Valuing variable annuity guarantees with the multivariate Esscher transform. (English) Zbl 1228.91044

Summary: Variable annuities are usually sold with a range of guarantees that protect annuity holders from some downside market risk. Although it is common to see variable annuity guarantees written on multiple funds, existing pricing methods are, by and large, based on stochastic processes for one single asset only. In this article, we fill this gap by developing a multivariate valuation framework. First, we consider a multivariate regime-switching model for modeling returns on various assets at the same time. We then identify a risk-neutral probability measure for use with the model under consideration. This is accomplished by a multivariate extension of the regime-switching conditional Esscher transform. We further extend our results to the situation when the guarantee being valued is linked to equity indexes measured in foreign currencies. In particular, we derive a probability measure that is risk-neutral from the perspective of domestic investors. Finally, we illustrate our results with a hypothetical variable annuity guarantee.


91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
Full Text: DOI


[1] Bacinello, A.R.; Millossovich, P.; Olivieri, A.; Pitacco, E., Variable annuities: a unifying valuation approach, Insurance: mathematics and economics, (2011)
[2] Bakshi, G.; Cao, C.; Chen, Z., Pricing and hedging long-term options, Journal of econometrics, 94, 227-318, (1999) · Zbl 0989.91041
[3] Bühlmann, H.; Delbaen, F.; Embrechts, P.; Shiryaev, A.N., No-arbitrage, change of measure and conditional esscher transforms, CWI quarterly, 9, 291-317, (1996) · Zbl 0943.91037
[4] Carmona, R.; Durrleman, V., Generalizing the black – scholes formula to multivariate contingent claims, Journal of computational finance, 9, 43-67, (2006)
[5] Elliott, R.J.; Aggoun, L.; Moore, J.B., Hidden Markov models: estimation and control, (1994), Springer Berlin, Heidelbery, New York
[6] Elliott, R.J.; Chan, L.; Siu, T.K., Option pricing and esscher transform under regime switching, Annals of finance, 1, 423-432, (2005) · Zbl 1233.91270
[7] Gerber, H.U.; Shiu, E.S.W., Option pricing by esscher transforms, Transactions of the society of actuaries, 46, 99-191, (1994)
[8] Hardy, M.R., Investment guarantees: modeling and risk management for equity-linked life insurance, (2003), John Wiley & Sons Hoboken, New Jersey · Zbl 1092.91042
[9] Hardy, M.R., A regime switching model of long term stock returns, North American actuarial journal, 3, 185-211, (2001)
[10] Milevsky, M.A.; Salisbury, T.S., Financial valuation of guaranteed minimum withdrawal benefits, Insurance: mathematics and economics, 38, 21-38, (2006) · Zbl 1116.91048
[11] Miyahara, Y., Geometric levy process and MEMM: pricing model and related estimation problems, Asia-Pacific financial markets, 8, 45-61, (2001) · Zbl 1070.91012
[12] Siu, T.K., Fair valuation of participating policies with surrender options and regime switching, Insurance: mathematics and economics, 37, 533-552, (2005) · Zbl 1129.60062
[13] Siu, T.K.; Lau, J.W.; Yang, H., Pricing participating products under a generalized jump-diffusion, Journal of applied mathematics and stochastic analysis, 2008, (2008) · Zbl 1141.91386
[14] Spiegel, C., 2010. Overview of the North American market landscape. In: The Society of Actuaries Equity-Based Insurance Guarantees Conference, Tokyo, Japan.
[15] Wyman, Oliver, 2007. VA VA voom: variable annuities are in pole position to meet the requirements of the European asset protection market. Available at http://www.mmc.com/knowledgecenter/OliverWymanVariableAnnuities.pdf.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.