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Solution Of The Extended Cir Term Structure And Bond Option Valuation

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  • Yoosef Maghsoodi
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    Abstract

    The extended Cox-Ingersoll-Ross (ECIR) models of interest rates allow for time-dependent parameters in the CIR square-root model. This article presents closed-form pathwise unique solutions of these unsolved stochastic differential equations (s.d.e.s) in terms of functionals of their driving Brownian motion and parameters. It is shown that quadratics in solution of linear s.d.e.s solve the ECIR model if and only if the "dimension" of the model is a positive integer and that this solution can be achieved by construction of a pathwise unique "generalized" Ornstein-Uhlenbeck process from the ECIR Brownian motion. For real valued dimensions an extension of the time-change theorem of Dubins and Schwarz (1965) is presented and applied to show that a lognormal process solves the model through a stochastic time change. Pathwise equivalence to a rescaled time-changed Bessel square process is also established. These novel results are applied to characterize zero-hitting time and to produce transition density and zero-hitting conditions for the ECIR spot rate. the CIR term structure is then extended to ECIR under no arbitrage, and its solutions and the transition density are represented under a new ECIR martingale measure. the findings are employed to derive a closed-form ECIR bond option valuation formula which generalizes that obtained by CIR (1985). Copyright 1996 Blackwell Publishers.

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    Bibliographic Info

    Article provided by Wiley Blackwell in its journal Mathematical Finance.

    Volume (Year): 6 (1996)
    Issue (Month): 1 ()
    Pages: 89-109

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    Handle: RePEc:bla:mathfi:v:6:y:1996:i:1:p:89-109

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    Cited by:
    1. Christian Gourieroux & Alain Monfort, 2007. "Quadratic Stochastic Intensity and Prospective Mortality Tables," Working Papers 2007-30, Centre de Recherche en Economie et Statistique.
    2. Antonio Mannolini & Carlo Mari & Roberto Ren�, 2008. "Pricing caps and floors with the extended CIR model," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 13(4), pages 386-400.
    3. Angelos Dassios & Jayalaxshmi Nagaradjasarma, 2006. "The square-root process and Asian options," Quantitative Finance, Taylor & Francis Journals, vol. 6(4), pages 337-347.
    4. Egorov, Alexei V. & Li, Haitao & Xu, Yuewu, 2003. "Maximum likelihood estimation of time-inhomogeneous diffusions," Journal of Econometrics, Elsevier, vol. 114(1), pages 107-139, May.
    5. Choi, Seungmoon, 2013. "Closed-form likelihood expansions for multivariate time-inhomogeneous diffusions," Journal of Econometrics, Elsevier, vol. 174(2), pages 45-65.
    6. Griselda Deelstra, 2000. "Long-term returns in stochastic interest rate models: applications," ULB Institutional Repository 2013/7590, ULB -- Universite Libre de Bruxelles.
    7. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742, October.
    8. Dahl, Mikkel & Moller, Thomas, 2006. "Valuation and hedging of life insurance liabilities with systematic mortality risk," Insurance: Mathematics and Economics, Elsevier, vol. 39(2), pages 193-217, October.
    9. Angelos Dassios & Jayalaxshmi Nagaradjasarma, 2011. "Pricing of Asian options on interest rates in the CIR model," LSE Research Online Documents on Economics 32084, London School of Economics and Political Science, LSE Library.
    10. Griselda Deelstra, 1999. "Yield option pricing in the generalized Cox-Ingersoll-Ross Model," ULB Institutional Repository 2013/7592, ULB -- Universite Libre de Bruxelles.
    11. Guarin, Alexander & Liu, Xiaoquan & Ng, Wing Lon, 2014. "Recovering default risk from CDS spreads with a nonlinear filter," Journal of Economic Dynamics and Control, Elsevier, vol. 38(C), pages 87-104.
    12. D. Duffie & D. Filipovic & W. Schachermayer, 2002. "Affine Processes and Application in Finance," NBER Technical Working Papers 0281, National Bureau of Economic Research, Inc.
    13. Li, Da-Ye & Nishimura, Yusaku & Men, Ming, 2014. "Fractal markets: Liquidity and investors on different time horizons," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 407(C), pages 144-151.
    14. Erik Schlogl & Lutz Schlogl, 2000. "A square root interest rate model fitting discrete initial term structure data," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(3), pages 183-209.
    15. Guo, Zhi Jun, 2008. "A note on the CIR process and the existence of equivalent martingale measures," Statistics & Probability Letters, Elsevier, vol. 78(5), pages 481-487, April.
    16. Dahl, Mikkel, 2004. "Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 113-136, August.

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