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Gain-loss pricing under ambiguity of measure

Published online by Cambridge University Press:  08 November 2008

Mustafa Ç. Pınar*
Department of Industrial Engineering, Bilkent University, 06800 Ankara, Turkey.
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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.

Research Article
© EDP Sciences, SMAI, 2008

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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 nemirovs/Lect_OptI-II.pdf.
Ben-Tal, A. and Teboulle, M., An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance 17 (2007) 449476. CrossRef
Bernardo, A.E. and Ledoit, O., Gain, loss and asset pricing. J. Political Economy 81 (2000) 637654.
Bertsimas, D. and Popescu, I., On the relation between option and stock prices: An optimization approach. Oper. Res. 50 (2002) 358374. CrossRef
Bertsimas, D. and Popescu, I., Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15 (2005) 780804. CrossRef
Black, F. and Scholes, M., The pricing of options and corporate liabilities. J. Political Economy 108 (1973) 144172.
A. Brooke, D. Kendrick and A. Meeraus, GAMS: A User's Guide. The Scientific Press, San Fransisco, California (1992).
Calafiore, G., Ambiguous risk measures and optimal robust portfolios. SIAM J. Optim. 18 (2007) 853877. CrossRef
Cont, R., Model uncertainty and its impact on the pricing of derivative instruments. Math. Finance 16 (2006) 519547. CrossRef
Csiszar, I., Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungarica 2 (1967) 299318.
d'Aspremont, A. and El Ghaoui, L., Static arbitrage bounds on basket option prices. Math. Programming 106 (2006) 467489. CrossRef
Eichhorn, A. and Römisch, W., Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16 (2005) 6995. CrossRef
El Ghaoui, L., Oks, M. and Oustry, F., Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Oper. Res. 51 (2003) 543556. CrossRef
Epstein, L.G., A definition of uncertainty aversion. Rev. Economic Studies 65 (1999) 579608. CrossRef
M.C. Ferris and T.S. Munson, Interfaces to PATH 3.0: Design, implementation and usage. Technical Report, University of Wisconsin, Madison (1998).
H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, De Gruyter Studies in Mathematics 27. Second Edition, Berlin (2004).
Harrison, J.M. and Kreps, D.M., Martingales and arbitrage in multiperiod securities markets. J. Economic Theory 20 (1979) 381408. CrossRef
Harrison, J.M. and Pliska, S.R., Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11 (1981) 215260. CrossRef
A.J. King and L.A. Korf, Martingale Pricing Measures in Incomplete Markets via Stochastic Programming Duality in the Dual of ${\cal L}^{\infty}$ . Technical Report (2001).
Korf, L.A., Stochastic programming duality: ${\cal L}^{\infty}$ multipliers for unbounded constraints with an application to mathematical finance. Math. Programming 99 (2004) 241259. CrossRef
S. Kullback, Information Theory and Statistics. Wiley, New York (1959)
H.J. Landau, Moments in mathematics, in Proc. Sympos. Appl. Math. 37, H.J. Landau Ed., AMS, Providence, RI (1987).
I.R. Longarela, A simple linear programming approach to gain, loss and asset pricing. Topics in Theoretical Economics 2 (2002) Article 4.
T.R. Rockafellar, Conjugate Duality and Optimization. SIAM, Philadelphia (1974).
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).
Ruszczyński, A. and Shapiro, A., Optimization of convex risk functions. Math. Oper. Res. 31 (2006) 433452. CrossRef
Shapiro, A., On duality theory of convex semi-infinite programming. Optimization 54 (2005) 535543. CrossRef
Shapiro, A. and Ahmed, S., On a class of stochastic minimax programs. SIAM J. Optim. 14 (2004) 12371249. CrossRef
Shapiro, A. and Kleywegt, A., Minimax analysis of stochastic problems. Optim. Methods Software 17 (2002) 523542. CrossRef
Smith, J.E., Generalized Chebychev inequalities: Theory and applications in decision analysis. Oper. Res. 43 (1995) 807825. CrossRef
Sh. Tian and R.J.-B. Wets, Pricing Contingent Claims: A Computational Compatible Approach. Technical Report, Department of Mathematics, University of California, Davis (2006).