Markov Chain Monte Carlo for Risk Measures

Yuya Suzuki, Thorbjörn Gudmundsson

2014

Abstract

In this paper, we consider random sums with heavy-tailed increments. By the term random sum, we mean a sum of random variables where the number of summands is also random. Our interest is to construct an efficient method to calculate tail-based risk measures such as quantiles and conditional expectation (expected shortfalls). When assuming extreme quantiles and heavy-tailed increments, using standard Monte Carlo method can be inefficient. In previous works, there exists an efficient method to sample rare-events (tail-events) using a Markov chain Monte Carlo (MCMC) with a given threshold. We apply the sampling method to estimate statistics based on tail-information, with a given rare-event probability. The performance is compared with other methods by some numerical results in the case increments follow Pareto distributions. We also show numerical results with Weibull, and Log-Normal distributions. Our proposed method is shown to be efficient especially in cases of extreme tails.

References

  1. Blanchet, J. H. and Liu, J. (2008). State-dependent importance sampling for regularly varying random walks. Advances in Applied Probability, pages 1104-1128.
  2. Dupuis, P., Leder, K., and Wang, H. (2007). Importance sampling for sums of random variables with regularly varying tails. ACM T. Model Comput. S., 17(3).
  3. Embrechts, P., Furrer, H., and Kaufmann, R. (2003). Quantifying regulatory capital for operational risk. Derivatives Use, Trading & Regulation, 9(3):217-233.
  4. Fischer, M. J., Bevilacqua Masi, D. M., Gross, D., Shortle, J., and Brill, P. H. (2001). Using quantile estimates in simulating internet queues with pareto service times. In Proceedings of the 33nd conference on Winter simulation, pages 477-485. IEEE Computer Society.
  5. Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical science, pages 457-472.
  6. Gross, D., Shortle, J. F., Fischer, M. J., and Masi, D. M. B. (2002). Difficulties in simulating queues with pareto service. In Simulation Conference, 2002. Proceedings of the Winter, volume 1, pages 407-415. IEEE.
  7. Gudmundsson, T. and Hult, H. (2012). Markov chain monte carlo for computing rare-event probabilities for a heavy-tailed random walk. arXiv preprint.
  8. Hastings, W. K. (1970). Monte carlo sampling methods using markov chains and their applications. Biometrika, 57(1):97-109.
  9. Hult, H. and Svensson, J. (2009). Efficient calculation of risk measures by importance sampling-the heavy tailed case. Preprint, KTH.
  10. McNeil, A. J., Frey, R., and Embrechts, P. (2010). Quantitative risk management: concepts, techniques, and tools. Princeton university press.
  11. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). Equation of state calculations by fast computing machines. The journal of chemical physics, 21:1087.
  12. Rubino, G., Tuffin, B., et al. (2009). Rare event simulation using Monte Carlo methods. Wiley Online Library.
  13. Sees Jr, J. C. and Shortle, J. F. (2002). Simulating m/g/1 queues with heavy-tailed service. In Simulation Conference, 2002. Proceedings of the Winter, volume 1, pages 433-438. IEEE.
Download


Paper Citation


in Harvard Style

Suzuki Y. and Gudmundsson T. (2014). Markov Chain Monte Carlo for Risk Measures . In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH, ISBN 978-989-758-038-3, pages 330-338. DOI: 10.5220/0005035303300338


in Bibtex Style

@conference{simultech14,
author={Yuya Suzuki and Thorbjörn Gudmundsson},
title={Markov Chain Monte Carlo for Risk Measures},
booktitle={Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},
year={2014},
pages={330-338},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005035303300338},
isbn={978-989-758-038-3},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,
TI - Markov Chain Monte Carlo for Risk Measures
SN - 978-989-758-038-3
AU - Suzuki Y.
AU - Gudmundsson T.
PY - 2014
SP - 330
EP - 338
DO - 10.5220/0005035303300338