Term Structure Equations Under Benchmark Framework

El Qalli, Yassine (2009): Term Structure Equations Under Benchmark Framework. Unpublished.

Abstract

This paper makes use of an integrated benchmark modeling framework that allows us to derive term structure equations for bond and forward prices. The benchmark or numeraire is chosen to be the growth optimal portfolio (GOP). For deterministic short rate the solution of the bond term structure equation coincides with the explicit formula obtained in Platen(2005). The resulting term structure equations are used to explain moves in bond and forward prices by introducing GOP as a factor and therefore constructing a hedge portfolio for bond consisting of units of the GOP and the saving account. The paper also derives an affine term structure equation for forward price in term of the GOP factor. In the case of stochastic short rate we restrict our selves to give only a term structure equation for the bond price.

• Item Type: MPRA Paper
• Language: English
• Keywords: Term structure, Benchmark approach, GOP, Forward price, bond.
• Subjects: E - Macroeconomics and Monetary Economics > E4 - Money and Interest Rates > E43 - Determination of Interest Rates; Term Structure of Interest Rates; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing; Futures Pricing

References

[1] Bjork, T. (2004) Arbitrage Theory in Continuous Time, Oxford University Press, New York.

[2] Eddahbi, M., El Qalli, Y. (2008) Real World pricing for Forward Contracts, In proceeding of “1st Hispano-Moroccan Days on Applied Mathematics and Statistics.” 29-31 December 2008. Tetouan Morocco.

[3] El Karoui, N., Geman, H. & Rochet, J.C. (1995) Changes of numeraire, changes of probability measures and pricing options. J. Appl. Prob. 32, 443-458.

[4] Geman, H. (1989) The importance of the forward neutral probability in stochastic approach of interest rates. Working paper, ESSEC.

[5] Jamshidian, F. (1987)Pricing of contingent claums in the one factor term structure model. Working paper, Merrill Lynch Capital Markets

[6] Kelly, J.R. (1956) A new interpretation of information rate. Bell Syst. Techn. J. Fixed income 1:54-61.

[7] Long. (1991) The numeraire portfolio. J. Financial Economics 26:29-69.

[8] Platen, E. (2002) Arbitrage in continuous complete market. Adv. in Appl. Probab. 34(3):540- 558.

[9] Platen, E. (2005) An alternative interest rate terme structure. Int. J. Theor. Appl. Finance 8(6):717-735.

[10] Platen, E. & Heath, D. (2006) Benchmark approach to quantitative finance. Springer finance. Springer