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Long-Term Returns in Stochastic Interest Rate Models: Applications

Published online by Cambridge University Press:  29 August 2014

Griselda Deelstra
Griselda Deelstra*
Affiliation:
Ensae, Crest and VUB
*
ENSAE, CREST, Timbre J120, 3, Av. Pierre Larousse, 92245 Malakoff CEDEX, France
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Abstract

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We extend the Cox-Ingersoll-Ross (1985) model of the short interest rate by assuming a stochastic reversion level, which better reflects the time dependence caused by the cyclical nature of the economy or by expectations concerning the future impact of monetary policies. In this framework, we have studied the convergence of the long-term return by using the theory of generalised Bessel-square processes. We emphasize the applications of the convergence results. A limit theorem proves evidence of the use of a Brownian motion with drift instead of the integral . For practice, however, this approximation turns out to be only appropriate when there are no explicit formulae and calculations are very time-consuming.

Keywords

Interest ratesCox-Ingersoll-Ross modelStochastic reversion levelGeneralised Bessel-square processesConvergenceBond pricesLife insurance
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 30 , Issue 1 , May 2000 , pp. 123 - 140
Copyright
Copyright © International Actuarial Association 2000

References

Beekman, J. and Fuelling, C. (1991) Extra randomness in certain annuity modes. Insurance: Mathematics and Economics 10, 275–287.Google Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Society of Actuaries, Schaumburg.Google Scholar
Brennan, M.J. and Schwartz, E.S. (1982) An equilibrium Model of Bond Pricing and a Test of Market Efficiency. Journal of Financial and Quantitative Analysis 17 (3), 301–329.CrossRefGoogle Scholar
Brown, R.H. and Schaefer, S.M. (1994) The term structure of real interest rates and the Cox, Ingersoll and Ross model. Journal of Financial Economics 35, 3–42.CrossRefGoogle Scholar
Chan, K.C., Karolyi, G.A., Longstaff, F.A. and Sanders, A.B. (1992) An Empirical Comparison of Alternative Models of the Short-Term Interest Rate. Journal of Finance 47, 1209–1227.Google Scholar
Chen, R. and Scott, L. (1992) Pricing interest rate options in a two-factor Cox-Ingersoll-Ross model of the term structure. Review of Financial Studies 5, 613–636.CrossRefGoogle Scholar
Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica 58, 385–407.CrossRefGoogle Scholar
Courtadon, G.The Pricing of Options on Default-Free Bonds. Financial and Quantitative Analysis 17, 75–100.CrossRefGoogle Scholar
Deelstra, G. (1995) Long-term returns in stochastic interest rate models. Ph.D. thesis, VUB Brussels.CrossRefGoogle Scholar
Deelstra, G. (2000) Yield Options in the Generalised Cox-Ingersoll-Ross Model. In revision.Google Scholar
Deelstra, G. and Delbaen, F. (1995a) Long-term returns in stochastic interest rate models. Insurance: Mathematics and Economics 17, 163–169.Google Scholar
Deelstra, G. and Delbaen, F. (1995b) Long-term returns in stochastic interest rate models: Convergence in law. Stochastics and Stochastics Reports 55, 253–277.CrossRefGoogle Scholar
Delbaen, F. and Shirakawa, H. (1996) Squared Bessel processes and their applications to the square root interest rate model. Working paper.Google Scholar
Dufresne, D. (1990) The distribution of a perpetuity, with applications to risk theory and pension funding. Scandinavian Actuarial Journal, 39–79.CrossRefGoogle Scholar
Dybvig, P.H., Ingersoll, J.E. and Ross, S.A. (1986) Do Interest Rates Converge?Working paper.Google Scholar
Dybvig, P.H., Ingersoll, J.E. and Ross, S.A. (1996) Long Forward and zero-coupon rates can Never Fall. Journal of Business 69, 1–25.CrossRefGoogle Scholar
El Karoui, N., Frachot, N. and Geman, H. (1998) On the behavior of long zero coupon rates in a no arbitrage framework. Journal of Derivatives Research 1, 351–369.Google Scholar
Giacotto, C. (1986) Stochastic Modelling of Interest Rates: Actuarial vs. Equilibrium Approach. Journal of Risk and Insurance 53 (3), 435–453.CrossRefGoogle Scholar
Goovaerts, M. and Teunen, M. (1995) Evaluation of interest randomness for pension valuation. Proceedings of the International Congress of Actuaries, Brussels, 689–709.Google Scholar
Goovaerts, M., Labie, E. and Vanneste, M. (1994) The Distributions of Annuities. Insurance: Mathematics and Economics 15, 37–48.Google Scholar
Hicks, J.R. (1937) Mr. Keynes and the ‘Classics’; A suggested interpretation. Econometrica 5, 147–159.CrossRefGoogle Scholar
Hogan, M. (1993) Problems in certain two-factor term structure models. Annals of Applied Probability 3, 576–581.CrossRefGoogle Scholar
Maghsoodi, Y. (1996) Solution of the extended CIR term structure and bond option valuation. Mathematical Finance 6, 89–109.CrossRefGoogle Scholar
Milevsky, M. (1997) The present value of a stochastic perpetuity and the Gamma distribution. Insurance: Mathematics and Economics 20, 243–250.Google Scholar
Parker, G. (1992) An application of Stochastic Interest Rates Models in Life Assurance. Ph.D. thesis, Herioi-Watt University.Google Scholar
Parker, G. (1993) Distribution of the Present Value of Future Cash Flows. Proceedings of the 3rd AFIR International Colloquium, Rome, 831–843.Google Scholar
Parker, G. (1994) Moments of the present value of a portfolio of policies. Scandinavian Actuarial Journal, 53–67.Google Scholar
Parker, G. (1997) Stochastic analysis of the interaction between investment and insurance risks. North American Actuarial Journal 1 (2), 55–84.CrossRefGoogle Scholar
Pearson, N.D. and Sun, T.-S. (1994) Exploiting the conditional density in estimating the term structure: an application of the Cox, Ingersoll and Ross model. Journal of Finance 49, 1279–1304.CrossRefGoogle Scholar
Pitman, J. and Yor, M. (1982) A decomposition of Bessel Bridges. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 59, 425–457.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer-Verlag Berlin, Heidelberg, New York.CrossRefGoogle Scholar
Tice, J. and Webber, N. (1997) An nonlinear model of the term structure of interest rates. Mathematical Finance 7, 177–209.CrossRefGoogle Scholar
Yao, Y. (1998) Term structure models and asymptotic long rate. Proceedings of the Second International Congress on Insurance: Mathematics and Economics.Google Scholar
Yor, M. (1992) On some exponential functionals of Brownian motion. Advances in Applied Probability 24, 509–531.CrossRefGoogle Scholar