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Pricing of equity indexed annuity under fractional Brownian motion model. (English) Zbl 07022267
Summary: Fractional Brownian motion with Hurst exponent $$H \in(1 / 2, 1)$$ is a good candidate for modeling financial time series with long-range dependence and self-similarity. The main purpose of this paper is to address the valuation of equity indexed annuity (EIA) designs under the market driven by fractional Brownian motion. As a result, this paper presents an explicit pricing expression for point-to-point EIA design and bounds for the pricing of high-water-marked EIA design. Some numerical examples are given to illustrate the impact of the parameters involved in the pricing problems.
##### MSC:
 62 Statistics 91 Game theory, economics, finance, and other social and behavioral sciences
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##### References:
 [1] Marrion, J., 4th quarter index annuity sales [2] Tiong, S., Valuing equity-indexed annuities, North American Actuarial Journal, 4, 4, 149-163, (2000) · Zbl 1083.62545 [3] Boyle, P.; Tian, W., The design of equity-indexed annuities, Insurance: Mathematics & Economics, 43, 3, 303-315, (2008) · Zbl 1152.91484 [4] Gerber, H. U.; Shiu, E. S. W., Pricing lookback options and dynamic guarantees, North American Actuarial Journal, 7, 1, 48-67, (2003) · Zbl 1084.91507 [5] Hardy, M., Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance, (2003), Ontario, Canada: John Wiley & Sons, Ontario, Canada · Zbl 1092.91042 [6] Jaimungal, S., Pricing and hedging equity indexed annuities with variance gamma deviates [7] Kijima, M.; Wong, T., Pricing of ratchet equity-indexed annuities under stochastic interest rates, Insurance: Mathematics & Economics, 41, 3, 317-338, (2007) · Zbl 1141.91457 [8] Lee, H., Pricing equity-indexed annuities with path-dependent options, Insurance: Mathematics & Economics, 33, 3, 677-690, (2003) · Zbl 1103.91368 [9] Lin, X. S.; Tan, K. S., Valuation of equity-indexed annuities under stochastic interest rates, North American Actuarial Journal, 7, 4, 72-91, (2003) · Zbl 1084.60530 [10] Moore, K. S., Optimal surrender strategies for equity-indexed annuity investors, Insurance: Mathematics & Economics, 44, 1, 1-18, (2009) · Zbl 1156.91379 [11] Biffis, E., Affine processes for dynamic mortality and actuarial valuations, Insurance: Mathematics & Economics, 37, 3, 443-468, (2005) · Zbl 1129.91024 [12] Biffis, E.; Denuit, M.; Devolder, P., Stochastic mortality under measure changes · Zbl 1226.91022 [13] Hainaut, D.; Devolder, P., Mortality modelling with Lévy processes, Insurance: Mathematics & Economics, 42, 1, 409-418, (2008) · Zbl 1141.91516 [14] Jalen, L.; Mamon, R., Valuation of contingent claims with mortality and interest rate risks, Mathematical and Computer Modelling, 49, 9-10, 1893-1904, (2009) · Zbl 1171.91349 [15] Qian, L.; Wang, W.; Wang, R.; Tang, Y., Valuation of equity-indexed annuity under stochastic mortality and interest rate, Insurance: Mathematics & Economics, 47, 2, 123-129, (2010) · Zbl 1231.91446 [16] Aít-Sahalia, Y., Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 3, 527-560, (1996) · Zbl 0844.62094 [17] Andrew, L., Long-term memory in stock price, Econometrica, 59, 1279-1313, (1991) · Zbl 0781.90023 [18] Granger, C. W. J., A typical spectral shape of an economic variable, Econometrica, 34, 150-161, (1966) [19] Sottinen, T.; Valkeila, E., Fractional Brownain motion as a model in finance, Report [20] Willinger, W.; Taqqu, M.; Teverovsky, V., Stock market prices and long-range dependence, Finance and Stochastics, 3, 1-13, (1999) · Zbl 0924.90029 [21] Dudley, R.; Norvaisa, R., An Introduction to P-Variation and Young Integrals with Emphasis on Sample Functions of Stochastic Processes, (1998), MaPhySto, Department of Mathematical Sciences, University of Aarhus · Zbl 0937.28001 [22] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion. I. Theory, SIAM Journal on Control and Optimization, 38, 2, 582-612, (2000) · Zbl 0947.60061 [23] Hu, Y.; Øksendal, B., Fractional white noise calculus and applications to finance, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6, 1, 1-32, (2003) · Zbl 1045.60072 [24] Norros, I.; Valkeila, E.; Virtamo, J., An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli, 5, 4, 571-587, (1999) · Zbl 0955.60034 [25] Jarrow, R. A.; Protter, P.; Sayit, H., No arbitrage without semimartingales, The Annals of Applied Probability, 19, 2, 596-616, (2009) · Zbl 1172.60027 [26] Soner, H. M.; Shreve, S. E.; Cvitanić, J., There is no nontrivial hedging portfolio for option pricing with transaction costs, The Annals of Applied Probability, 5, 2, 327-355, (1995) · Zbl 0837.90012 [27] Narayan, O., Exact asymptotic queue length distribution for fractional Brownian traffic, Advances in Performance Analysis, 1, 39-63, (1998) [28] Michna, Z., Self-similar processes in collective risk theory, Journal of Applied Mathematics and Stochastic Analysis, 11, 4, 429-448, (1998) · Zbl 0937.60029 [29] Ferebee, B., An asymptotic expansion for one-sided Brownian exit densities, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 63, 1, 1-15, (1983) · Zbl 0488.60086 [30] Asmussen, S., Ruin Probabilities, (2000), Singapore: World Scientific Publishing, Singapore
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