Bernoulli

  • Bernoulli
  • Volume 23, Number 3 (2017), 1481-1517.

Saddlepoint methods for conditional expectations with applications to risk management

Sojung Kim and Kyoung-Kuk Kim

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The paper derives saddlepoint expansions for conditional expectations in the form of $\mathsf{E}[\overline{X}|\overline{\mathbf{Y}}=\mathbf{a}]$ and $\mathsf{E}[\overline{X}|\overline{\mathbf{Y}}\geq\mathbf{a}]$ for the sample mean of a continuous random vector $(X,\mathbf{Y}^{\top})$ whose joint moment generating function is available. Theses conditional expectations frequently appear in various applications, particularly in quantitative finance and risk management. Using the newly developed saddlepoint expansions, we propose fast and accurate methods to compute the sensitivities of risk measures such as value-at-risk and conditional value-at-risk, and the sensitivities of financial options with respect to a market parameter. Numerical studies are provided for the accuracy verification of the new approximations.

Article information

Source
Bernoulli Volume 23, Number 3 (2017), 1481-1517.

Dates
Received: July 2015
Revised: September 2015
First available in Project Euclid: 17 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1489737615

Digital Object Identifier
doi:10.3150/15-BEJ774

Zentralblatt MATH identifier
06714309

Keywords
conditional expectation risk management saddlepoint approximation sensitivity estimation

Citation

Kim, Sojung; Kim, Kyoung-Kuk. Saddlepoint methods for conditional expectations with applications to risk management. Bernoulli 23 (2017), no. 3, 1481--1517. doi:10.3150/15-BEJ774. https://projecteuclid.org/euclid.bj/1489737615


Export citation

References

  • [1] Barndorff-Nielsen, O. and Cox, D.R. (1979). Edgeworth and saddle-point approximations with statistical applications. J. R. Stat. Soc. Ser. B. Stat. Methodol. 41 279–312.
  • [2] Booth, J., Hall, P. and Wood, A. (1992). Bootstrap estimation of conditional distributions. Ann. Statist. 20 1594–1610.
  • [3] Broda, S.A. and Paolella, M.S. (2010). Saddlepoint approximation of expected shortfall for transformed means. Preprint. Available at http://hdl.handle.net/11245/1.327329.
  • [4] Butler, R.W. (2007). Saddlepoint Approximations with Applications. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge Univ. Press.
  • [5] Butler, R.W. and Bronson, D.A. (2002). Bootstrapping survival times in stochastic systems by using saddlepoint approximations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 64 31–49.
  • [6] Butler, R.W. and Wood, A.T.A. (2004). Saddlepoint approximation for moment generating functions of truncated random variables. Ann. Statist. 32 2712–2730.
  • [7] Carr, P. and Madan, D. (2009). Saddlepoint methods for option pricing. J. Comput. Finance 13 49–61.
  • [8] Daniels, H.E. (1954). Saddlepoint approximations in statistics. Ann. Math. Stat. 25 631–650.
  • [9] Daniels, H.E. (1987). Tail probability approximations. Int. Stat. Rev. 55 37–48.
  • [10] Daniels, H.E. and Young, G.A. (1991). Saddlepoint approximation for the Studentized mean, with an application to the bootstrap. Biometrika 78 169–179.
  • [11] Feuerverger, A. and Wong, A.C.M. (2000). Computation of value-at-risk for nonlinear portfolios. Journal of Risk. 3 37–55.
  • [12] Glasserman, P. and Kim, K.-K. (2009). Saddlepoint approximations for affine jump-diffusion models. J. Econom. Dynam. Control 33 15–36.
  • [13] Gordy, M.B. (2002). Saddlepoint approximation of CreditRisk$^{+}$. J. Bank. Financ. 26 1335–1353.
  • [14] Hong, L.J. (2009). Estimating quantile sensitivities. Oper. Res. 57 118–130.
  • [15] Hong, L.J. and Liu, G. (2009). Simulating sensitivities of conditional value-at-risk. Manage. Sci. 55 281–293.
  • [16] Huang, X. and Oosterlee, C.W. (2011). Saddlepoint approximations for expectations and an application to CDO pricing. SIAM J. Financial Math. 2 692–714.
  • [17] Jensen, J.L. (1995). Saddlepoint Approximations. Oxford Statistical Science Series 16. New York: Oxford Univ. Press.
  • [18] Kolassa, J. and Li, J. (2010). Multivariate saddlepoint approximations in tail probability and conditional inference. Bernoulli 16 1191–1207.
  • [19] Kolassa, J.E. (2006). Series Approximation Methods in Statistics, 3rd ed. New York: Springer.
  • [20] Kolassa, J.E. (1996). Higher-order approximations to conditional distribution functions. Ann. Statist. 24 353–364.
  • [21] Kolassa, J.E. (2003). Multivariate saddlepoint tail probability approximations. Ann. Statist. 31 274–286.
  • [22] Li, J. (2008). Multivariate Saddlepoint Tail probability Approximations, for Conditional and Unconditional Distributions, based on the Signed Root of the Log Likelihood Ratio Statistic. Ph.D. dissertation, Rutgers Univ., Dept. Statistics and Biostatistics.
  • [23] Liu, G. and Hong, L.J. (2011). Kernel estimation of the Greeks for options with discontinuous payoffs. Oper. Res. 59 96–108.
  • [24] Lugannani, R. and Rice, S. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables. Adv. in Appl. Probab. 12 475–490.
  • [25] Martin, R. (2006). The saddlepoint method and portfolio optionalities. Risk 19 93–95.
  • [26] Martin, R., Thompson, K. and Browne, C. (2001). VAR: Who contributes and how much? Risk 14 99–102.
  • [27] McCullagh, P. (1987). Tensor Methods in Statistics. Monographs on Statistics and Applied Probability. London: Chapman & Hall.
  • [28] Muromachi, Y. (2004). A conditional independence approach for portfolio risk evaluation. Journal of Risk 7 27–53.
  • [29] Reid, N. (1988). Saddlepoint methods and statistical inference. Statist. Sci. 3 213–238.
  • [30] Reid, N. (2003). Asymptotics and the theory of inference. Ann. Statist. 31 1695–1731.
  • [31] Rogers, L.C.G. and Zane, O. (1999). Saddlepoint approximations to option prices. Ann. Appl. Probab. 9 493–503.
  • [32] Tasche, D. (2008). Capital allocation to business units and sub-portfolios: The Euler principle. In Pillar II in the New Basel Accord: The Challenge of Economic Capital 423–453. London: Risk books.
  • [33] Temme, N.M. (1982). The uniform asymptotic expansion of a class of integrals related to cumulative distribution functions. SIAM J. Math. Anal. 13 239–253.
  • [34] Tierney, L. and Kadane, J.B. (1986). Accurate approximations for posterior moments and marginal densities. J. Amer. Statist. Assoc. 81 82–86.
  • [35] Wang, S. (1990). Saddlepoint approximations for bivariate distributions. J. Appl. Probab. 27 586–597.
  • [36] Watson, G.N. (1948). Theory of Bessel Functions. Cambridge: Cambridge Univ. Press.