Motivated by the observation that the gain-loss criterion, while offering economically meaningful prices of contingent claims, is sensitive to the reference measure governing the underlying stock price process (a situation referred to as ambiguity of measure), we propose a gain-loss pricing model robust to shifts in the reference measure. Using a dual representation property of polyhedral risk measures we obtain a one-step, gain-loss criterion based theorem of asset pricing under ambiguity of measure, and illustrate its use.

Keywords: contingent claim, pricing, gain-loss ratio, hedging, martingales, stochastic programming, risk measures

@article{COCV_2010__16_1_132_0, author = {P{\i}nar, Mustafa \c{C}.}, title = {Gain-loss pricing under ambiguity of measure}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {132--147}, publisher = {EDP-Sciences}, volume = {16}, number = {1}, year = {2010}, doi = {10.1051/cocv:2008068}, mrnumber = {2598092}, zbl = {1186.91219}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008068/} }

TY - JOUR AU - Pınar, Mustafa Ç. TI - Gain-loss pricing under ambiguity of measure JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 132 EP - 147 VL - 16 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008068/ DO - 10.1051/cocv:2008068 LA - en ID - COCV_2010__16_1_132_0 ER -

%0 Journal Article %A Pınar, Mustafa Ç. %T Gain-loss pricing under ambiguity of measure %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 132-147 %V 16 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008068/ %R 10.1051/cocv:2008068 %G en %F COCV_2010__16_1_132_0

Pınar, Mustafa Ç. Gain-loss pricing under ambiguity of measure. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 132-147. doi : 10.1051/cocv:2008068. http://archive.numdam.org/articles/10.1051/cocv:2008068/

[1] Optimization I-II, Convex Analysis, Nonlinear Programming, Nonlinear Programming Algorithms, Lecture Notes. Technion, Israel Institute of Technology (2004), available for download at http://www2.isye.gatech.edu/ nemirovs/Lect_OptI-II.pdf.

and ,[2] An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance 17 (2007) 449-476. | Zbl

and ,[3] Gain, loss and asset pricing. J. Political Economy 81 (2000) 637-654.

and ,[4] On the relation between option and stock prices: An optimization approach. Oper. Res. 50 (2002) 358-374. | Zbl

and ,[5] Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15 (2005) 780-804. | Zbl

and ,[6] The pricing of options and corporate liabilities. J. Political Economy 108 (1973) 144-172. | Zbl

and ,[7] GAMS: A User's Guide. The Scientific Press, San Fransisco, California (1992).

, and ,[8] Ambiguous risk measures and optimal robust portfolios. SIAM J. Optim. 18 (2007) 853-877. | Zbl

,[9] Model uncertainty and its impact on the pricing of derivative instruments. Math. Finance 16 (2006) 519-547. | Zbl

,[10] Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungarica 2 (1967) 299-318. | Zbl

,[11] Static arbitrage bounds on basket option prices. Math. Programming 106 (2006) 467-489. | Zbl

and ,[12] Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16 (2005) 69-95. | Zbl

and ,[13] Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Oper. Res. 51 (2003) 543-556. | Zbl

, and ,[14] A definition of uncertainty aversion. Rev. Economic Studies 65 (1999) 579-608. | Zbl

,[15] Interfaces to PATH 3.0: Design, implementation and usage. Technical Report, University of Wisconsin, Madison (1998). | Zbl

and ,[16] Stochastic Finance: An Introduction in Discrete Time, De Gruyter Studies in Mathematics 27. Second Edition, Berlin (2004). | Zbl

and ,[17] Martingales and arbitrage in multiperiod securities markets. J. Economic Theory 20 (1979) 381-408. | Zbl

and ,[18] Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11 (1981) 215-260. | Zbl

and ,[19] Martingale Pricing Measures in Incomplete Markets via Stochastic Programming Duality in the Dual of ${\mathcal{L}}^{\infty}$. Technical Report (2001).

and ,[20] Stochastic programming duality: ${\mathcal{L}}^{\infty}$ multipliers for unbounded constraints with an application to mathematical finance. Math. Programming 99 (2004) 241-259. | Zbl

,[21] Information Theory and Statistics. Wiley, New York (1959) | Zbl

,[22] Moments in mathematics, in Proc. Sympos. Appl. Math. 37, H.J. Landau Ed., AMS, Providence, RI (1987). | Zbl

,[23] A simple linear programming approach to gain, loss and asset pricing. Topics in Theoretical Economics 2 (2002) Article 4.

,[24] Conjugate Duality and Optimization. SIAM, Philadelphia (1974). | Zbl

,[25] Optimization of risk measures, in Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene Eds., Springer, London (2005). | Zbl

and ,[26] Optimization of convex risk functions. Math. Oper. Res. 31 (2006) 433-452.

and ,[27] On duality theory of convex semi-infinite programming. Optimization 54 (2005) 535-543. | Zbl

,[28] On a class of stochastic minimax programs. SIAM J. Optim. 14 (2004) 1237-1249. | Zbl

and ,[29] Minimax analysis of stochastic problems. Optim. Methods Software 17 (2002) 523-542. | Zbl

and ,[30] Generalized Chebychev inequalities: Theory and applications in decision analysis. Oper. Res. 43 (1995) 807-825. | Zbl

,[31] Pricing Contingent Claims: A Computational Compatible Approach. Technical Report, Department of Mathematics, University of California, Davis (2006).

and ,*Cited by Sources: *