## Pricing ratchet equity-indexed annuities with early surrender risk in a CIR$$++$$ model.(English)Zbl 1412.91058

Summary: In this article we propose a lattice algorithm for pricing simple ratchet equity-indexed annuities (EIAs) with early surrender risk and global minimum contract value when the asset value depends on the CIR$$++$$ stochastic interest rates. In addition we present an asymptotic expansion technique that permits us to obtain a first-order approximation formula for the price of simple ratchet EIAs without early surrender risk and without a global minimum contract value. Numerical comparisons show the reliability of the proposed methods.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 91G60 Numerical methods (including Monte Carlo methods) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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### References:

 [1] Alfonsi, A., High Order Discretization Schemes for the CIR Process: Application to Affine Term Structure and Heston Models, Mathematics of Computation, 79, 209-237, (2010) · Zbl 1198.60030 [2] Amin, K.; Khanna, A., Convergence of American Option Values from Discrete-to Continuous-Time Financial Models, Mathematical Finance, 4, 289-304, (1994) · Zbl 0884.90012 [3] Appolloni, E.; Caramellino, L.; Zanette, A., A Robust Tree Method for Pricing American Options with CIR Stochastic Interest Rate, (2013) [4] Brigo, D.; Mercurio, F., Interest Rate Models—Theory and Practice, (2006), Berlin: Springer, Berlin [5] Costabile, M.; Gaudenzi, M.; Massabï¿, I.; Zanette, A., Evaluating Fair Premiums of Equity-Linked Policies with Surrender Option in a Bivariate Model, Insurance: Mathematics and Economics, 41, 317-338, (2009) [6] Cox, J.; Ross, S. A.; Rubinstein, M., Option Pricing: A Simplified Appoach, Journal of Financial Economics, 7, 229-264, (1979) [7] Cox, J. C.; Ingersoll, J.; Ross, S., A Theory of the Term Structure of Interest Rates, Econometrica, 53, 385-407, (1985) · Zbl 1274.91447 [8] Ethier, S. N.; Kurtz, T., Markov Processes: Characterization and Convergence, (1986), New York: John Wiley & Sons, New York · Zbl 0592.60049 [9] Gaudenzi, M.; Lepellere, M. A.; Zanette, A., The Singular Point Method for Pricing Path-Dependent Options, Journal of Computational Finance, 14, 29-56, (2010) · Zbl 1284.91571 [10] Gerber, H.; Shiu, E., Pricing Lookback Options and Dynamic Guarantees, North American Actuarial Journal, 7, 48-67, (2003) · Zbl 1084.91507 [11] Hardy, M. R., Investment Guarantees: Modelling and Risk Management for Equity-Linked Life Insurance, (2003), New York: Wiley, New York · Zbl 1092.91042 [12] Hull, J.; White, A., Numerical Procedures for Implementing Term Structure Models. I., Journal of Derivatives, 2, 7-16, (1994) [13] Annuity Fact Book: A Guide to Information, Trends, and Data in the Annuity Industry, (2009), Washington, DC: Insured Retirement Institute., Washington, DC [14] Kijima, M.; Wong, T., Pricing of Ratchet Equity-Indexed Annuities Under Stochastic Interest Rate, Insurance: Mathematics and Economics, 41, 317-338, (2007) · Zbl 1141.91457 [15] Kim, Y.; Kunitomo, N., Pricing options under Stochastic Interest Rates: A New Approach, Asia-pacific Financial Markets, 6, 49-70, (1999) · Zbl 1157.91363 [16] Kunitomo, N.; Takahashi, A., On Validity of the Asymptotic Expansion Approach in Contingent Claim Analysis, Annals of Applied Probability, 13, 914-952, (2003) · Zbl 1091.91037 [17] Kushner, H.; Dupuis, P. G., Numerical Methods for Stochastic Control Problems in Continous Time, (1992), Berlin: Springer Verlag, Berlin [18] Lin, X. D.; Tan, K. S., Valuation of Equity-Indexed Annuities under Stochastic Interest Rates, North American Auctuarial Journal, 6, 72-91, (2003) · Zbl 1084.60530 [19] Nelson, D. B.; Ramaswamy, K., Simple Binomial Processes as Diffusion Approximations in Financial Models, Review of Financial Studies, 3, 393-430, (1990) [20] Palmer, B. A., Equity-Indexed Annuities: Fundamental Concepts and Issues, Report of Insurance Information Institute, (2006) [21] Wei, J., Valuing American Equity Options with a Stochastic Interest Rate: A Note, Journal of Financial Engineering, 2, 195-206, (1996) [22] Wilmott, P., Cliquet Options and Volatility Models, Wilmott Magazine, 3, 78-83, (2002)
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