Abstract
Portfolio risk estimation in volatile markets requires employing fat-tailed models for financial returns combined with copula functions to capture asymmetries in dependence and an appropriate downside risk measure. In this survey, we discuss how these three essential components can be combined together in a Monte Carlo based framework for risk estimation and risk capital allocation with the average value-at-risk measure (AVaR). AVaR is the average loss provided that the loss is larger than a predefined value-at-risk level. We consider in some detail the AVaR calculation and estimation and investigate the stochastic stability.
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References
Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1998). Coherent measures of risk. Mathematical Finances, 6, 203–228.
Bibby, B. M., & Sorensen, M. (2003). Hyperbolic processes in finance. In S. Rachev (Ed.), Handbook of heavy-tailed distributions in finance (pp. 212–248). Amsterdam: Elsevier.
Bradley, B., & Taqqu, M. S. (2003). Financial risk and heavy tails. In S. Rachev (Ed.), Handbook of heavy-tailed distributions in finance (pp. 35–103). Amsterdam: Elsevier.
Dokov, S., Stoyanov, S., & Rachev, S. (2008). Computing VaR and AVaR of skewed t distribution. Journal of Applied Functional Analysis, 3, 189–209.
Embrechts, P., McNeil, A., & Straumann, D. (2002). Correlation and dependence in risk management: properties and pitfalls. In M. Dempster (Ed.), Risk management: Value at risk and beyond (vol. 10(3), pp. 341–352). Cambridge: Cambridge University Press.
Embrechts, P., Lindskog, F., & McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (Ed.), Handbook of heavy-tailed distributions in finance (pp. 329–384). Amsterdam: Elsevier.
Fama, E. (1963). Mandelbrot and the stable Paretian hypothesis. Journal of Business, 36, 420–429.
Fama, E. (1965). The behavior of stock market prices. Journal of Business, 38, 34–105.
Hurst, S. H., Platen, E., & Rachev, S. (1997). Subordinated market index models. A comparison. Financial Engineering and the Japanese Markets, 4, 97–124.
Janicki, A., & Weron, A. (1994). Simulation and chaotic behavior of alpha-stable stochastic processes. New York: Marcel Dekker.
Kim, Y., Rachev, S., Bianchi, M., & Fabozzi, F. (2008). Financial market models with Lévy processes and time-varying volatility. Journal of Banking and Finance, 32(7), 1363–1378.
Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 26, 394–419.
Mittnik, S., & Rachev, S. (1999). Stable non-Gaussian models in finance and econometrics. Mathematical and Computer Modeling, 10–12.
Pflug, G. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In S. Uryasev (Ed.), Probabilistic constrained optimization: Methodology and applications. Dordrecht: Kluwer Academic.
Platen, E., & Rendek, R. (2007), Empirical evidence on Student-t log-returns of diversified world stock indices. Research paper series 194, Quantitative Finance Research Centre, University of Technology, Sydney.
Rachev, S. (Ed.) (2003). Handbook of heavy-tailed distributions in finance. Amsterdam: Elsevier.
Rachev, S., & Mittnik, S. (2000). Stable Paretian models in finance. Series in financial economics. Chichester: Wiley.
Rachev, S., Fabozzi, F., & Menn, C. (2005). Fat tails and skewed asset returns distributions. New York: Wiley Finance.
Rachev, S., Stoyanov, S., Biglova, A., & Fabozzi, F. (2006). An empirical examination of daily stock return distributions for U.S. stocks. In Data analysis and decision making. Berlin: Springer.
Rachev, S., Martin, D., Racheva-Iotova, B., & Stoyanov, S. (2007). Stable ETL optimal portfolios and extreme risk management. In G. Bol, S. Rachev, & R. Wuerth (Eds.), Risk assessment: Decisions in banking and finance. Berlin: Springer.
Rachev, S., Stoyanov, S., & Fabozzi, F. (2008). Advanced stochastic models, risk assessment, and portfolio optimization: The ideal risk, uncertainty, and performance measures. New York: Wiley Finance.
Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26(7), 1443–1471.
Samorodnitsky, G., & Taqqu, M. S. (1994). Stable non-Gaussian random processes. New York/London: Chapman & Hall.
Sklar, A. (1996). Random variables, distribution functions and copulas—a personal look backward and forward. In L. Rüschendorf, B. Schweizer, & M. D. Taylor (Eds.), Distributions with fixed marginals and related topics (pp. 1–14). Hayward: Institute of Mathematical Statistics.
Stoyanov, S., & Rachev, S. (2008a). Asymptotic distribution of the sample average value-at-risk. Journal of Computational Analysis and Applications, 10, 465–483.
Stoyanov, S., & Rachev, S. (2008b). Asymptotic distribution of the sample average value-at-risk in the case of heavy-tailed returns. Journal of Applied Functional Analysis, 3, 443–461.
Stoyanov, S., Samorodnitsky, G., Rachev, S., & Ortobelli, S. (2006). Computing the portfolio conditional value-at-risk in the α-stable case. Probability and Mathematical Statistics, 26, 1–22.
Sun, W., Rachev, S., Stoyanov, S., & Fabozzi, F. (2008). Multivariate skewed student’s t copula in the analysis of nonlinear and asymmetric dependence in the German equity market. Studies in Nonlinear Dynamics and Econometrics, 12(2), 3.
Zhang, Y., & Rachev, S. (2006). Risk attribution and portfolio performance measurement. Journal of Applied Functional Analysis, 4(1), 373–402.
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Stoyanov, S.V., Racheva-Iotova, B., Rachev, S.T. et al. Stochastic models for risk estimation in volatile markets: a survey. Ann Oper Res 176, 293–309 (2010). https://doi.org/10.1007/s10479-008-0468-1
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DOI: https://doi.org/10.1007/s10479-008-0468-1