Find out how to access previewonly content
Date:
23 Jan 2013
Setvalued average value at risk and its computation
 Andreas H. Hamel,
 Birgit Rudloff,
 Mihaela Yankova
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
New versions of the setvalued average value at risk for multivariate risks are introduced by generalizing the wellknown certainty equivalent representation to the setvalued case. The first ’regulator’ version is independent from any market model whereas the second version, called the market extension, takes trading opportunities into account. Essential properties of both versions are proven and an algorithmic approach is provided which admits to compute the values of both versions over finite probability spaces. Several examples illustrate various features of the theoretical constructions.
 Acerbi, C, Tasche, D (2002) On the coherence of expected shortfall. Journal of Banking and Finance 26: pp. 14871503 CrossRef
 Artzner, P, Delbaen, F, KochMedina, P (2009) Risk measures and efficient use of capital. Astin Bull. 39: pp. 101116 CrossRef
 Benson, HP (1998) An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global Optim. 13: pp. 124 CrossRef
 Burgert, C, Rüschendorf, L (2006) Consistent risk measures for portfolio vectors. Insurance Math. Econom. 38: pp. 289297 CrossRef
 Ekeland, I, Galichon, A, Henry, M (2012) Comonotonic measures of multivariate risks. Mathematical Finance 22: pp. 109132 CrossRef
 Ekeland, I, Schachermayer, W (2011) Law invariant risk measures in $$L^\infty (R_d)$$. Statistics and Risk Modeling 28: pp. 195225 CrossRef
 Föllmer, H, Schied, A (2011) Stochastic Finance. Walter de Gruyter & Co., Berlin CrossRef
 Hamel, A, Heyde, F (2010) Duality for setvalued measures of risk. SIAM Journal on Financ. Mathematics 1: pp. 6695 CrossRef
 Hamel, A, Heyde, F, Rudloff, B (2011) Setvalued risk measures for conical market models. Mathematics and Financial Economics 5: pp. 128 CrossRef
 Hamel, A., Löhne, A., Rudloff, B.: A benson type algorithm for linear vector optimization and applications. Submitted for publication (2013)
 Hodges, S, Neuberger, A (1989) Optimal replication of contingent claims under transaction costs. Rev. Futures Markets 8: pp. 222239
 Jaschke, S, Küchler, U (2001) Coherent risk measures and gooddeal bounds. Finance Stoch. 5: pp. 181200 CrossRef
 Jouini, E, Meddeb, M, Touzi, N (2004) Vectorvalued coherent risk measures. Finance and Stochastics 8: pp. 531552
 Kabanov, YM (1999) Hedging and liquidation under transaction costs in currency markets. Finance and Stochastics 3: pp. 237248 CrossRef
 Kabanov, YM, Safarian, M (2009) Markets with Transaction Costs: Mathematical Theory. Springer, Springer Finance
 Korn, R, Müller, S (2009) The decoupling approach to binomial pricing of multiasset options. J. Comput. Finance 12: pp. 130
 Löhne, A.: Vector Optimization with Infimum and Supremum. Springer, (2011)
 Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. ArXiv eprints (2011). Submitted for publication
 Löhne, A., Rudloff, B.: On solvency cones and their duals in markets with transaction costs. Working paper (2013)
 Pflug, G.C.: Some remarks on the valueatrisk and the conditional valueatrisk. In: Probabilistic constrained optimization, Nonconvex Optim. Appl., vol. 49, pp. 272–281. Kluwer Acad. Publ., Dordrecht (2000)
 Rockafellar, RT, Uryasev, SP (2000) Optimization of conditional valueatrisk. Journal of risk 2: pp. 2142
 Schachermayer, W (2004) The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Mathematical Finance 14: pp. 1948 CrossRef
 Yankova, M.: Setvalued Average Value at Risk with random transaction costs (2011). Senior thesis, Princeton University
 Title
 Setvalued average value at risk and its computation
 Journal

Mathematics and Financial Economics
Volume 7, Issue 2 , pp 229246
 Cover Date
 20130301
 DOI
 10.1007/s1157901300949
 Print ISSN
 18629679
 Online ISSN
 18629660
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Average value at risk
 Setvalued risk measures
 Coherent risk measures
 Transaction costs
 Benson’s algorithm
 91B30
 46N10
 26E25
 46A20
 C61
 G32
 Industry Sectors
 Authors

 Andreas H. Hamel ^{(1)}
 Birgit Rudloff ^{(2)}
 Mihaela Yankova ^{(3)}
 Author Affiliations

 1. Department of Mathematical Sciences, Yeshiva University, New York, NY, USA
 2. ORFE, BCF, Princeton University, Princeton, NJ, USA
 3. Barclays Capital, New York, NY, USA