CVaR-minimising hedging by a smoothing method

Minimisation of a conditional value-at-risk (CVaR) is a non-smooth stochastic minimisation problem with the non-smoothness caused by a plus function in the integrand of the objective function. We study the performance of several smoothing approximations of the plus function in the CVaR minimisation, using an example of a one period CVaR-minimising hedging. The smooth plus function that outperforms others is identified. The convergence of the solution of the smoothed CVaR minimisation problem with increase in the sample size and with decrease in the smoothing parameter is illustrated. The performance of the one period CVaR-minimising hedging and delta-gamma hedging are compared in terms of several commonly used performance criteria. Numerical simulations show that the magnitude and the probability of large losses are smaller in the CVaR-minimising hedge, compared to the delta-gamma hedge. This often occurs at the expense of a deteriorated expected return and increased variance of the hedge. We identify the situations when the CVaR-minimising hedge outperforms the delta-gamma hedge according to all performance criteria. References S. Alexander, T. F. Coleman, and Y. Li, Derivative portfolio hedging based on cvar. In G. Szego, editor, Risk Measures for the 21st Century, pages 339--363. London: Wiley, 2004. S. Alexander, T. F. Coleman, and Y. Li, Minimizing cvar and var for a portfolio of derivatives. Journal of Banking and Finance, 30(2), 2006, 583--605. doi:10.1016/j.jbankfin.2005.04.012 B. Chen, and P. T. Harker, A non-interior-point continuation method for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications, 14, 1993, 1168--1190. doi:10.1137/0614081 C. Chen, and O. L. Mangasarian, Smoothing method for convex inequalities and linear complementary problems, Mathematical Programming, 71(1), 1995, 51--69. doi:10.1007/BF01592244 C. Chen, and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complimentarity problems, Computational Optimization and Applications, 5(2), 1996, 97--138. doi:10.1007/BF00249052 C. Kanzow, Some tools allowing interior-point methods to become noninterior, Technical Report, Institute of Applied Mathematics, University of Hamburg, Germany, 1994. H. Mausser, and D. Rosen, Beyond var: from measuring risk to managing risk. {ALGO Research Quarterly}, 1(2), 1998, 5--20. J. Peng, A smoothing function and its applications. In M. Fukushima, and L. Qi, editors, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pages 293--316. Kluwer, Dordrecht, 1998. J.-S. Pinar, and S. A. Zenios, On smoothing exact penalty functions for convex constrained optimization, SIAM J. Optimization, 4, 1994, 486--511. doi:10.1137/0804027 R. T. Rockafellar, and S. Uryasev, Optimization of Conditional value at Risk, Journal of Risk, 2(3), 2000, 21--41. R. T. Rockafellar, and S. Uryasev, Conditional Value at Risk for General Loss Distributions, Journal of Banking and Finance, 26(7), 2002, 1443--1471. doi:10.1016/S0378-4266(02)00271-6 S. Smale, Algorithm for solving equations, In Proceedings of the International Congress of Mathematicians, pages 172--195, Amer. Math. Soc., Providence, 1987. T. Tarnopolskaya, J. Tabak, and F. R. de Hoog, L-curve for hedging instrument selection in cvar-minimizing portfolio hedging, In R. S. Anderssen, R. D. Braddock and L. T. H. Newham, editors, 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation, pages 1559--1565, July 2009. http://www.mssanz.org.au/modsim09/D11/tarnopolskaya_D11.pdf. T. Tarnopolskaya, and Z. Zhu, A robust hedging strategy via cvar minimization, Proceedings of the First Chinese Forum on Intelligent Finance (CFIF-I 2009), Beijing, February 2009. S. P. Uryasev, and R. T. Rockafellar, Conditional value-at-risk: Optimization approach. Stochastic Optimization: Algorithms and Applications, 54, 2001, 411--435. H. Xu, and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming, 119(2), 2009, 371--401. doi:10.1007/S10107-008-0214-0 Y. Yamai, and T. Yosiba, Value-at-risk versus expected shortfall: a practical perspective, Journal of Banking and Finance, 29, 2005, 997--1015. doi:10.1016/j.jbankfin.2004.08.010 I. Zang, A smoothing-out technique for min-max optimization, Mathematical Programming, 19, 1980, 61--77. doi:10.1007/BF0158162