Gain-loss pricing under ambiguity of measure
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 132-147.

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.

DOI: 10.1051/cocv:2008068
Classification: 91B28, 90C90, 90C25
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] A. Ben-Tal and A. Nemirovski, 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.

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

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

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

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

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

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

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

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

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

[11] A. D'Aspremont and L. El Ghaoui, Static arbitrage bounds on basket option prices. Math. Programming 106 (2006) 467-489. | Zbl

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

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

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

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

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

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

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

[19] A.J. King and L.A. Korf, Martingale Pricing Measures in Incomplete Markets via Stochastic Programming Duality in the Dual of . Technical Report (2001).

[20] L.A. Korf, Stochastic programming duality: multipliers for unbounded constraints with an application to mathematical finance. Math. Programming 99 (2004) 241-259. | Zbl

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

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

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

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

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

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

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

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

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

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

[31] Sh. Tian and R.J.-B. Wets, Pricing Contingent Claims: A Computational Compatible Approach. Technical Report, Department of Mathematics, University of California, Davis (2006).

Cited by Sources: