An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit
Tóm tắt
In this paper we explore an identity in distribution of hitting times of a finite variation process (integrated geometric Brownian motion) and a diffusion process (geometric Brownian motion with affine drift), both of which arise from various applications in financial mathematics. We develop semi-analytical solutions to fair charges of variable annuity guaranteed minimum withdrawal benefit from both a policyholder’s perspective and an insurer’s perspective. The pricing framework from the policyholder’s perspective was known previously in the literature only by numerical methods, whereas the insurer’s pricing method was used in the industry but only with Monte Carlo simulations. While comparing their similarities and differences, we prove under the assumption of no friction cost the two pricing approaches are equivalent. In the presence of friction cost, the semi-analytic solutions in this paper lead to a fast and accurate algorithm for determining rider charges and other management fees.
Tài liệu tham khảo
Buchholz, H.: Die Konfluente Hypergeometrische Funktion mit Besonderer Berücksichtigung ihrer Anwendungen. Ergebnisse der angewandten Mathematik. Bd. 2. Springer, Berlin (1953)
Carr, P., Schröder, M.: Bessel processes, the integral of geometric Brownian motion, and Asian options. Teor. Veroyatnost. i Primenen. 48(3), 503–533 (2003)
Chen, Z., Forsyth, P.A.: A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB). Numer. Math. 109(4), 535–569 (2008)
Dai, M., Kwok, Y.K., Zong, J.: Guaranteed minimum withdrawal benefit in variable annuities. Math. Finance 18(4), 595–611 (2008)
Doetsch, G.: Introduction to the theory and application of the Laplace transformation. Springer, New York (1974)
Dufresne, D.: Weak convergence of random growth processes with applications to insurance. Insurance Math. Econom. 8(3), 187–201 (1989)
Dufresne, D.: The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuar. J. 1–2, 39–79 (1990)
Dufresne, D.: The integral of geometric Brownian motion. Adv. Appl. Probab. 33(1), 223–241 (2001)
Feng, R., Volkmer, H.W.: Analytical calculation of risk measures for variable annuity guaranteed benefits. Insurance Math. Econom. 51(3), 636–648 (2012)
Feng, R., Volkmer, H.W.: Spectral methods for the calculation of risk measures for variable annuity guaranteed benefits. Astin Bull. 44(3), 653–681 (2014)
Forsyth, P.A., Vetzal, K.R. Numerical methods for nonlinear PDEs in finance. In: Handbook of computational finance, Springer Handbook of Computational Statistic, pages 503–528. Springer, Heidelberg (2012)
Geman, H., Yor, M.: Bessel processes, asian options, and perpetuities. Math. Finance 3(4), 349–375 (1993)
Gjessing, H.K., Paulsen, J.: Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71(1), 123–144 (1997)
Lewis, A.L.: Applications of eigenfunction expansions in continuous-time finance. Math. Finance 8(4), 349–383 (1998)
Linetsky, V.: Spectral expansions for Asian (average price) options. Oper. Res. 52(6), 856–867 (2004)
Metzler, A.: The laplace transform of hitting times of integrated geometric brownian motion. J. Appl. Probab. 50(1), 295–299 (2013)
Milevsky, M.A., Salisbury, T.S.: Financial valuation of guaranteed minimum withdrawal benefits. Insurance Math. Econom. 38(1), 21–38 (2006)
Norberg, R.: Ruin problems with assets and liabilities of diffusion type. Stoch. Process. Appl. 81(2), 255–269 (1999)
Olver, F.W.: Asymptotics and Special Functions. Academic Press, New York (1974)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC (2010)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and series. Vol. 2. Gordon & Breach Science Publishers, New York. Special functions, Translated from the Russian by N. M. Queen (1986)
Schwaerzler, R.: The method of control variates applied to the estimation of value-at-risk for variable annuities. Master’s thesis, University of Wisconsin-Milwaukee (2012)
Yor, M.: On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24(3), 509–531 (1992)
Yor, M.: Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin (2001)