Efficient Greek calculation of variable annuity portfolios for dynamic hedging: a two-level metamodeling approach. (English) Zbl 1414.91188

Summary: The financial risk associated with the guarantees embedded in variable annuities cannot be addressed adequately by traditional actuarial techniques. Dynamical hedging is used in practice to mitigate the financial risk arising from variable annuities. However, a major challenge of dynamical hedging is to calculate the dollar Deltas of a portfolio of variable annuities within a short time interval so that rebalancing can be done on a timely basis. In this article, we propose a two-level metamodeling approach to efficiently estimating the partial dollar Deltas of a portfolio of variable annuities under a multiasset framework. The first-level metamodel is used to estimate the partial dollar Deltas at some well-chosen market levels, and the second-level metamodel is used to estimate the partial dollar Deltas at the current market level based on the precalculated partial dollar Deltas. Our numerical results show that the proposed approach performs well in terms of accuracy and speed.


91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI


[1] Alam, F. M.; McNaught, K. R.; Ringrose, T. J., A comparison of experimental designs in the development of a neural network simulation metamodel, Simulation Modelling Practice and Theory, 12, 7, 559-578, (2004)
[2] Ankenman, B.; Nelson, B. L.; Staum, J., Stochastic kriging for simulation metamodeling, Operations Research, 58, 2, 371-382, (2010) · Zbl 1342.62134
[3] Bacinello, A. R.; Millossovich, P.; Montealegre, A., The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours, Scandinavian Actuarial Journal, 2014, 1-20, (2014)
[4] Barton, R., Metamodeling: a state of the art review, Winter Simulation Conference Proceedings, 237-244, (1994)
[5] Bauer, D.; Ha, H., A least-squares Monte Carlo approach to the calculation of capital requirements, (2013)
[6] Bauer, D.; Kling, A.; Russ, J., A universal pricing framework for guaranteed minimum benefits in variable annuities, ASTIN Bulletin, 38, 2, 621-651, (2008) · Zbl 1274.91399
[7] Bélanger, A.; Forsyth, P.; Labahn, G., Valuing the guaranteed minimum death benefit clause with partial withdrawals, Applied Matehmatical Finance, 16, 6, 451-496, (2009) · Zbl 1189.91066
[8] Box, G. E. P.; Draper, N. R., Response Surfaces, Mixtures, and Ridge Analyses, (2007), Hoboken, NJ: Wiley, Hoboken, NJ · Zbl 1267.62006
[9] Caballero, W.; Giraldo, R.; Mateu, J., A universal kriging approach for spatial functional data, Stochastic Environmental Research and Risk Assessment, 27, 7, 1553-1563, (2013)
[10] Carnell, R., lhs: Latin Hypercube Samples, (2012), Vienna: R Foundation, Vienna
[11] Cressie, N., Statistics for Spatial Data, (1993), ed. Hoboken, NJ: Wiley, ed. Hoboken, NJ
[12] Das, R. N., Robust Response Surfaces, Regression, and Positive Data Analyses, (2014), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 1306.62009
[13] Friedman, L. W., The Simulation Metamodel, (1996), Norwell, MA: Kluwer, Norwell, MA
[14] Gan, G., Data Clustering in C++: An Object-Oriented Approach, (2011), Boca Raton, FL: Chapman & Hall/CRC Press, Boca Raton, FL · Zbl 1221.68002
[15] Gan, G., Application of data clustering and machine learning in variable annuity valuation, Insurance: Mathematics and Economics, 53, 3, 795-801, (2013) · Zbl 1290.91086
[16] Gan, G., A multi-asset Monte Carlo simulation model for the valuation of variable annuities, Proceedings of the Winter Simulation Conference, 3162-3163, (2015)
[17] Gan, G.; Lin, X. S., Valuation of large variable annuity portfolios under nested simulation: A functional data approach, Insurance: Mathematics and Economics, 62, 1, 138-150, (2015) · Zbl 1318.91112
[18] Gerber, H.; Shiu, E., Pricing lookback options and dynamic guarantees, North American Actuarial Journal, 7, 1, 48-67, (2003) · Zbl 1084.91507
[19] Huang, H.; Milevsky, M.; Salisbury, T., Optimal initiation of a GLWB in a variable annuity: No arbitrage approach, Insurance: Mathematics and Economics, 56, 102-111, (2014) · Zbl 1304.91113
[20] Isaaks, E.; Srivastava, R., An Introduction to Applied Geostatistics, (1990), Oxford: Oxford University Press, Oxford
[21] Kleijnen, J. P., Kriging metamodeling in simulation: A review, European Journal of Operational Research, 192, 3, 707-716, (2009) · Zbl 1157.90544
[22] Kleijnen, J. P. C., A comment on Blanning’s “metamodel for sensitivity analysis: The regression metamodel in simulation, Interfaces, 5, 3, 21-23, (1975)
[23] Kou, S. G.; Wang, H., Option pricing under a double exponential jump diffusion model, Management Science, 50, 9, 1178-1192, (2004)
[24] Liefvendahl, M.; Stocki, R., A study on algorithms for optimization of Latin hypercubes, Journal of Statistical Planning and Inference, 136, 9, 3231-3247, (2006) · Zbl 1094.62084
[25] Lin, X. S.; Tan, K. S.; Yang, H. L., Pricing annuity guarantees under a regime-switching model, North American Actuarial Journal, 13, 316-338, (2009)
[26] Loeppky, J. L.; Sacks, J.; Welch, W. J., Choosing the sample size of a computer experiment: A practical guide, Technometrics, 51, 4, 366-376, (2009)
[27] Merton, R. C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 1, 125-144, (1976) · Zbl 1131.91344
[28] Milevsky, M.; Posner, S., The titanic option: Valuation of the guaranteed minimum death benefit in variable annuities and mutual funds, Journal of Risk and Insurance, 68, 1, 93-128, (2001)
[29] Minasny, B.; McBratney, A. B., A conditioned latin hypercube method for sampling in the presence of ancillary information, Computers & Geosciences, 32, 9, 1378-1388, (2006)
[30] Moon, H.; Dean, A.; Santner, T., Algorithms for generating maximin latin hypercube and orthogonal designs, Journal of Statistical Theory and Practice, 5, 1, 81-98, (2011) · Zbl 05902637
[31] Ng, A. C.-Y.; Li, J. S.-H., Pricing and hedging variable annuity guarantees with multiasset stochastic investment models, North American Actuarial Journal, 17, 1, 41-62, (2013)
[32] Reynolds, C.; Man, S., Nested stochastic pricing: The time has come, Product Matters! – Society of Actuaries, 71, 16-20, (2008)
[33] Roudier, P., clhs: a R Package for Conditioned Latin Hypercube Sampling, (2011), Vienna: R Foundation, Vienna
[34] Viana, F., Things you wanted to know about the Latin hypercube design and were afraid to ask, 10th World Congress on Structural and Multidisciplinary Optimization, (2013), Orlando, FL
[35] Zhu, H.; Liu, L.; Long, T.; Peng, L., A novel algorithm of maximin latin hypercube design using successive local enumeration, Engineering Optimization, 44, 5, 551-564, (2012)
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.