662$ $t{\text{-}}stat$ $(30.04)$ $(12.79)$ $2018$ $0.039580$ $0.500025$ $t{\text{-}}stat$ $(109.82)$ $(47.43)$ $2019$ $0.014905$ $0.469397$ $t{\text{-}}stat$ $(48.38)$ $(20.64)$
 $b$ $c$ $2017$ $0.008962$ $0.475662$ $t{\text{-}}stat$ $(30.04)$ $(12.79)$ $2018$ $0.039580$ $0.500025$ $t{\text{-}}stat$ $(109.82)$ $(47.43)$ $2019$ $0.014905$ $0.469397$ $t{\text{-}}stat$ $(48.38)$ $(20.64)$
Maximal risk charges
 Modulated No Modulation $2017$ $0.01884$ $0.00946$ $2018$ $0.07916$ $0.02087$ $2019$ $0.03175$ $0.02169$
 Modulated No Modulation $2017$ $0.01884$ $0.00946$ $2018$ $0.07916$ $0.02087$ $2019$ $0.03175$ $0.02169$
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