Fundamental Theorems of Asset Pricing for Good Deal Bounds

We prove fundamental theorems of asset pricing for good deal bounds in incomplete markets. These theorems relate arbitrage-freedom and uniqueness of prices for over-the-counter derivatives to existence and uniqueness of a pricing kernel that is consistent with market prices and the acceptance set of good deals. They are proved using duality of convex optimization in locally convex linear topological spaces. The concepts investigated are closely related to convex and coherent risk measures, exact functionals, and coherent lower previsions in the theory of imprecise probabilities.

[1]  R. Tyrrell Rockafellar.  Conjugate Duality and Optimization , 1974 .

[2]  Non-Arbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results , 1997 .

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

[4]  Alexander Schied,et al.  Robust Preferences and Convex Measures of Risk , 2002 .

[5]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[6]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[7]  M. Frittelli.  The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets , 2000 .

[8]  Hélyette Geman,et al.  Pricing and hedging in incomplete markets , 2001 .

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

[10]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[11]  Stefan Jaschke,et al.  Coherent risk measures, valuation bounds, and (my,p)-portfolio optimization , 1999 .

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

[13]  Philip Protter,et al.  Liquidity Risk and Arbitrage Pricing Theory , 2004 .

[14]  F. Smithies.  Linear Operators , 2019, Nature.

[15]  Robert Jarrow,et al.  Hedging contingent claims on semimartingales , 1999, Finance Stochastics.

[16]  Marco Frittelli,et al.  Introduction to a theory of value coherent with the no-arbitrage principle , 2000, Finance Stochastics.

[17]  Stefan Jaschke,et al.  Coherent Risk Measures , Valuation Bounds , and ( μ , ρ )-Portfolio Optimization , 1999 .

[18]  Berend Roorda,et al.  Time Consistency Conditions for Acceptability Measures, with an Application to Tail Value at Risk , 2007 .

[19]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[20]  Sebastian Maaβ,et al.  Exact functionals and their core , 2002 .

[21]  J. Cochrane,et al.  Beyond Arbitrage: Good‐Deal Asset Price Bounds in Incomplete Markets , 2000, Journal of Political Economy.

[22]  Jeremy Staum,et al.  Pricing and Hedging in Incomplete Markets: Fundamental Theorems and Robust Utility Maximization , 2002 .

[24]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[25]  M. Schweizer.  Approximation pricing and the variance-optimal martingale measure , 1996 .

[26]  S. Hilden.  Convex measures of risk and trading constraints , 2003 .