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Backward stochastic difference equations for dynamic convex risk measures on a binomial tree

Part of: Stochastic analysis Mathematical finance

Published online by Cambridge University Press:  30 March 2016

Robert J. Elliott ,
Tak Kuen Siu  and
Samuel N. Cohen
Robert J. Elliott*
Affiliation:
University of Adelaide and University of Calgary
Tak Kuen Siu*
Affiliation:
Macquarie University and City University London
Samuel N. Cohen*
Affiliation:
University of Oxford
*
∗ Postal address: School of Mathematical Sciences, University of Adelaide, SA 5005, Australia. Email address: [email protected]
∗∗ Postal address: Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia.
∗∗∗ Postal address: Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK.
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Abstract

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Using backward stochastic difference equations (BSDEs), this paper studies dynamic convex risk measures for risky positions in a simple discrete-time, binomial tree model. A relationship between BSDEs and dynamic convex risk measures is developed using nonlinear expectations. The time consistency of dynamic convex risk measures is discussed in the binomial tree framework. A relationship between prices and risks is also established. Two particular cases of dynamic convex risk measures, namely risk measures with stochastic distortions and entropic risk measures, and their mathematical properties are discussed.

Keywords

Dynamic convex risk measureconditional nonlinear expectationbinomial treebackward stochastic difference equationstochastic distortion probability

MSC classification

Primary: 91G20: Derivative securities
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60H07: Stochastic calculus of variations and the Malliavin calculus
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 771 - 785
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203-228.Google Scholar
Artzner, P. et al. (2002). Coherent multiperiod risk measurement. Preprint, Department of Mathematics, ETH, Zürich.Google Scholar
Barrieu, P. and El Karoui, N. (2004). Optimal derivatives design under dynamic risk measures. In Mathematics of Finance (Contemp. Math. 351), American Mathematical Society, Providence, RI, pp. 13-25.Google Scholar
Bion-Nadal, J. (2004). Conditional risk measure and robust representation of convex conditional risk measures. CMAP Preprint 557, Ecole Polytechnique, Paris.Google Scholar
Boyle, P., Siu, T. K. and Yang, H. (2002). Risk and probability measures. Risk 15 53-57.Google Scholar
Cheridito, P. and Stadje, M. (2013). BSDEs and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness. Bernoulli 19 1047-1085.Google Scholar
Cherny, A. and Madan, D. (2009). New measures of performance evaluation. Rev. Financ. Stud. 22 2571-2606.CrossRefGoogle Scholar
Cohen, S. N. and Elliott, R. J. (2009). A general theory of backward stochastic difference equations. Presentation Notes. University of Adelaide and University of Calgary.Google Scholar
Cohen, S. N. and Elliott, R. J. (2010). A general theory of finite state backward stochastic difference equations. Stoch. Process. Appl. 120 442-466.Google Scholar
Cohen, S. N. and Elliott, R. J. (2011). Backward stochastic difference equations with finite states. In Stochastic Analysis with Financial Applications (Progr. Prob. 65), Birkhäuser, Basel, pp. 33-42.Google Scholar
Coquet, F., Hu, Y., Mémin, J. and Peng, S. (2002). Filtration-consistent nonlinear expectations and related g-expectations. Prob. Theory Relat. Fields 123 1-27.Google Scholar
Cvitanić, J. and Karatzas, I. (1999). On dynamic measures of risk. Finance Stoch. 3 451-482.Google Scholar
Delbaen, F. (2006). The structure of m-stable sets and in particular of the set of risk neutral measures. In Memoriam Paul-André Meyer (Lecture Notes Math. 1874), Springer, Berlin, pp. 215-258.Google Scholar
Delbaen, F., Peng, S. and Rosazza-Gianin, E. (2010). Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14, 449–472.Google Scholar
Detlefsen, K. and Scandolo, G. (2005). Conditional and dynamic convex risk measures. Finance Stoch. 9 539-561.Google Scholar
Eberlein, E. and Madan, D. B. (2009). Hedge fund performance: sources and measures. Internat. J. Theoret. Appl. Finance 12 267-282.Google Scholar
El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71.Google Scholar
Elliott, R. J. and Kopp, P. E. (2005). Mathematics of Financial Markets, 2nd edn. Springer, New York.Google Scholar
Elliott, R. J. and Siu, T. K. (2012). A BSDE approach to convex risk measures for derivative securities. Stoch. Anal. Appl. 30 1083-1101.Google Scholar
Elliott, R. J. and Siu, T. K. (2013). Reflected backward stochastic differential equations, convex risk measures and American options. Stoch. Anal. Appl. 31 1077-1096.Google Scholar
Elliott, R. J. and Yang, H. (1994). How to count and guess well: discrete adaptive filters. Appl. Math. Optimization 30 51-78.Google Scholar
Elliott, R. J., Siu, T. K. and Chan, L. (2008). A PDE approach for risk measures for derivatives with regime switching. Ann. Finance 4 55-74.Google Scholar
Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. Finance Stoch. 6 429-447.Google Scholar
Föllmer, H. and Schied, A. (2004). Stochastic Finance: An Introduction in Discrete Time (De Gruyter Studies Math. 27), 2nd edn. De Gruyter, Berlin.Google Scholar
Frittelli, M. and Rosazza-Gianin, E. (2002). Putting order in risk measures. J. Banking Finance 26, 1473-1486.Google Scholar
Frittelli, M. and Rosazza-Gianin, E. (2004). Dynamic convex risk measures. In Risk Measures for the 21st Century, ed. Szegö, G., John Wiley, Chichester, pp. 227–248.Google Scholar
Jobert, A. and Rogers, L. C. G. (2008). Valuations and dynamic convex risk measures. Math. Finance 18 1-22.Google Scholar
Klöppel, S. and Schweizer, M. (2007). Dynamic indifference valuation via convex risk measures. Math. Finance 17 599-627.Google Scholar
Kupper, M. and Schachermayer, W. (2009). Representation results for law invariant time consistent functions. Math. Financial Econom. 2 189-210.Google Scholar
Peng, S. (1997). Backward SDE and related g-expectation. In Backward Stochastic Differential Equations (Pitman Res. Notes Math. Ser. 364), Longman, Harlow, pp. 141-159.Google Scholar
Privault, N. (2009). Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Springer, Berlin.Google Scholar
Rosazza-Gianin, E. (2006). Risk measures via g-expectations. Insurance Math. Econom. 39 19-34.Google Scholar
Siu, T. K. (2012). Functional Itô's calculus and dynamic convex risk measures for derivative securities. Commun. Stoch. Anal. 6 339-358.Google Scholar
Siu, T. K. and Yang, H. (1999). Subjective risk measures: Bayesian predictive scenarios analysis. Insurance Math. Econom. 25 157-169.Google Scholar
Siu, T. K. and Yang, H. (2000). A PDE approach to risk measures of derivatives. Appl. Math. Finance 7 211-228.CrossRefGoogle Scholar
Siu, T. K., Tong, H. and Yang, H. (2001). Bayesian risk measures for derivatives via random Esscher transform. N. Amer. Actuarial J. 5 78-91.CrossRefGoogle Scholar
Wang, S. S. (2000). A class of distortion operators for pricing financial and insurance risks. J. Risk Insurance 67 15-36.Google Scholar
Wang, S. S. and Young, V. R. (1998). Risk-adjusted credibility premiums using distorted probabilities. Scand. Actuarial J. 1998 143-165.Google Scholar
Wang, T. (1999). A class of dynamic risk measures. Preprint. Faculty of Commerce and Business Administration, University of British Columbia.Google Scholar