Risk Measures and Nonlinear Expectations

Affiliation(s)

School of Mathematics, Shandong University, Jinan, China; Department of Financial Engineering, Ajou University, Suwon, Korea.

Department of Mathematics, Donghua University, Shanghai, China.

Department of Statistical and Actuarial Science, The University of Western Ontario London, Ontario, Canada.

School of Mathematics, Shandong University, Jinan, China; Department of Financial Engineering, Ajou University, Suwon, Korea.

Department of Mathematics, Donghua University, Shanghai, China.

Department of Statistical and Actuarial Science, The University of Western Ontario London, Ontario, Canada.

ABSTRACT

Coherent and convex risk measures, Choquet expectation and Peng’s *g-*expectation are all generalizations of mathematical expectation. All have been widely used to assess financial riskiness under uncertainty. In this paper, we investigate differences amongst these risk measures and expectations. For this purpose, we constrain our attention of coherent and convex risk measures, and Choquet expectation to the domain of *g-*expectation. Some differences among coherent and convex risk measures and Choquet expectations are accounted for in the framework of *g-*expectations. We show that in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality. In mathematical finance, risk measures and Choquet expectations are typically used in the pricing of contingent claims over families of measures. The different risk measures will typically yield different pricing. In this paper, we show that the coherent pricing is always less than the corresponding Choquet pricing. This property and inequality fails in general when one uses pricing by convex risk measures. We also discuss the relation between static risk measure and dynamic risk measure in the framework of *g-*expectations. We show that if *g-*expectations yield coherent (convex) risk measures then the corresponding conditional *g-*expectations or equivalently the dynamic risk measure is also coherent (convex). To prove these results, we establish a new converse of the comparison theorem of *g-*expectations.

Cite this paper

Z. Chen, K. He and R. Kulperger, "Risk Measures and Nonlinear Expectations,"*Journal of Mathematical Finance*, Vol. 3 No. 3, 2013, pp. 383-391. doi: 10.4236/jmf.2013.33039.

Z. Chen, K. He and R. Kulperger, "Risk Measures and Nonlinear Expectations,"

References

[1] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, “Coherent Measures of Risk,” Mathematical Finance, Vol. 9, No. 3, 1999, pp. 203-228. doi:10.1111/1467-9965.00068

[2] F. Delbaen, “Coherent Risk Measures on General Probability,” In: K. Sandmann and P. Schonbucher, Eds., Advances in Finance and Stochastics, Springer-Verlag, Berlin, 2002, pp. 1-37. doi:10.1007/978-3-662-04790-3_1

[3] H. Follmer and A. Schied, “Convex Measures of Risk and Trading Constraints,” Finance and Stochastics, Vol. 6, No. 4, 2002, pp. 429-447. doi:10.1007/s007800200072

[4] M. Frittelli and E. R. Gianin, “Putting Order in Risk Measures,” Journal of Banking Finance, Vol. 26, No. 7, 2002, pp. 1474-1486. doi:10.1016/S0378-4266(02)00270-4

[5] S. G. Peng, “BSDE and Related g-Expectation,” Pitman Research Notes in Mathematics Series, Vol. 364, 1997, pp. 141-159.

[6] E. Pardoux and S. G. Peng, “Adapted Solution of a Backward Stochastic Differential Equation,” Systems and Control Letters, Vol. 14, No. 1, 1990, pp. 55-61. doi:10.1016/0167-6911(90)90082-6

[7] G. Choquet, “Theory of Capacities,” Annales de l'institut Fourier (Grenoble), Vol. 5, 1953, pp. 131-195. doi:10.5802/aif.53

[8] L. Jiang, “Convexity, Translation Invariance and Subadditivity for g-Expectations and Related Risk Measures,” Annals of Applied Probability, Vol. 18, No. 1, 2008, pp. 245-258. doi:10.1214/105051607000000294

[9] P. Briand, F. Coquet, Y. Hu, J. Mémin and S. G. Peng, “A Converse Comparison Theorem for BSDEs and Related Problems of g-Expectation,” Electronic Communications in Probability, Vol. 5, 2000, pp. 101-117. doi:10.1214/ECP.v5-1025

[10] B. Roorda, J. M. Schumacher and J. Engwerda, “Coherent Acceptability Measures in Multi-Period Models,” Mathematical Finance, Vol. 15, No. 4, 2005, pp. 589-612. doi:10.1111/j.1467-9965.2005.00252.x

[11] P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, “Multiperiod Risk and Coherent Multiperiod Risk Measurement,” E.T.H.Zürich, Preprint, 2002.

[12] P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, “Coherent Multiperiod Risk Adjusted Values and Bellman’s Principle,” Annals of Operations Research, Vol. 152, No. 1, 2007, pp. 5-22. doi:10.1007/s10479-006-0132-6

[13] M. Frittelli and E. R. Gianin, “Dynamic Convex Risk Measures,” In: G. Szego, Ed., Risk Measures for the 21st Century, John Wiley, Hoboken, 2004, pp. 227-248.

[14] F. Coquet, Y. Hu, J. Mémin and S. G. Peng, “A General Converse Comparison Theorem for Backward Stochastic Differential Equations,” Comptes Rendus de l’Académie des Sciences Paris, Series I, Vol. 333, 2001, pp. 577-581.

[15] Z. J. Chen, R. Kulperger and L. Jiang, “Jensen’s Inequality for g-Expectation: Part 1,” Comptes Rendus de l’Académie des Sciences Paris, Series I, Vol. 337, 2003, pp. 725-730.

[1] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, “Coherent Measures of Risk,” Mathematical Finance, Vol. 9, No. 3, 1999, pp. 203-228. doi:10.1111/1467-9965.00068

[2] F. Delbaen, “Coherent Risk Measures on General Probability,” In: K. Sandmann and P. Schonbucher, Eds., Advances in Finance and Stochastics, Springer-Verlag, Berlin, 2002, pp. 1-37. doi:10.1007/978-3-662-04790-3_1

[3] H. Follmer and A. Schied, “Convex Measures of Risk and Trading Constraints,” Finance and Stochastics, Vol. 6, No. 4, 2002, pp. 429-447. doi:10.1007/s007800200072

[4] M. Frittelli and E. R. Gianin, “Putting Order in Risk Measures,” Journal of Banking Finance, Vol. 26, No. 7, 2002, pp. 1474-1486. doi:10.1016/S0378-4266(02)00270-4

[5] S. G. Peng, “BSDE and Related g-Expectation,” Pitman Research Notes in Mathematics Series, Vol. 364, 1997, pp. 141-159.

[6] E. Pardoux and S. G. Peng, “Adapted Solution of a Backward Stochastic Differential Equation,” Systems and Control Letters, Vol. 14, No. 1, 1990, pp. 55-61. doi:10.1016/0167-6911(90)90082-6

[7] G. Choquet, “Theory of Capacities,” Annales de l'institut Fourier (Grenoble), Vol. 5, 1953, pp. 131-195. doi:10.5802/aif.53

[8] L. Jiang, “Convexity, Translation Invariance and Subadditivity for g-Expectations and Related Risk Measures,” Annals of Applied Probability, Vol. 18, No. 1, 2008, pp. 245-258. doi:10.1214/105051607000000294

[9] P. Briand, F. Coquet, Y. Hu, J. Mémin and S. G. Peng, “A Converse Comparison Theorem for BSDEs and Related Problems of g-Expectation,” Electronic Communications in Probability, Vol. 5, 2000, pp. 101-117. doi:10.1214/ECP.v5-1025

[10] B. Roorda, J. M. Schumacher and J. Engwerda, “Coherent Acceptability Measures in Multi-Period Models,” Mathematical Finance, Vol. 15, No. 4, 2005, pp. 589-612. doi:10.1111/j.1467-9965.2005.00252.x

[11] P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, “Multiperiod Risk and Coherent Multiperiod Risk Measurement,” E.T.H.Zürich, Preprint, 2002.

[12] P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, “Coherent Multiperiod Risk Adjusted Values and Bellman’s Principle,” Annals of Operations Research, Vol. 152, No. 1, 2007, pp. 5-22. doi:10.1007/s10479-006-0132-6

[13] M. Frittelli and E. R. Gianin, “Dynamic Convex Risk Measures,” In: G. Szego, Ed., Risk Measures for the 21st Century, John Wiley, Hoboken, 2004, pp. 227-248.

[14] F. Coquet, Y. Hu, J. Mémin and S. G. Peng, “A General Converse Comparison Theorem for Backward Stochastic Differential Equations,” Comptes Rendus de l’Académie des Sciences Paris, Series I, Vol. 333, 2001, pp. 577-581.

[15] Z. J. Chen, R. Kulperger and L. Jiang, “Jensen’s Inequality for g-Expectation: Part 1,” Comptes Rendus de l’Académie des Sciences Paris, Series I, Vol. 337, 2003, pp. 725-730.