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December  2021, 6(4): 267-300. doi: 10.3934/puqr.2021014

Conditional coherent risk measures and regime-switching conic pricing

Engel John C Dela Vega 1, and  Robert J Elliott 1,2,,

1. 

UniSA Business, University of South Australia, SA 5000 Adelaide, Australia
2. 

Haskayne School of Business, University of Calgary, Calgary, Alberta, T2N 1N4, Canada

Robert J Elliott E-mail: relliott@ucalgary.ca

Received  December 13, 2020 Accepted  October 13, 2021 Published  November 2021

Fund Project: The authors would like to thank the Australian Research Council and NSERC for continuing support. The authors are also grateful to the referees for carefully reviewing the manuscript and providing valuable feedback.

This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function. A model is then developed for the bid and ask prices of a European-type asset by a conic formulation. The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. The bid and ask prices of a European-type asset are then characterized using conic quantization.

Keywords: Conditional coherent risk measures, Markov chains, Regime-switching, Conic finance, European options, Distortion functions, Conic quantization.
Mathematics Subject Classification: 60J27, 91G70, 91J20.
Citation: Engel John C Dela Vega, Robert J Elliott. Conditional coherent risk measures and regime-switching conic pricing. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 267-300. doi: 10.3934/puqr.2021014
References:
[1]

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.  Google Scholar

[2]

Artzner, P., Delbaen, F., Eber, J. M., Heath, D. and Ku, H., Coherent multiperiod risk adjusted values and Bellman’s principle, Annals of Operations Research, 2007, 152(1): 5−22. doi: 10.1007/s10479-006-0132-6.  Google Scholar

[3]

Cheridito, P., Delbaen, F. and Kupper, M., Coherent and convex monetary risk measures for bounded càdlàg processes, Stochastic Processes and their Applications, 2004, 112(1): 1−22. doi: 10.1016/j.spa.2004.01.009.  Google Scholar

[4]

Cheridito, P., Delbaen, F. and Kupper, M., Coherent and convex monetary risk measures for unbounded càdlàg processes, Finance and Stochastics, 2006, 10(3): 427−448. doi: 10.1007/s00780-004-0150-7.  Google Scholar

[5]

Cheridito, P., Delbaen, F. and Kupper, M., Dynamic monetary risk measures for bounded discrete-time processes, Electronic Journal of Probability, 2006, 11(3): 57−106. Google Scholar

[6]

Cherny, A. and Madan, D. B., New measure for performance evaluation, The Review of Financial Studies, 2009, 22(7): 2571−2606. doi: 10.1093/rfs/hhn081.  Google Scholar

[7]

Delbaen, F., Coherent risk measures on general probability spaces, In: Sandmann K, Schönbucher PJ (eds.), Advances in Finance and Stochastics, Springer, 2002. Google Scholar

[8]

Detlefsen, K. and Scandolo, G., Conditional and dynamic convex risk measures, Finance and Stochastics, 2005, 9(4): 539−561. doi: 10.1007/s00780-005-0159-6.  Google Scholar

[9]

Dufour, F. and Elliott, R. J., Filtering with discrete state observations, Applied Mathematics and Optimization, 1999, 40(2): 259−272. doi: 10.1007/s002459900125.  Google Scholar

[10]

Dunford, N. and Schwartz, J. T., Linear Operators Part I: General Theory, Interscience Publishers, New York, 1958. Google Scholar

[11]

Elliott, R. J., Aggoun, L. and Moore, J. B., Hidden Markov Models: Estimation and Control, 1st ed., Springer, 1995. Google Scholar

[12]

Elliott, R. J., Chan, L. and Siu, T. K., Option pricing and Esscher transform under regime switching, Annals of Finance, 2005, 1(14): 423−432. Google Scholar

[13]

Epstein, L. G. and Schneider, M., Recursive multiple-priors, Journal of Economic Theory, 2003, 113(1): 1−31. Google Scholar

[14]

Fiorin, L. and Schoutens, W., Conic quantization: Stochastic volatility and market implied liquidity, Quantitative Finance, 2020, 20(4): 531−542. doi: 10.1080/14697688.2019.1687928.  Google Scholar

[15]

Föllmer, H. and Schied, A., Convex measures of risk and trading constraints, Finance and Stochastics, 2002, 6(4): 429−447. doi: 10.1007/s007800200072.  Google Scholar

[16]

Föllmer, H. and Schied, A., Robust prefernces and convex measures of risk, In: Sandmann K, Schönbucher PJ (eds.) Advances in Finance and Stochastics, Springer, Berlin, Heidenberg, 2002. Google Scholar

[17]

Föllmer, H. and Schied, A., Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. Google Scholar

[18]

Graf, S. and Luschgy, H., Foundations of Quantization for Probability Distributions, Springer, New York, 2000. Google Scholar

[19]

Inoue, A., On the worst conditional expectation, Journal of Mathematical Analysis and Applications, 2003, 286(1): 237−247. doi: 10.1016/S0022-247X(03)00477-3.  Google Scholar

[20]

Kopycka, D., Dynamic risk measures, robust representation and examples, Master’s thesis, The Netherlands: VU University Amsterdam and Poland: Jagiellonian University, 2009. Google Scholar

[21]

Kusuoka, S., On law invariant coherent risk measures, In: Kusuoka S., Maruyama T. (eds.), Advances in Mathematical Economics, Springer, 2001. Google Scholar

[22]

Madan, D. B. and Cherny, A., 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.  Google Scholar

[23]

Madan, D. B. and Schoutens, W., Applied Conic Finance, 1st ed., Cambridge University Press, 2020. Google Scholar

[24]

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. doi: 10.1007/s00780-017-0339-1.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

Artzner, P., Delbaen, F., Eber, J. M., Heath, D. and Ku, H., Coherent multiperiod risk adjusted values and Bellman’s principle, Annals of Operations Research, 2007, 152(1): 5−22. doi: 10.1007/s10479-006-0132-6.  Google Scholar

[3]

Cheridito, P., Delbaen, F. and Kupper, M., Coherent and convex monetary risk measures for bounded càdlàg processes, Stochastic Processes and their Applications, 2004, 112(1): 1−22. doi: 10.1016/j.spa.2004.01.009.  Google Scholar

[4]

Cheridito, P., Delbaen, F. and Kupper, M., Coherent and convex monetary risk measures for unbounded càdlàg processes, Finance and Stochastics, 2006, 10(3): 427−448. doi: 10.1007/s00780-004-0150-7.  Google Scholar

[5]

Cheridito, P., Delbaen, F. and Kupper, M., Dynamic monetary risk measures for bounded discrete-time processes, Electronic Journal of Probability, 2006, 11(3): 57−106. Google Scholar

[6]

Cherny, A. and Madan, D. B., New measure for performance evaluation, The Review of Financial Studies, 2009, 22(7): 2571−2606. doi: 10.1093/rfs/hhn081.  Google Scholar

[7]

Delbaen, F., Coherent risk measures on general probability spaces, In: Sandmann K, Schönbucher PJ (eds.), Advances in Finance and Stochastics, Springer, 2002. Google Scholar

[8]

Detlefsen, K. and Scandolo, G., Conditional and dynamic convex risk measures, Finance and Stochastics, 2005, 9(4): 539−561. doi: 10.1007/s00780-005-0159-6.  Google Scholar

[9]

Dufour, F. and Elliott, R. J., Filtering with discrete state observations, Applied Mathematics and Optimization, 1999, 40(2): 259−272. doi: 10.1007/s002459900125.  Google Scholar

[10]

Dunford, N. and Schwartz, J. T., Linear Operators Part I: General Theory, Interscience Publishers, New York, 1958. Google Scholar

[11]

Elliott, R. J., Aggoun, L. and Moore, J. B., Hidden Markov Models: Estimation and Control, 1st ed., Springer, 1995. Google Scholar

[12]

Elliott, R. J., Chan, L. and Siu, T. K., Option pricing and Esscher transform under regime switching, Annals of Finance, 2005, 1(14): 423−432. Google Scholar

[13]

Epstein, L. G. and Schneider, M., Recursive multiple-priors, Journal of Economic Theory, 2003, 113(1): 1−31. Google Scholar

[14]

Fiorin, L. and Schoutens, W., Conic quantization: Stochastic volatility and market implied liquidity, Quantitative Finance, 2020, 20(4): 531−542. doi: 10.1080/14697688.2019.1687928.  Google Scholar

[15]

Föllmer, H. and Schied, A., Convex measures of risk and trading constraints, Finance and Stochastics, 2002, 6(4): 429−447. doi: 10.1007/s007800200072.  Google Scholar

[16]

Föllmer, H. and Schied, A., Robust prefernces and convex measures of risk, In: Sandmann K, Schönbucher PJ (eds.) Advances in Finance and Stochastics, Springer, Berlin, Heidenberg, 2002. Google Scholar

[17]

Föllmer, H. and Schied, A., Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. Google Scholar

[18]

Graf, S. and Luschgy, H., Foundations of Quantization for Probability Distributions, Springer, New York, 2000. Google Scholar

[19]

Inoue, A., On the worst conditional expectation, Journal of Mathematical Analysis and Applications, 2003, 286(1): 237−247. doi: 10.1016/S0022-247X(03)00477-3.  Google Scholar

[20]

Kopycka, D., Dynamic risk measures, robust representation and examples, Master’s thesis, The Netherlands: VU University Amsterdam and Poland: Jagiellonian University, 2009. Google Scholar

[21]

Kusuoka, S., On law invariant coherent risk measures, In: Kusuoka S., Maruyama T. (eds.), Advances in Mathematical Economics, Springer, 2001. Google Scholar

[22]

Madan, D. B. and Cherny, A., 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.  Google Scholar

[23]

Madan, D. B. and Schoutens, W., Applied Conic Finance, 1st ed., Cambridge University Press, 2020. Google Scholar

[24]

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. doi: 10.1007/s00780-017-0339-1.  Google Scholar

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