Tradable measure of risk

Pospisil, Libor and Vecer, Jan and Xu, Mingxin (2007): Tradable measure of risk.

Abstract

The main idea of this paper is to introduce Tradeable Measures of Risk as an objective and model independent way of measuring risk. The present methods of risk measurement, such as the standard Value-at-Risk supported by BASEL II, are based on subjective assumptions of future returns. Therefore two different models applied to the same portfolio can lead to different values of a risk measure. In order to achieve an objective measurement of risk, we introduce a concept of {\em Realized Risk} which we define as a directly observable function of realized returns. Predictive assessment of the future risk is given by {\em Tradeable Measure of Risk} -- the price of a financial contract which pays its holder the Realized Risk for a certain period. Our definition of the Realized Risk payoff involves a Weighted Average of Ordered Returns, with the following special cases: the worst return, the empirical Value-at-Risk, and the empirical mean shortfall. When Tradeable Measures of Risk of this type are priced and quoted by the market (even of an experimental type), one does not need a model to calculate values of a risk measure since it will be observed directly from the market. We use an option pricing approach to obtain dynamic pricing formulas for these contracts, where we make an assumption about the distribution of the returns. We also discuss the connection between Tradeable Measures of Risk and the axiomatic definition of Coherent Measures of Risk.

• Item Type: MPRA Paper
• Language: English
• Keywords: dynamic risk measure; conditional value-at-risk; shortfall
• Subjects: G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing; G - Financial Economics > G2 - Financial Institutions and Services > G28 - Government Policy and Regulation

References

\bibitem{AcerbiTasche}{{\sc Acerbi, C., D. Tasche (2002)}: ``On the coherence of expected shortfall," {\em Journal of Banking and Finance}, {\bf 26}, 1487--1503.} % \bibitem{ArtznerDelbaenEberHeath1}{{\sc Artzner, P., F. Delbaen, J.-M. Eber, D. Heath (1997)}: ``Thinking coherently," {\em Risk}, {\bf 10}, 68--71.} % \bibitem{ArtznerDelbaenEberHeath2}{{\sc Artzner, P., F. Delbaen, J.-M. Eber, D. Heath (1999)}: ``Coherent Measures of Risk," {\em Mathematical Finance}, {\bf 9}, 203--228.} % \bibitem{ArtznerDelbaenEberHeathKu}{{\sc Artzner, P., F. Delbaen, J.-M. Eber, D. Heath, H. Ku (2007)}: ``Coherent multiperiod risk adjusted values and Bellman's principle," {\em Annals of Operations Research}, {\bf 152}, 5--22.} % \bibitem{CarrMadanGemanYor}{{\sc Carr, P., D. Madan, H. Geman, M. Yor (2005)}: ``Pricing options on realized variance," {\em Finance and Stochastics}, {\bf 9}, 453--475.} % \bibitem{CheriditoDelbaenKupper}{{\sc Cheridito, P., F., Delbaen, M. Kupper (2006)}: ``Coherent and convex monetary risk measures for unbounded c\`{a}dl\`{a}g processes," {\em Finance and Stochastics}, {\bf 10}, 427--448.} % \bibitem{Cherny1}{{\sc Cherny, A. (2006):} ``Weighted V@R and its properties," {\em Finance and Stochastics,} {\bf 10}, 367--393.} % \bibitem{ContTankov}{{\sc Cont, R., P. Tankov (2004)}: ``Financial modelling with jump processes," {\em Chapman \& Hall/CRC financial mathematics series}.} % \bibitem{Delbaen}{{\sc Delbaen, F. (2002)}: ``Coherent risk measures on general probability spaces," {\em Adavances in Finance and Stochastics - Essays in Honour of Dieter Sondermann}, Springer.} % \bibitem{FollmerSchied1}{{\sc F\"{o}llmer, H., A. Schied (2002)}: ``Convex measures of risk and trading constraints," {\em Finance and Stochastics}, {\bf 6}, 429--447.} % \bibitem{FollmerSchied2}{{\sc F\"{o}llmer, H., A. Schied (2004)}: {\em Stochastic finance - an introduction in discrete time, 2nd Edition,} Walter de Gruyter, Studies in Mathematics, {\bf 27}.} % \bibitem{FrittelliScandolo}{{\sc Frittelli, M., G. Scandolo (2006)}: ``Risk measures and capital requirements for processes," {\em Mathematical Finance}, {\bf 16}, 589--612.} % \bibitem{HeydeKouPeng}{{\sc Heyde, C. C., S. G. Kou, X. H. Peng (2006)}: ``What is a good risk measure: bridging the gaps between data, coherent risk measure, and isurance risk measure," preprint, Columbia University.} % \bibitem{Jarrow}{{\sc Jarrow, R. (2002)}: ``Put option premiums and coherent risk measures," {\em Mathematical Finance,} {\bf 12}, 135--142.} % \bibitem{KloppelSchweizer}{{\sc Kl\"{o}ppel, S., M. Schweizer (2006)}: ``Dynamic utility indifference valuation via convex risk measures," to appear in {\em Mathematical Finance}.} % \bibitem{Kusuoka}{{\sc Kusuoka, S. (2001)}: ``On law invariant coherent risk measures," {\em Advances in Mathematical Economics}, {\bf 3}, 83--95.} % \bibitem{Miura}{{\sc Miura, R. (1992)}: ``A note on look-back options based on order statistics," {\em Hitotsubashi Journal of Commerce and Management,} {\bf 27}, 15-28.} % \bibitem{Riedel}{{\sc Riedel, F. (2004)}: ``Dynamic coherent risk measures," {\em Stochastic Processes and Their Applications}, {\bf 112}, 185--200.}

\bibitem{RockafellarUryasev1}{{\sc Rockafellar, R. T., S. Uryasev (2000)}: ``Optimization of Conditional Value-at-Risk," {\em The Journal of Risk}, {\bf 2, 4}, 21--51.} \bibitem{V}{{\sc Vecer, J. (2006):} "Maximum Drawdown and Directional Trading", Risk, Vol. 19, No. 12, 88-92.} % \bibitem{WangYoungPanjer}{{\sc Wang, S. S., V. R. Young, H. H. Panjer (1997):} ``Axiomatic characterization of insurance prices," {\em Insurance: Mathematics and Economics}, {\bf 21}, 173--183.} % \bibitem{Weber}{{\sc Weber, S. (2006)}: ``Distribution-invariant risk measures, information, and dynamic consistency," {\em Mathematical Finance}, {\bf 16}, 419--442.} % \bibitem{Yaari}{{\sc Yaari, M. E. (1987):} ``The dual theory of the choice under risk," {\em Econometrica}, {\bf 55}, 95--115. }