Coherent risk measures and good-deal bounds

Abstract. The relation between coherent risk measures, valuation bounds, and certain classes of portfolio optimization problems is established. One of the key results is that coherent risk measures are essentially equivalent to generalized arbitrage bounds, named “good deal bounds” by Cerny and Hodges (1999). The results are economically general in the sense that they work for any cash stream spaces, be it in dynamic trading settings, one-step models, or deterministic cash streams. They are also mathematically general as they work in (possibly infinite-dimensional) linear spaces.The valuation theory presented seems to fill a gap between arbitrage valuation on the one hand and utility maximization (or equilibrium theory) on the other hand. “Coherent” valuation bounds strike a balance in that the bounds can be sharp enough to be useful in the practice of pricing and still be generic, i.e., somewhat independent of personal preferences, in the way many coherent risk measures are somewhat generic.

[1]  E. Jouini,et al.  Martingales and Arbitrage in Securities Markets with Transaction Costs , 1995 .

[2]  Gerold Studer,et al.  Maximum Loss for Measurement of Market Risk , 1997 .

[3]  J. Cochrane,et al.  Beyond Arbitrage: 'Good Deal' Asset Price Bounds in Incomplete Markets , 1996 .

[4]  N. Karoui,et al.  Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market , 1995 .

[5]  Alexandre Grothendieck,et al.  Topological vector spaces , 1973 .

[6]  S. D. Hodges,et al.  A Model for Bond Portfolio Improvement , 1977, Journal of Financial and Quantitative Analysis.

[7]  Walter Schachermayer,et al.  A bipolar theorem for $L^0_+(\Om, \Cal F, \P)$ , 1999 .

[8]  Aleš Černý,et al.  The Theory of Good-Deal Pricing in Financial Markets , 1998 .

[9]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[10]  S. Jaschke,et al.  Higher Order Forward Rate Agreements and the Smoothness of the Term Structure , 1998 .

[11]  H. H. Schaefer,et al.  Topological Vector Spaces , 1967 .

[12]  J. Ingersoll.  Theory of Financial Decision Making , 1987 .

[13]  E. Jouini,et al.  ARBITRAGE IN SECURITIES MARKETS WITH SHORT-SALES CONSTRAINTS , 1995 .

[14]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[15]  Leonard Rogers,et al.  Robust Hedging of Barrier Options , 2001 .

[16]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[17]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .