Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T07:52:27.492Z Has data issue: false hasContentIssue false
  • English
  • Français

CALCULATING VARIABLE ANNUITY LIABILITY “GREEKS” USING MONTE CARLO SIMULATION

Published online by Cambridge University Press:  05 January 2015

Mark J. Cathcart ,
Hsiao Yen Lok ,
Alexander J. McNeil  and
Steven Morrison
Mark J. Cathcart
Affiliation:
Standard Life Group, Edinburgh EH1 2DH, UK E-mail: [email protected]
Hsiao Yen Lok
Affiliation:
Heriot-Watt University, Edinburgh, Maxwell Institute for Mathematical Sciences, Edinburgh EH14 4AS, UK E-mail: [email protected]
Alexander J. McNeil*
Affiliation:
Heriot-Watt University, Edinburgh, Maxwell Institute for Mathematical SciencesEdinburgh EH14 4AS, UK
Steven Morrison
Affiliation:
Moody's Analytics, Edinburgh EH3 8RD, UK E-mail: [email protected]
*
E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The implementation of hedging strategies for variable annuity products requires the calculation of market risk sensitivities (or “Greeks”). The complex, path-dependent nature of these products means that these sensitivities are typically estimated by Monte Carlo methods. Standard market practice is to use a “bump and revalue” method in which sensitivities are approximated by finite differences. As well as requiring multiple valuations of the product, this approach is often unreliable for higher-order Greeks, such as gamma, and alternative pathwise (PW) and likelihood-ratio estimators should be preferred. This paper considers a stylized guaranteed minimum withdrawal benefit product in which the reference equity index follows a Heston stochastic volatility model in a stochastic interest rate environment. The complete set of first-order sensitivities with respect to index value, volatility and interest rate and the most important second-order sensitivities are calculated using PW, likelihood-ratio and mixed methods. It is observed that the PW method delivers the best estimates of first-order sensitivities while mixed estimation methods deliver considerably more accurate estimates of second-order sensitivities; moreover there are significant computational gains involved in using PW and mixed estimators rather than simple BnR estimators when many Greeks have to be calculated.

Keywords

Stochastic simulationMonte Carlo estimationvariable annuityGreekssensitivitiesHeston stochastic volatility modelpathwise methodlikelihood-ratio methodstochastic interest rates
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 2 , May 2015 , pp. 239 - 266
Copyright
Copyright © ASTIN Bulletin 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersen, L. (2008) Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance, 11, 1–42.CrossRefGoogle Scholar
Bacinello, A.R., Millossovich, P., Olivieri, A. and Potacco, E. (2011) Variable annuities: A unifying valuation approach. Insurance: Mathematics and Economics, 49, 285–297.Google Scholar
Bauer, D.K., Kling, A. and Russ, J. (2008) A universal pricing framework for guaranteed minimum benefits in variable annuities. Astin Bulletin, 38, 621–650.CrossRefGoogle Scholar
Broadie, M. and Glasserman, P. (1996) Estimating security price derivatives using simulation. Journal of Economic Dynamics and Control, 21, 1323–1352.CrossRefGoogle Scholar
Broadie, M. and Kaya, O. (2004) Exact simulation of option Greeks under stochastic volatility and jump diffusion models. In Proceedings of the 2004 Winter Simulation Conference, Washington DC.Google Scholar
Chan, J.H. and Joshi, M.S. (2013) Fast and accurate long-stepping simulation of the Heston stochastic volatility model. Journal of Computational Finance, 16, 47–97.CrossRefGoogle Scholar
Chen, Z., Vetzal, K. and Forsyth, P. (2008) The effect of modelling parameters on the value of GMWB guarantees. Insurance: Mathematics and Economics, 43, 165–173.Google Scholar
Dai, M., Kwok, Y.K. and Zong, J. (2008) Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance, 18, 595–611.CrossRefGoogle Scholar
Glasserman, P. (2003) Monte Carlo} Methods in Financial Engineering. New York: Springer.CrossRefGoogle Scholar
Heston, S.L. (1993) A closed form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327–343.CrossRefGoogle Scholar
Hobbs, C., Krishnaraj, B., Liu, Y. and Musselman, J. (2009) Calculation of variable annuity market sensitivities using a pathwise methodology. Life and Pensions, September, 40–44.Google Scholar
Jaimungal, S., Donnelly, R. and Rubisov, D.H. (2014) Valuing GWBs with stochastic interest rates and volatility. Quantitative Finance, 14, 369–382.Google Scholar
Kling, A., Ruez, F. and Russ, J. (2011) The impact of stochastic volatility on pricing, hedging, and hedge efficiency of withdrawal benefit guarantees in variable annuities. Astin Bulletin, 41, 511–545.Google Scholar
Ledlie, M.C., Corry, D.P., Finkelstein, G. S., Ritchie, K.S. and Wilson, D.C.E. (2008) Variable annuities. British Actuarial Journal, 14, 327–389.CrossRefGoogle Scholar
Milevsky, M.A. and Salisbury, T.S. (2006) Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics, 38, 21–38.Google Scholar
Peng, J., Leung, K.S. and Kwok, Y.K. (2012) Pricing guaranteed minimum withdrawal benefits under stochastic interest rate. Quantitative Finance, 12, 933–941.CrossRefGoogle Scholar