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A Two-Factor Model for Low Interest Rate Regimes

Author

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  • Shane Miller
  • Eckhard Platen

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Abstract

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. Copyright Springer Science + Business Media, Inc. 2004

Suggested Citation

  • Shane Miller & Eckhard Platen, 2004. "A Two-Factor Model for Low Interest Rate Regimes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 107-133, March.
  • Handle: RePEc:kap:apfinm:v:11:y:2004:i:1:p:107-133
    DOI: 10.1007/s10690-005-4251-x
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    References listed on IDEAS

    as
    1. David Heath & Eckhard Platen, 2004. "Understanding the Implied Volatility Surface for Options on a Diversified Index," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 55-77, March.
    2. Black, Fischer, 1995. " Interest Rates as Options," Journal of Finance, American Finance Association, vol. 50(5), pages 1371-1376, December.
    3. repec:wsi:ijtafx:v:08:y:2005:i:06:n:s0219024905003244 is not listed on IDEAS
    4. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters,in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    5. Leif Andersen & Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(1), pages 1-32.
    6. Eckhard Platen, 2003. "Modeling the Volatility and Expected Value of a Diversified World Index," Research Paper Series 103, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July.
    8. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    9. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    10. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    11. Hull, John & White, Alan, 1993. "One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(02), pages 235-254, June.
    12. Damiano Brigo & Fabio Mercurio, 2001. "A deterministic-shift extension of analytically-tractable and time-homogeneous short-rate models," Finance and Stochastics, Springer, vol. 5(3), pages 369-387.
    13. Eckhard Platen, 2003. "An Alternative Interest Rate Term Structure Model," Research Paper Series 97, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    15. repec:wsi:ijtafx:v:07:y:2004:i:04:n:s0219024904002499 is not listed on IDEAS
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    Citations

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    Cited by:

    1. Nicola Bruti-Liberati & Christina Nikitopoulos-Sklibosios & Eckhard Platen, 2010. "Real-world jump-diffusion term structure models," Quantitative Finance, Taylor & Francis Journals, vol. 10(1), pages 23-37.
    2. Ashkan Nikeghbali & Eckhard Platen, 2008. "On Honest Times in Financial Modeling," Research Paper Series 229, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Shane Miller & Eckhard Platen, 2010. "Real-World Pricing for a Modified Constant Elasticity of Variance Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(2), pages 147-175.
    4. Hardy Hulley & Shane Miller & Eckhard Platen, 2005. "Benchmarking and Fair Pricing Applied to Two Market Models," Research Paper Series 155, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Eckhard Platen, 2004. "Capital Asset Pricing for Markets with Intensity Based Jumps," Research Paper Series 143, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. repec:wsi:ijtafx:v:11:y:2008:i:08:n:s0219024908005056 is not listed on IDEAS
    7. Nicola Bruti-Liberati & Christina Nikitopoulos-Sklibosios & Eckhard Platen, 2007. "Pricing under the Real-World Probability Measure for Jump-Diffusion Term Structure Models," Research Paper Series 198, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. Shane Miller & Eckhard Platen, 2008. "Analytic Pricing of Contingent Claims Under the Real-World Measure," Research Paper Series 216, Quantitative Finance Research Centre, University of Technology, Sydney.
    9. Jan Baldeaux & Fung & Katja Ignatieva & Eckhard Platen, 2015. "A Hybrid Model for Pricing and Hedging of Long-dated Bonds," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(4), pages 366-398, September.

    More about this item

    Keywords

    fair pricing; growth optimal portfolio; interest rate caps; interest rate term structure; total market price for risk;

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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