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Conditional coherent risk measures and regime-switching conic pricing
1. | UniSA Business, University of South Australia, SA 5000 Adelaide, Australia |
2. | Haskayne School of Business, University of Calgary, Calgary, Alberta, T2N 1N4, Canada |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
Delbaen, F., Coherent risk measures on general probability spaces, In: Sandmann K, Schönbucher PJ (eds.), Advances in Finance and Stochastics, Springer, 2002. |
[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. |
[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. |
[10] |
Dunford, N. and Schwartz, J. T., Linear Operators Part I: General Theory, Interscience Publishers, New York, 1958. |
[11] |
Elliott, R. J., Aggoun, L. and Moore, J. B., Hidden Markov Models: Estimation and Control, 1st ed., Springer, 1995. |
[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. |
[13] |
Epstein, L. G. and Schneider, M., Recursive multiple-priors, Journal of Economic Theory, 2003, 113(1): 1−31. |
[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. |
[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. |
[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. |
[17] |
Föllmer, H. and Schied, A., Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. |
[18] |
Graf, S. and Luschgy, H., Foundations of Quantization for Probability Distributions, Springer, New York, 2000. |
[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. |
[20] |
Kopycka, D., Dynamic risk measures, robust representation and examples, Master’s thesis, The Netherlands: VU University Amsterdam and Poland: Jagiellonian University, 2009. |
[21] |
Kusuoka, S., On law invariant coherent risk measures, In: Kusuoka S., Maruyama T. (eds.), Advances in Mathematical Economics, Springer, 2001. |
[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. |
[23] |
Madan, D. B. and Schoutens, W., Applied Conic Finance, 1st ed., Cambridge University Press, 2020. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
Delbaen, F., Coherent risk measures on general probability spaces, In: Sandmann K, Schönbucher PJ (eds.), Advances in Finance and Stochastics, Springer, 2002. |
[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. |
[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. |
[10] |
Dunford, N. and Schwartz, J. T., Linear Operators Part I: General Theory, Interscience Publishers, New York, 1958. |
[11] |
Elliott, R. J., Aggoun, L. and Moore, J. B., Hidden Markov Models: Estimation and Control, 1st ed., Springer, 1995. |
[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. |
[13] |
Epstein, L. G. and Schneider, M., Recursive multiple-priors, Journal of Economic Theory, 2003, 113(1): 1−31. |
[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. |
[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. |
[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. |
[17] |
Föllmer, H. and Schied, A., Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. |
[18] |
Graf, S. and Luschgy, H., Foundations of Quantization for Probability Distributions, Springer, New York, 2000. |
[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. |
[20] |
Kopycka, D., Dynamic risk measures, robust representation and examples, Master’s thesis, The Netherlands: VU University Amsterdam and Poland: Jagiellonian University, 2009. |
[21] |
Kusuoka, S., On law invariant coherent risk measures, In: Kusuoka S., Maruyama T. (eds.), Advances in Mathematical Economics, Springer, 2001. |
[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. |
[23] |
Madan, D. B. and Schoutens, W., Applied Conic Finance, 1st ed., Cambridge University Press, 2020. |
[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. |
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