# 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
##### References:
 [1] Ahimud, Y. and Mendelson, H., Asset pricing and the bid-Ask spread, Journal of Financial Economics, 1986, 17(2): 223−249. doi: 10.1016/0304-405X(86)90065-6. [2] Andersen, T. W. and Darling, D. A., Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes, The Annals of Mathematical Statistics, 1952, 23(2): 193−212. doi: 10.1214/aoms/1177729437. [3] Artzner, P., Delbaen, F., Eber, J. M. and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203−228. doi: 10.1111/1467-9965.00068. [4] Back, K. and Baruch, S., Information in securities markets: Kyle meets glosten and milgrom, Econometrica, 2004, 72(2): 433−465. doi: 10.1111/j.1468-0262.2004.00497.x. [5] Becherer, D. and Davis, M. H. A., Arrow-Debreu prices, In: Cont, R. (ed.), Encyclopedia of Quantitative Finance, 2010. https://doi.org/10.1002/9780470061602.eqf04018. [6] Carr, P., Geman, H., Madan, D. B. and Yor, M., The fine structure of asset returns: An empirical investigation, Journal of Business, 2002, 75(2): 305−332. doi: 10.1086/338705. [7] Cherny, A. and Madan, D. B., New measures for performance evaluation, Review of Financial Studies, 2009, 22(7): 2571−2606. doi: 10.1093/rfs/hhn081. [8] Cherny, A. and Madan, D. B., Markets as a counterparty: An introduction to conic finance, International Journal of Theoretical and Applied Finance, 2010, 13(8): 1149−1177. doi: 10.1142/S0219024910006157. [9] Constantinedes, G. M., Capital market equilibrium with transactions costs, Journal of Political Economy, 1986, 94(4): 842−862. doi: 10.1086/261410. [10] Copeland, T. E. and Galai, D., Information effects on the bid-ask spread, The Journal of Finance, 1983, 38(5): 1457−1469. doi: 10.1111/j.1540-6261.1983.tb03834.x. [11] Dalang, R. C., Morton, A. and Willinger, W., Equivalent martingale measures and no-arbitrage in stochastic securities market models, Stochastics and Stochastic Reports, 1990, 29(2): 185−201. doi: 10.1080/17442509008833613. [12] Delbaen, F. and Schachermayer, W., A general version of the fundamental theorem of asset pricing, The Mathematische Annalen, 1994, 300(1): 463−520. doi: 10.1007/BF01450498. [13] Demsetz, H., The costs of transacting, The Quarterly Journal of Economics, 1968, 82(1): 33−53. doi: 10.2307/1882244. [14] Easley, D. and O’Hara, M., Price, trade size, and information in securities markets, Journal of Financial Economics, 1987, 19(1): 69−90. [15] Eberlein, E., Madan, D. B., Pistorius, M. and Yor, M., Bid and ask prices as non-linear continuous time G-expectations based on distortions, Mathematics and Financial Economics, 2014, 8(3): 265−289. doi: 10.1007/s11579-014-0117-1. [16] Eberlein, E., Madan, D. B., Pistorius, M., Schoutens, W. and Yor, M., Two price economies in continuous time, Annals of Finance, 2014, 10(1): 71−100. doi: 10.1007/s10436-013-0228-3. [17] Elliott, R. J., Chan, L. and Siu, T. K., Risk measures for derivatives with Markov-modulated pure jump processes, Asia-Pacific Financial Markets, 2006, 13(2): 129−149. [18] Elliott, R. J. and Siu, T. K., Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 2013, 20(1): 1−25. doi: 10.1080/1350486X.2012.655929. [19] Elliott, R. J., Siu, T. K. and Yang, H., Filtering a Markov modulated random measure, IEEE Transactions on Automatic Control, 2010, 55(1): 74−88. doi: 10.1109/TAC.2009.2034227. [20] Elliott, R. J. and Osakwe, C. J. U., Option pricing for pure jump processes with Markov switching compensators, Finance and Stochastics, 2006, 10(2): 250−275. doi: 10.1007/s00780-006-0004-6. [21] Ferson, W., Empirical Asset Pricing: Models and Methods, MIT Press, Boston, MA, 2019. [22] Föllmer, H. and Schied, A., Convex measures of risk and trading constraints, Finance and Stochastics, 2002, 6(4): 429−447. doi: 10.1007/s007800200072. [23] Föllmer, H. and Schied, A., Stochastic Finance, 2nd ed., De Gruyter, Berlin, 2004. [24] Glosten, L. and Milgrom, P. R., Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, Journal of Financial Economics, 1985, 14(1): 71−100. doi: 10.1016/0304-405X(85)90044-3. [25] Harrison, J. and Kreps, D., Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 1979, 20(3): 381−408. doi: 10.1016/0022-0531(79)90043-7. [26] Harrison, J. M. and Pliska, S. R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and their Applications, 1981, 11(3): 215−260. doi: 10.1016/0304-4149(81)90026-0. [27] Ho, T. and Stoll, H. R., Optimal dealer pricing under transactions costs and return uncertainty, Journal of Financial Economics, 1981, 9: 47−73. doi: 10.1016/0304-405X(81)90020-9. [28] Ho, T. and Stoll, H. R., The dynamics of dealer markets under competition, The Journal of Finance, 1983, 38(4): 1053−1074. doi: 10.1111/j.1540-6261.1983.tb02282.x. [29] Jarrow, R. A., A characterization theorem for unique risk neutral probability measures, Economics Letters, 1986, 22(1): 61−65. doi: 10.1016/0165-1765(86)90143-6. [30] Jouini, E. and Kallal, H., Martingale and arbitrage in securities markets with transaction cost, Journal of Economic Theory, 1995, 66(1): 178−197. doi: 10.1006/jeth.1995.1037. [31] Khintchine, A. Y., Limit laws of sums of independent random variables, ONTI, Moscow, Russian, 1938. [32] Küchler, U. and Tappe, S., Bilateral gamma distributions and processes in financial mathematics, Stochastic Processes and their Applications, 2008, 118(2): 261−283. doi: 10.1016/j.spa.2007.04.006. [33] Kusuoka, S., On law invariant coherent risk measures, In: Kusuoka, S. and Maruyama, T. (eds), Advances in Mathematical Economics, 2001, 3: 83–95. [34] Lévy, P., Théorie de l’Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937. [35] Lo, A. W., Mamaysky, H. and Wang, J., Asset prices and trading volume under fixed transaction costs, Journal of Political Economy, 2004, 112(5): 1054−1090. doi: 10.1086/422565. [36] Madan, D. B., Asset pricing theory for two price economies, Annals of Finance, 2015, 11(1): 1−35. doi: 10.1007/s10436-014-0255-8. [37] Madan, D. B., Benchmarking in two price financial markets, Annals of Finance, 2016, 12(2): 201−219. doi: 10.1007/s10436-016-0278-4. [38] Madan, D. B., Efficient estimation of expected stock returns, Finance Research Letters, 2017, 23: 31−38. doi: 10.1016/j.frl.2017.08.001. [39] Madan, D. B., Instantaneous portfolio theory, Quantitative Finance, 2018, 18(8): 1345−1364. doi: 10.1080/14697688.2017.1420210. [40] Madan, D. B., Schoutens, W. and Wang, K., Measuring and monitoring the efficiency of markets, International Journal of Theoretical and Applied Finance, 2017, 20(8): 1750051. doi: 10.1142/S0219024917500510. [41] Madan, D. B., Schoutens, W. and Wang, K., Bilateral multiple gamma returns: Their risks and rewards, International Journal of Financial Engineering, 2020, 7(1): 2050008. doi: 10.1142/S2424786320500085. [42] Madan, D. B. and Schoutens, W., Equilibrium asset returns in financial markets, International Journal of Theoretical and Applied Finance, 2019, 22(2): 1850063. doi: 10.1142/S0219024918500632. [43] Madan, D. B. and Schoutens, W., Financial Valuation: A Nonlinear Restoration, Submitted to press, 2020. [44] Madan, D. B. and Seneta, E., The variance gamma (V. G.) model for share market returns, The Journal of Business, 1990, 63(4): 511−524. doi: 10.1086/296519. [45] Madan, D. B. and Wang, K., Asymmetries in financial returns, International Journal of Financial Engineering, 2017, 4(4): 1750045. doi: 10.1142/S2424786317500451. [46] Madan, D. B., Pistorius, M. and Stadje, M., On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation, Finance and Stochastics, 2017, 21(4): 1073−1102. [47] Madan, D. B., Pistorius, M. and Stadje, M., Convergence of BSΔEs driven by random walks to BSDEs: The case of (in)finite activity jumps with general driver, Stochastic Processes and their Applications, 2016, 126(5): 1553−1584. doi: 10.1016/j.spa.2015.11.013. [48] Madan, D. B., Schoutens, W. and Wang, K., Measuring and monitoring the efficiency of markets, International Journal of Theoretical and Applied Finance, 2017, 20(8): 1750051. doi: 10.1142/S0219024917500510. [49] Peng, S., Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, In: Frittelli, M. and Runggaldier, W. (eds.), Stochastic Methods in Finance, Lecture Notes in Mathematics, 2004, 1856: 165−253. [50] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer Nature, Berlin, 2010. [51] Ross, S. A., A simple approach to the valuation of risky streams, The Journal of Business, 1978, 51(3): 453–475. [52] Royal, A. J. and Elliott, R. J., Asset prices with regime-switching variance gamma dynamics, In: Bensoussan, A. and Zhang, Q. (eds.), Handbook of Numerical Analysis, 2007, 15: 685–711. [53] Stoll, H. R., The pricing of security dealer services: An empirical study of nasdaq stocks, The Journal of Finance, 1978, 33(4): 1153−1172. doi: 10.1111/j.1540-6261.1978.tb02054.x. [54] Sundaram, R. K., Equivalent martingale measures and risk-neutral pricing, The Journal of Derivatives, 1997, 5(1): 85−98. doi: 10.3905/jod.1997.407984.

show all references

##### References:
 [1] Ahimud, Y. and Mendelson, H., Asset pricing and the bid-Ask spread, Journal of Financial Economics, 1986, 17(2): 223−249. doi: 10.1016/0304-405X(86)90065-6. [2] Andersen, T. W. and Darling, D. A., Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes, The Annals of Mathematical Statistics, 1952, 23(2): 193−212. doi: 10.1214/aoms/1177729437. [3] Artzner, P., Delbaen, F., Eber, J. M. and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203−228. doi: 10.1111/1467-9965.00068. [4] Back, K. and Baruch, S., Information in securities markets: Kyle meets glosten and milgrom, Econometrica, 2004, 72(2): 433−465. doi: 10.1111/j.1468-0262.2004.00497.x. [5] Becherer, D. and Davis, M. H. A., Arrow-Debreu prices, In: Cont, R. (ed.), Encyclopedia of Quantitative Finance, 2010. https://doi.org/10.1002/9780470061602.eqf04018. [6] Carr, P., Geman, H., Madan, D. B. and Yor, M., The fine structure of asset returns: An empirical investigation, Journal of Business, 2002, 75(2): 305−332. doi: 10.1086/338705. [7] Cherny, A. and Madan, D. B., New measures for performance evaluation, Review of Financial Studies, 2009, 22(7): 2571−2606. doi: 10.1093/rfs/hhn081. [8] Cherny, A. and Madan, D. B., Markets as a counterparty: An introduction to conic finance, International Journal of Theoretical and Applied Finance, 2010, 13(8): 1149−1177. doi: 10.1142/S0219024910006157. [9] Constantinedes, G. M., Capital market equilibrium with transactions costs, Journal of Political Economy, 1986, 94(4): 842−862. doi: 10.1086/261410. [10] Copeland, T. E. and Galai, D., Information effects on the bid-ask spread, The Journal of Finance, 1983, 38(5): 1457−1469. doi: 10.1111/j.1540-6261.1983.tb03834.x. [11] Dalang, R. C., Morton, A. and Willinger, W., Equivalent martingale measures and no-arbitrage in stochastic securities market models, Stochastics and Stochastic Reports, 1990, 29(2): 185−201. doi: 10.1080/17442509008833613. [12] Delbaen, F. and Schachermayer, W., A general version of the fundamental theorem of asset pricing, The Mathematische Annalen, 1994, 300(1): 463−520. doi: 10.1007/BF01450498. [13] Demsetz, H., The costs of transacting, The Quarterly Journal of Economics, 1968, 82(1): 33−53. doi: 10.2307/1882244. [14] Easley, D. and O’Hara, M., Price, trade size, and information in securities markets, Journal of Financial Economics, 1987, 19(1): 69−90. [15] Eberlein, E., Madan, D. B., Pistorius, M. and Yor, M., Bid and ask prices as non-linear continuous time G-expectations based on distortions, Mathematics and Financial Economics, 2014, 8(3): 265−289. doi: 10.1007/s11579-014-0117-1. [16] Eberlein, E., Madan, D. B., Pistorius, M., Schoutens, W. and Yor, M., Two price economies in continuous time, Annals of Finance, 2014, 10(1): 71−100. doi: 10.1007/s10436-013-0228-3. [17] Elliott, R. J., Chan, L. and Siu, T. K., Risk measures for derivatives with Markov-modulated pure jump processes, Asia-Pacific Financial Markets, 2006, 13(2): 129−149. [18] Elliott, R. J. and Siu, T. K., Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 2013, 20(1): 1−25. doi: 10.1080/1350486X.2012.655929. [19] Elliott, R. J., Siu, T. K. and Yang, H., Filtering a Markov modulated random measure, IEEE Transactions on Automatic Control, 2010, 55(1): 74−88. doi: 10.1109/TAC.2009.2034227. [20] Elliott, R. J. and Osakwe, C. J. U., Option pricing for pure jump processes with Markov switching compensators, Finance and Stochastics, 2006, 10(2): 250−275. doi: 10.1007/s00780-006-0004-6. [21] Ferson, W., Empirical Asset Pricing: Models and Methods, MIT Press, Boston, MA, 2019. [22] Föllmer, H. and Schied, A., Convex measures of risk and trading constraints, Finance and Stochastics, 2002, 6(4): 429−447. doi: 10.1007/s007800200072. [23] Föllmer, H. and Schied, A., Stochastic Finance, 2nd ed., De Gruyter, Berlin, 2004. [24] Glosten, L. and Milgrom, P. R., Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, Journal of Financial Economics, 1985, 14(1): 71−100. doi: 10.1016/0304-405X(85)90044-3. [25] Harrison, J. and Kreps, D., Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 1979, 20(3): 381−408. doi: 10.1016/0022-0531(79)90043-7. [26] Harrison, J. M. and Pliska, S. R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and their Applications, 1981, 11(3): 215−260. doi: 10.1016/0304-4149(81)90026-0. [27] Ho, T. and Stoll, H. R., Optimal dealer pricing under transactions costs and return uncertainty, Journal of Financial Economics, 1981, 9: 47−73. doi: 10.1016/0304-405X(81)90020-9. [28] Ho, T. and Stoll, H. R., The dynamics of dealer markets under competition, The Journal of Finance, 1983, 38(4): 1053−1074. doi: 10.1111/j.1540-6261.1983.tb02282.x. [29] Jarrow, R. A., A characterization theorem for unique risk neutral probability measures, Economics Letters, 1986, 22(1): 61−65. doi: 10.1016/0165-1765(86)90143-6. [30] Jouini, E. and Kallal, H., Martingale and arbitrage in securities markets with transaction cost, Journal of Economic Theory, 1995, 66(1): 178−197. doi: 10.1006/jeth.1995.1037. [31] Khintchine, A. Y., Limit laws of sums of independent random variables, ONTI, Moscow, Russian, 1938. [32] Küchler, U. and Tappe, S., Bilateral gamma distributions and processes in financial mathematics, Stochastic Processes and their Applications, 2008, 118(2): 261−283. doi: 10.1016/j.spa.2007.04.006. [33] Kusuoka, S., On law invariant coherent risk measures, In: Kusuoka, S. and Maruyama, T. (eds), Advances in Mathematical Economics, 2001, 3: 83–95. [34] Lévy, P., Théorie de l’Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937. [35] Lo, A. W., Mamaysky, H. and Wang, J., Asset prices and trading volume under fixed transaction costs, Journal of Political Economy, 2004, 112(5): 1054−1090. doi: 10.1086/422565. [36] Madan, D. B., Asset pricing theory for two price economies, Annals of Finance, 2015, 11(1): 1−35. doi: 10.1007/s10436-014-0255-8. [37] Madan, D. B., Benchmarking in two price financial markets, Annals of Finance, 2016, 12(2): 201−219. doi: 10.1007/s10436-016-0278-4. [38] Madan, D. B., Efficient estimation of expected stock returns, Finance Research Letters, 2017, 23: 31−38. doi: 10.1016/j.frl.2017.08.001. [39] Madan, D. B., Instantaneous portfolio theory, Quantitative Finance, 2018, 18(8): 1345−1364. doi: 10.1080/14697688.2017.1420210. [40] Madan, D. B., Schoutens, W. and Wang, K., Measuring and monitoring the efficiency of markets, International Journal of Theoretical and Applied Finance, 2017, 20(8): 1750051. doi: 10.1142/S0219024917500510. [41] Madan, D. B., Schoutens, W. and Wang, K., Bilateral multiple gamma returns: Their risks and rewards, International Journal of Financial Engineering, 2020, 7(1): 2050008. doi: 10.1142/S2424786320500085. [42] Madan, D. B. and Schoutens, W., Equilibrium asset returns in financial markets, International Journal of Theoretical and Applied Finance, 2019, 22(2): 1850063. doi: 10.1142/S0219024918500632. [43] Madan, D. B. and Schoutens, W., Financial Valuation: A Nonlinear Restoration, Submitted to press, 2020. [44] Madan, D. B. and Seneta, E., The variance gamma (V. G.) model for share market returns, The Journal of Business, 1990, 63(4): 511−524. doi: 10.1086/296519. [45] Madan, D. B. and Wang, K., Asymmetries in financial returns, International Journal of Financial Engineering, 2017, 4(4): 1750045. doi: 10.1142/S2424786317500451. [46] Madan, D. B., Pistorius, M. and Stadje, M., On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation, Finance and Stochastics, 2017, 21(4): 1073−1102. [47] Madan, D. B., Pistorius, M. and Stadje, M., Convergence of BSΔEs driven by random walks to BSDEs: The case of (in)finite activity jumps with general driver, Stochastic Processes and their Applications, 2016, 126(5): 1553−1584. doi: 10.1016/j.spa.2015.11.013. [48] Madan, D. B., Schoutens, W. and Wang, K., Measuring and monitoring the efficiency of markets, International Journal of Theoretical and Applied Finance, 2017, 20(8): 1750051. doi: 10.1142/S0219024917500510. [49] Peng, S., Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, In: Frittelli, M. and Runggaldier, W. (eds.), Stochastic Methods in Finance, Lecture Notes in Mathematics, 2004, 1856: 165−253. [50] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer Nature, Berlin, 2010. [51] Ross, S. A., A simple approach to the valuation of risky streams, The Journal of Business, 1978, 51(3): 453–475. [52] Royal, A. J. and Elliott, R. J., Asset prices with regime-switching variance gamma dynamics, In: Bensoussan, A. and Zhang, Q. (eds.), Handbook of Numerical Analysis, 2007, 15: 685–711. [53] Stoll, H. R., The pricing of security dealer services: An empirical study of nasdaq stocks, The Journal of Finance, 1978, 33(4): 1153−1172. doi: 10.1111/j.1540-6261.1978.tb02054.x. [54] Sundaram, R. K., Equivalent martingale measures and risk-neutral pricing, The Journal of Derivatives, 1997, 5(1): 85−98. doi: 10.3905/jod.1997.407984.
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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