L. Jeff Hong
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Abstract
Value-at-risk (VaR) and conditional value-at-risk (CVaR) are two widely used risk measures of large losses and are employed in the financial industry for risk management purposes. In practice, loss distributions typically do not have closed-form expressions, but they can often be simulated (i.e., random observations of the loss distribution may be obtained by running a computer program). Therefore, Monte Carlo methods that design simulation experiments and utilize simulated observations are often employed in estimation, sensitivity analysis, and optimization of VaRs and CVaRs. In this article, we review some of the recent developments in these methods, provide a unified framework to understand them, and discuss their applications in financial risk management.
- C. Acerbi and D. Tasche. 2002. On the coherence of expected shortfall. Journal of Banking and Finance 26, 1487--1503.Google Scholar
Cross Ref
- S. Alexander, T. F. Coleman, and Y. Li. 2006. Minimizing CVaR and VaR for a portfolio of derivatives. Journal of Banking and Finance 30, 583--605.Google ScholarCross Ref
- F. Andersson, H. Mausser, D. Rosen, and S. Uryasev. 2001. Credit risk optimization with conditional value-at-risk criterion. Mathematical Programming 89, 273--291.Google ScholarCross Ref
- B. Ankenman, B. L. Nelson, and J. Staum. 2010. Stochastic kriging for simulation metamodeling. Operations Research 58, 2, 371--382. Google ScholarDigital Library
- P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. 1999. Coherent measures of risk. Mathematical Finance 9, 3, 203--228.Google ScholarCross Ref
- S. Asmussen and P. W. Glynn. 2007. Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google ScholarCross Ref
- A. N. Avramidis and J. R. Wilson. 1998. Correlation-induction techniques for estimating quantiles in simulation experiments. Operations Research 46, 574--591. Google ScholarDigital Library
- R. Bahadur. 1966. A note on quantiles in large samples. Annals of Mathematical Statistics, 37, 577--580.Google ScholarCross Ref
- O. A. Bardou, N. Frikha, and G. Pagès. 2009. Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling. Monte Carlo Methods and Applications 15, 3, 173--210.Google ScholarCross Ref
- Basel Committee of Banking Supervision of the Bank for International Settlements. 2012. Fundamental review of the trading book. Consultative Document.Google Scholar
- H. G. Basova, R. T. Rockafellar, and J. O. Royset. 2011. A computational study of the buffered failure probability in reliability-based design optimization. In Proceedings of the 11th Conference on Application of Statistics and Probability in Civil Engineering.Google Scholar
- A. Bassamboo, S. Juneja, and A. Zeevi. 2008. Portfolio credit risk with extremal dependence: Asymptotic analysis and efficient simulation. Operations Research 56, 593--606. Google ScholarDigital Library
- D. P. Bertsekas. 2007. Dynamic Programming and Optimal Control (Vol. 2). Athena Scientific, Belmont, MA. Google ScholarDigital Library
- D. Bosq. 1998. Nonparametric Statistics for Stochastic Processes. New York: Springer.Google Scholar
- M. Broadie, Y. Du, and C. C. Moallemi. 2010. Efficient risk estimation via nested sequential simulation. Management Science, 57, 1172--1194. Google ScholarDigital Library
- M. Broadie, Y. Du, and C. C. Moallemi. 2011. Risk estimation via regression. Working paper, Graduate School of Business, Columbia University.Google Scholar
- M. Broadie and P. Glasserman. 1996. Estimating security price derivatives using simulation. Management Science, 42, 269--285. Google ScholarDigital Library
- G. Calafiore and M. C. Campi. 2005. Uncertain convex programs: Randomized solutions and confidence levels. Mathematical Programming 102, 25--46. Google ScholarDigital Library
- G. Calafiore and M. C. Campi. 2006. The scenario approach to robust control design. IEEE Transactions on Automatic Control 51, 742--753.Google ScholarCross Ref
- J. F. Carrière. 1996. Valuation of early-exercise price for options using simulations and nonparametric regression. Insurance: Mathematics and Economics 19, 19--30.Google ScholarCross Ref
- A. Charnes, W. W. Cooper, and G. H. Symonds. 1958. Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Management Science 4, 235--263. Google ScholarDigital Library
- F. Chu and M. K. Nakayama. 2012. Confidence intervals for quantiles when applying variance-reduction techniques. ACM Transactions on Modeling and Computer Simulation 22, 2, Article 10. Google ScholarDigital Library
- S. Deng, K. Giesecke, and T. L. Lai. 2012. Sequential importance sampling and resampling for dynamic portfolio credit risk. Operations Research 60, 78--91. Google ScholarDigital Library
- D. P. De Farias and B. Van Roy. 2004. On constraint sampling in the linear programming approach to approximate dynamic programming. Mathematics of Operations Research 29, 3, 462--478. Google ScholarDigital Library
- D. Duffie. 2001. Dynamic Asset Pricing Theory. Princeton University Press, Princeton, NJ.Google Scholar
- L. El Ghaoui, M. Oks, and F. Oustry. 2003. Worst case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research 51, 543--556. Google ScholarDigital Library
- M. C. Fu, L. J. Hong, and J.-Q. Hu. 2009. Conditional Monte Carlo estimation of quantile sensitivities. Management Science 55, 2019--2027. Google ScholarDigital Library
- M. C. Fu and J.-Q. Hu. 1997. Conditional Monte Carlo, Gradient Estimation and Optimization Applications. Kluwer Academic Publishers, Boston.Google Scholar
- P. Glasserman. 1991. Gradient Estimation via Perturbation Analysis. Kluwer Academic Publishers, Boston, MA.Google Scholar
- P. Glasserman. 2004. Monte Carlo Methods in Financial Engineering. Springer, New York.Google Scholar
- P. Glasserman. 2005. Measuring marginal risk contributions of credit portfolios. Journal of Computational Finance 9, 1--41.Google ScholarCross Ref
- P. Glasserman. 2008. Calculating portfolio credit risk. In Financial Engineering: Handbooks in Operations Research and Management Science, J. Birge and V. Linetsky (Eds.). Vol. 15. Elsevier, North-Holland, The Netherlands.Google Scholar
- P. Glasserman, P. Heidelberger, and P. Shahabuddin. 1999. Importance sampling and stratification for Value-at-Risk. Computational Finance 1999, Y. S. Abu-Mostafa, B. Le Baron, A. W. Lo, and A. S. Weigend (Eds.). MIT Press, Cambridge, MA.Google Scholar
- P. Glasserman, P. Heidelberger, and P. Shahabuddin. 2000. Variance reduction techniques for estimating value-at-risk. Management Science 46, 1349--1364. Google ScholarDigital Library
- P. Glasserman, P. Heidelberger, and P. Shahabuddin. 2002. Portfolio value-at-risk with heavy-tailed risk factors. Mathematical Finance, 12, 239--269.Google ScholarCross Ref
- P. Glasserman, W. Kang, and P. Shahabuddin. 2007. Large deviations in multifactor portfolio credit risk. Mathematical Finance 17, 345--379.Google ScholarCross Ref
- P. Glasserman, W. Kang, and P. Shahabuddin. 2008. Fast simulation of multifactor portfolio credit risk. Operations Research 56, 1200--1217.Google ScholarCross Ref
- P. Glasserman and J. Li. 2005. Importance sampling for portfolio credit risk. Management Science 51, 1643--1656.Google ScholarCross Ref
- P. W. Glynn. 1996. Importance sampling for Monte Carlo estimation of quantiles. In Proceedings of 1996 Second International Workshop on Mathematical Methods in Stochastic Simulation and Experimental Design, St. Petersburg, Russia, 180--185.Google Scholar
- P. W. Glynn and D. L. Iglehart. 1989. Importance sampling for stochastic simulation. Management Science 35, 1367--1392. Google ScholarDigital Library
- M. Gordy and S. Juneja. 2010. Nested simulation in portfolio risk measurement. Management Science 56(10), 1833--1848. Google ScholarDigital Library
- G. Gupton, C. Finger, and M. Bhatia. 1997. CreditMetrics technical document. New York: J. P. Morgan & Co.Google Scholar
- B. Heidergott and W. Volk-Makarewicz. 2012. Sensitivity analysis of quantiles. Working paper.Google Scholar
- T. C. Hesterberg and B. L. Nelson. 1998. Control variates for probability and quantile estimation. Management Science 44, 1295--1312. Google ScholarDigital Library
- Y. C. Ho and X. R. Cao. 1983. Perturbation analysis and optimization of queueing networks. Journal of Optimization Theory and Applications 40, 4, 559--582. Google ScholarDigital Library
- T. Homem-de-Mello and G. Bayraksan. 2013. Monte Carlo sampling-based methods for stochastic optimization. Available at http://www.optimization-online.org/DB_FILE/2013/06/3920.pdf.Google Scholar
- L. J. Hong. 2009. Estimating quantile sensitivities. Operations Research 57, 118--130. Google ScholarDigital Library
- L. J. Hong and G. Liu. 2009. Simulating sensitivities of conditional value-at-risk. Management Science 55, 281--293. Google ScholarDigital Library
- L. J. Hong and G. Liu. 2010. Pathwise estimation of probability sensitivities through terminating or steady-state simulations. Operations Research 58, 357--370. Google ScholarDigital Library
- L. J. Hong, Y. Yang, and L. Zhang. 2011. Sequential convex approximations to joint chance constrained programs: A Monte Carlo approach. Operations Research 59, 617--630. Google ScholarDigital Library
- L. J. Hong, Z. Hu, and L. Zhang. 2012a. Conditional value-at-risk approximation to value-at-risk constrained programs: A remedy via Monte Carlo. INFORMS Journal on Computing 26, 2, 385--400. Google ScholarDigital Library
- L. J. Hong, S. Juneja, and G. Liu. 2012b. Kernel smoothing for nested estimation with application to portfolio risk measurement. Working paper.Google Scholar
- J. C. Hsu and B. L. Nelson. 1990. Control variates for quantile estimation. Management Science 36, 835--851. Google ScholarDigital Library
- Z. Hu and L. J. Hong. 2012. Kullback-Leibler divergence constrained distributionally robust optimization. Available at http://www.optimization-online.org/DB_FILE/2012/11/3677.pdf.Google Scholar
- Z. Hu, L. J. Hong, and A. M. C. So. 2013a. Distributionally robust probabilistic programs. Working paper. Available at http://www.optimization-online.org/DB_FILE/2013/09/4039.pdf.Google Scholar
- Z. Hu, L. J. Hong, and L. Zhang. 2013b. A smooth Monte Carlo approach to joint chance constrained program. IIE Transactions 45, 7, 716--735.Google ScholarCross Ref
- G. Iyengar and A. K. C. Ma. 2013. Fast gradient descent method for mean-CVaR optimization. Annals of Operations Research 205, 203--212.Google ScholarCross Ref
- P. Jorion. 2006. Value at Risk, (2nd ed.). McGraw-Hill, New York.Google Scholar
- P. Jorion. 2010. Financial Risk Manager Handbook (6th ed.). Wiley, New York.Google Scholar
- M. Kalkbrener. 2003. An axiomatic approach to capital allocation. Mathematical Finance 15, 425--437.Google ScholarCross Ref
- M. Kalkbrener, H. Lotter, and L. Overbeck. 2004. Sensible and efficient capital allocation for credit portfolios. Risk 17, 19--24.Google Scholar
- A. J. King and R. T. Rockafellar. 1993. Asymptotic theory for solutions in statistical estimation and stochastic programming. Mathematics of Operations Research 18, 1, 148--162. Google ScholarDigital Library
- S. G. Kou, X. Peng, and C. C. Heyde. 2013. External risk measures and Basel accords. Mathematics of Operations Research 38, 3, 393--417. Google ScholarDigital Library
- P. Krokhmal, J. Palmquist, and S. Uryasev. 2002. Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk 4, 11--27.Google Scholar
- A. Kurth and D. Tasche. 2003. Contributions to credit risk. Risk 16, 84--88.Google Scholar
- P. L’Ecuyer. 1990. A unified view of the IPA, SF, and LR gradient estimation techniques. Management Science 36, 11, 1364--1383. Google ScholarDigital Library
- H. Lan, B. L. Nelson, and J. Staum. 2010. A confidence interval procedure for expected shortfall risk measurement via two-level simulation. Operations Research 58, 5, 1481--1490. Google ScholarDigital Library
- A. M. Law and W. D. Kelton. 2000. Simulation Modeling & Analysis (3rd ed.). McGraw-Hill, New York. Google ScholarDigital Library
- S. H. Lee. 1998. Monte Carlo expectation of conditional quantiles. Ph.D. Thesis, Department of Operations Research, Stanford University.Google Scholar
- D. Li. 2000. On default correlation: A copula function approach. Journal of Fixed Income 9, 43--54.Google ScholarCross Ref
- C. Lim, H. D. Sherali, and S. Uryasev. 2010. Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Computational Optimization and Applications 46, 391--415. Google ScholarDigital Library
- A. E. Lim, J. G. Shanthikumar, and G. Y. Vahn. 2011. Conditional value-at-risk in portfolio optimization: Coherent but fragile. Operations Research Letters 39, 3, 163--171. Google ScholarDigital Library
- G. Liu. 2012. Simulating risk contributions of credit portfolios. Working paper, Department of Management Sciences, City University of Hong Kong.Google Scholar
- G. Liu and L. J. Hong. 2009. Kernel estimation of quantile sensitivities. Naval Research Logistics 56, 511--525.Google ScholarCross Ref
- M. Liu and J. Staum. 2010. Stochastic kriging for efficient nested simulation of expected shortfall. Journal of Risk 12, 3, 3--27.Google ScholarCross Ref
- F. A. Longstaff and S. Schwartz. 2001. Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies 14, 113--147.Google ScholarCross Ref
- J. Luedtke and S. Ahmed. 2008. A sample approximation approach for optimization with probabilistic constraints. SIAM Journal on Optimization 19, 2, 674--699. Google ScholarDigital Library
- A. J. McNeil, R. Frey, and P. Embrechts. 2005. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton.Google Scholar
- L. B. Miller and H. Wagner. 1965. Chance-constrained programming with joint constraints. Operations Research 13, 930--945. Google ScholarDigital Library
- K. Natarajan, D. Pachamanova, and M. Sim. 2008. Incorporating asymmetric distributional information in robust value-at-risk optimization. Management Science 54, 3, 573--585. Google ScholarDigital Library
- A. Nemirovski and A. Shapiro. 2005. Scenario approximation of chance constraints. Probabilistic and Randomized Methods for Design Under Uncertainty, G. Calafiore and F. Dabbene (Eds.). Springer, London, 3--48.Google Scholar
- A. Nemirovski and A. Shapiro. 2006. Convex approximations of chance constrained programs. SIAM Journal on Optimization 17, 969--996. Google ScholarDigital Library
- Y. Nesterov. 2005. Smooth minimization of nonsmooth functions. Mathematical Programming 103, 127--152. Google ScholarDigital Library
- W. Ogryczak and T. Śliwiński. 2011. On solving the dual for portfolio selection by optimizing conditional value at risk. Computational Optimization and Applications 50, 591--595.Google ScholarCross Ref
- B. K. Pagnoncelli, S. Ahmed, and A. Shapiro. 2009. Sample average approximation method for chance constrained programming: Theory and applications. Journal of Optimization Theory and Applications 142, 399--416.Google ScholarDigital Library
- G. Pflug. 2000. Some remarks on the value-at-risk and conditional value-at-risk. In Probabilistic Constrained Optimization: Methodology and Applications, S. Uryasev (Ed.). Kluwer, Dordrecht, The Netherlands, 272--281.Google Scholar
- A. Prékopa. 1970. On probabilistic constrained programming. Proceedings of the Princeton Symposium on Mathematical Programming, 113--138.Google Scholar
- R. T. Rockafellar and J. O. Royset. 2010. On buffered failure probability in design and optimization of structures. Reliability Engineering & System Safety 95, 499--510.Google ScholarCross Ref
- R. T. Rockafellar and S. Uryasev. 2000. Optimization of conditional value-at-risk. Journal of Risk, 2, 21--41.Google ScholarCross Ref
- R. T. Rockafellar and S. Uryasev. 2002. Conditional value-at-risk for general loss distributions. Journal of Banking and Finance 26, 1443--1471.Google ScholarCross Ref
- O. Scaillet. 2004. Nonparametric estimation and sensitivity analysis of expected shortfall. Mathematical Finance, 14, 115--129.Google ScholarCross Ref
- R. J. Serfling. 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York.Google Scholar
- A. Shapiro. 1993. Asymptotic behavior of optimal solutions in stochastic programming. Mathematics of Operations Research 18, 4, 829--845. Google ScholarDigital Library
- A. Shapiro, D. Dentcheva, and A. Ruszczyński. 2009. Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia.Google ScholarDigital Library
- L. Sun and L. J. Hong. 2010. Asymptotic representations for importance-sampling estimators of value-at-risk and conditional value-at-risk. Operations Research Letters 38, 246--251. Google ScholarDigital Library
- D. Tasche. 2009. Capital allocation for credit portfolios with kernel estimators. Quantitative Finance 9, 581--595.Google ScholarCross Ref
- A. A. Trindade, S. Uryasev, A. Shapiro, and G. Zrazhevsky. 2007. Financial prediction with constrained tail risk. Journal of Banking and Finance 31, 3524--3538.Google ScholarCross Ref
- J. N. Tsitsiklis and B. van Roy. 2001. Regression method for pricing complex American-style options. IEEE Transactions on Neural Networks, 12, 694--703. Google ScholarDigital Library
- H. Xu and D. Zhang. 2009. Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications. Mathematical Programming 119, 371--401. Google ScholarDigital Library
- S. Zymler, D. Kuhn, and B. Rustem. 2013. Worst-case value-at-risk of nonlinear portfolios. Management Science, 59(1), 172--188. Google ScholarDigital Library
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- Monte Carlo Methods for Value-at-Risk and Conditional Value-at-Risk: A Review
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