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Nonlinear Kalman Filtering in Affine Term Structure Models

Author

Listed:
  • Peter Christoffersen

    (University of Toronto - Rotman School of Management and CREATES)

  • Christian Dorion

    (HEC Montreal)

  • Kris Jacobs

    (University of Houston and Tilburg University)

  • Lotfi Karoui

    (Goldman, Sachs & Co.)

Abstract
When the relationship between security prices and state variables in dynamic term structure models is nonlinear, existing studies usually linearize this relationship because nonlinear fi?ltering is computationally demanding. We conduct an extensive investigation of this linearization and analyze the potential of the unscented Kalman ?filter to properly capture nonlinearities. To illustrate the advantages of the unscented Kalman ?filter, we analyze the cross section of swap rates, which are relatively simple non-linear instruments, and cap prices, which are highly nonlinear in the states. An extensive Monte Carlo experiment demonstrates that the unscented Kalman fi?lter is much more accurate than its extended counterpart in fi?ltering the states and forecasting swap rates and caps. Our fi?ndings suggest that the unscented Kalman fi?lter may prove to be a good approach for a number of other problems in fi?xed income pricing with nonlinear relationships between the state vector and the observations, such as the estimation of term structure models using coupon bonds and the estimation of quadratic term structure models.

Suggested Citation

  • Peter Christoffersen & Christian Dorion & Kris Jacobs & Lotfi Karoui, 2012. "Nonlinear Kalman Filtering in Affine Term Structure Models," CREATES Research Papers 2012-49, Department of Economics and Business Economics, Aarhus University.
    • Handle: RePEc:aah:create:2012-49
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    File URL: https://repec.econ.au.dk/repec/creates/rp/12/rp12_49.pdf
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    References listed on IDEAS

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    1. Duffie, Darrell & Singleton, Kenneth J, 1997. "An Econometric Model of the Term Structure of Interest-Rate Swap Yields," Journal of Finance, American Finance Association, vol. 52(4), pages 1287-1321, September.
    2. Qiang Dai & Kenneth J. Singleton, 2000. "Specification Analysis of Affine Term Structure Models," Journal of Finance, American Finance Association, vol. 55(5), pages 1943-1978, October.
    3. Pierre Collin‐Dufresne & Robert S. Goldstein, 2002. "Do Bonds Span the Fixed Income Markets? Theory and Evidence for Unspanned Stochastic Volatility," Journal of Finance, American Finance Association, vol. 57(4), pages 1685-1730, August.
    4. Li, Junye, 2011. "Sequential Bayesian Analysis of Time-Changed Infinite Activity Derivatives Pricing Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(4), pages 468-480.
    5. Backus, David & Foresi, Silverio & Mozumdar, Abon & Wu, Liuren, 2001. "Predictable changes in yields and forward rates," Journal of Financial Economics, Elsevier, vol. 59(3), pages 281-311, March.
    6. Bakshi, Gurdip & Carr, Peter & Wu, Liuren, 2008. "Stochastic risk premiums, stochastic skewness in currency options, and stochastic discount factors in international economies," Journal of Financial Economics, Elsevier, vol. 87(1), pages 132-156, January.
    7. Torben G. Andersen & Luca Benzoni, 2010. "Do Bonds Span Volatility Risk in the U.S. Treasury Market? A Specification Test for Affine Term Structure Models," Journal of Finance, American Finance Association, vol. 65(2), pages 603-653, April.
    8. Pearson, Neil D & Sun, Tong-Sheng, 1994. "Exploiting the Conditional Density in Estimating the Term Structure: An Application to the Cox, Ingersoll, and Ross Model," Journal of Finance, American Finance Association, vol. 49(4), pages 1279-1304, September.
    9. Jean-Sébastien Fontaine & René Garcia, 2012. "Bond Liquidity Premia," The Review of Financial Studies, Society for Financial Studies, vol. 25(4), pages 1207-1254.
    10. Aït-Sahalia, Yacine & Kimmel, Robert L., 2010. "Estimating affine multifactor term structure models using closed-form likelihood expansions," Journal of Financial Economics, Elsevier, vol. 98(1), pages 113-144, October.
    11. Duffee, Gregory R, 1999. "Estimating the Price of Default Risk," The Review of Financial Studies, Society for Financial Studies, vol. 12(1), pages 197-226.
    12. Egorov, Alexei V. & Hong, Yongmiao & Li, Haitao, 2006. "Validating forecasts of the joint probability density of bond yields: Can affine models beat random walk?," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 255-284.
    13. Peter Carr & Liuren Wu, 2010. "Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation," Journal of Financial Econometrics, Oxford University Press, vol. 8(4), pages 409-449, Fall.
    14. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    15. Pierre Collin-Dufresne & Christopher S. Jones & Robert S. Goldstein, 2004. "Can Interest Rate Volatility be Extracted from the Cross Section of Bond Yields? An Investigation of Unspanned Stochastic Volatility," NBER Working Papers 10756, National Bureau of Economic Research, Inc.
    16. Carr, Peter & Wu, Liuren, 2007. "Stochastic skew in currency options," Journal of Financial Economics, Elsevier, vol. 86(1), pages 213-247, October.
    17. Gregory R. Duffee, 2002. "Term Premia and Interest Rate Forecasts in Affine Models," Journal of Finance, American Finance Association, vol. 57(1), pages 405-443, February.
    18. Babbs, Simon H. & Nowman, K. Ben, 1999. "Kalman Filtering of Generalized Vasicek Term Structure Models," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(1), pages 115-130, March.
    19. Collin-Dufresne, Pierre & Goldstein, Robert S. & Jones, Christopher S., 2009. "Can interest rate volatility be extracted from the cross section of bond yields?," Journal of Financial Economics, Elsevier, vol. 94(1), pages 47-66, October.
    20. Feng Zhao & Robert Jarrow & Haitao Li, 2004. "Interest Rate Caps Smile Too! But Can the LIBOR Market Models Capture It?," Econometric Society 2004 North American Winter Meetings 431, Econometric Society.
    21. Chen, Ren-Raw & Cheng, Xiaolin & Fabozzi, Frank J. & Liu, Bo, 2008. "An Explicit, Multi-Factor Credit Default Swap Pricing Model with Correlated Factors," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 43(1), pages 123-160, March.
    22. Jefferson Duarte, 2004. "Evaluating an Alternative Risk Preference in Affine Term Structure Models," The Review of Financial Studies, Society for Financial Studies, vol. 17(2), pages 379-404.
    23. de Jong, Frank, 2000. "Time Series and Cross-Section Information in Affine Term-Structure Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 18(3), pages 300-314, July.
    24. Cheridito, Patrick & Filipovic, Damir & Kimmel, Robert L., 2007. "Market price of risk specifications for affine models: Theory and evidence," Journal of Financial Economics, Elsevier, vol. 83(1), pages 123-170, January.
    25. Hirsa, Ali & Neftci, Salih N., 2013. "An Introduction to the Mathematics of Financial Derivatives," Elsevier Monographs, Elsevier, edition 3, number 9780123846822.
    26. Junye Li, 2011. "Sequential Bayesian Analysis of Time-Changed Infinite Activity Derivatives Pricing Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(4), pages 468-480, October.
    27. Gurdip Bakshi & Liuren Wu, 2010. "The Behavior of Risk and Market Prices of Risk Over the Nasdaq Bubble Period," Management Science, INFORMS, vol. 56(12), pages 2251-2264, December.
    28. Robert Jarrow & Haitao Li & Feng Zhao, 2007. "Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile?," Journal of Finance, American Finance Association, vol. 62(1), pages 345-382, February.
    29. Pierre Collin‐Dufresne & Bruno Solnik, 2001. "On the Term Structure of Default Premia in the Swap and LIBOR Markets," Journal of Finance, American Finance Association, vol. 56(3), pages 1095-1115, June.
    30. Li, Junye, 2013. "An unscented Kalman smoother for volatility extraction: Evidence from stock prices and options," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 15-26.
    31. Darrell Duffie & Rui Kan, 1996. "A Yield‐Factor Model Of Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 379-406, October.
    32. Almeida, Caio & Graveline, Jeremy J. & Joslin, Scott, 2011. "Do interest rate options contain information about excess returns?," Journal of Econometrics, Elsevier, vol. 164(1), pages 35-44, September.
    33. Duan, Jin-Chuan & Simonato, Jean-Guy, 1999. "Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter," Review of Quantitative Finance and Accounting, Springer, vol. 13(2), pages 111-135, September.
    34. Samuel Thompson, 2008. "Identifying Term Structure Volatility from the LIBOR-Swap Curve," The Review of Financial Studies, Society for Financial Studies, vol. 21(2), pages 819-854, April.
    35. Haitao Li & Feng Zhao, 2006. "Unspanned Stochastic Volatility: Evidence from Hedging Interest Rate Derivatives," Journal of Finance, American Finance Association, vol. 61(1), pages 341-378, February.
    36. Gregory R. Duffee & Richard H. Stanton, 2012. "Estimation of Dynamic Term Structure Models," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-51.
    37. Feldhütter, Peter & Lando, David, 2008. "Decomposing swap spreads," Journal of Financial Economics, Elsevier, vol. 88(2), pages 375-405, May.
    38. Dai, Qiang & Singleton, Kenneth J., 2002. "Expectation puzzles, time-varying risk premia, and affine models of the term structure," Journal of Financial Economics, Elsevier, vol. 63(3), pages 415-441, March.
    39. Massoud Heidari & Liuren Wu, 2010. "Market Anticipation of Fed Policy Changes and the Term Structure of Interest Rates," Review of Finance, European Finance Association, vol. 14(2), pages 313-342.
    40. Chen, Ren-Raw & Scott, Louis, 2003. "Multi-factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model," The Journal of Real Estate Finance and Economics, Springer, vol. 27(2), pages 143-172, September.
    41. Ruslan Bikbov & Mikhail Chernov, 2009. "Unspanned Stochastic Volatility in Affine Models: Evidence from Eurodollar Futures and Options," Management Science, INFORMS, vol. 55(8), pages 1292-1305, August.
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    More about this item

    Keywords

    Kalman filtering; nonlinearity; term structure models; swaps; caps.;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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    This paper has been announced in the following NEP Reports:

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