Mathematical Finance pp 663-731 | Cite as

# Interest Rate Models

Chapter

First Online:

- 1.2k Downloads

## Abstract

From an abstract point of view interest rate markets fall into the reach of the general theory which is discussed in Chaps. 9– 13. They differ from equity markets just in the kind of assets, namely bonds, swaps, caps and other interest rate products. On closer observation, however, one faces new phenomena. For example, the natural choice of a numeraire is not obvious in markets where interest rates change randomly over time.

## References

- 26.T. Bielecki, M. Rutkowski,
*Credit Risk: Modelling, Valuation and Hedging*(Springer, Berlin, 2002)zbMATHGoogle Scholar - 29.T. Björk,
*Arbitrage Theory in Continuous Time*, 3rd edn. (Oxford Univ. Press, Oxford, 2009)zbMATHGoogle Scholar - 30.T. Björk, An overview of interest rate theory, in
*Handbook of Financial Time Series*, ed. by T. Andersen, R. Davis, J.-P. Kreiß, T. Mikosch (Springer, Berlin, 2009), pp. 615–651zbMATHCrossRefGoogle Scholar - 31.T. Björk, B. Christensen, Interest rate dynamics and consistent forward rate curves. Math. Finance
**9**(4), 323–348 (1999)MathSciNetzbMATHCrossRefGoogle Scholar - 32.T. Björk, L. Svensson, On the existence of finite-dimensional realizations for nonlinear forward rate models. Math. Finance
**11**(2), 205–243 (2001)MathSciNetzbMATHCrossRefGoogle Scholar - 33.T. Björk, G. Di Masi, Y. Kabanov, W. Runggaldier, Towards a general theory of bond markets. Finance Stochast.
**1**(2), 141–174 (1997)MathSciNetzbMATHCrossRefGoogle Scholar - 34.T. Björk, Y. Kabanov, W. Runggaldier, Bond market structure in the presence of marked point processes. Math. Finance
**7**(2), 211–239 (1997)MathSciNetzbMATHCrossRefGoogle Scholar - 39.A. Brace, D. Ga̧tarek, M. Musiela, The market model of interest rate dynamics. Math. Finance
**7**(2), 127–155 (1997)MathSciNetzbMATHCrossRefGoogle Scholar - 42.D. Brigo, F. Mercurio,
*Interest Rate Models—Theory and Practice*, 2nd edn. (Springer, Berlin, 2006)zbMATHGoogle Scholar - 44.D. Brody, L. Hughston, E. Mackie, Rational term structure models with geometric Lévy martingales. Stochastics
**84**(5–6), 719–740 (2012)MathSciNetzbMATHCrossRefGoogle Scholar - 46.R. Carmona, M. Tehranchi,
*Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective*(Springer, Berlin, 2006)zbMATHGoogle Scholar - 62.S. Crépey, T. Bielecki,
*Counterparty Risk and Funding*(CRC Press, Boca Raton, 2014)zbMATHGoogle Scholar - 83.E. Eberlein, W. Kluge, Exact pricing formulae for caps and swaptions in a Lévy term structure model. J. Comput. Finance
**9**(2), 99–125 (2006)CrossRefGoogle Scholar - 84.E. Eberlein, W. Kluge, Valuation of floating range notes in Lévy term-structure models. Math. Finance
**16**(2), 237–254 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 85.E. Eberlein, W. Kluge, Calibration of Lévy term structure models, in
*Advances in Mathematical Finance*(Birkhäuser, Boston, 2007), pp. 147–172zbMATHGoogle Scholar - 86.E. Eberlein, F. Özkan, The defaultable Lévy term structure: Ratings and restructuring. Math. Finance
**13**(2), 277–300 (2003)MathSciNetzbMATHCrossRefGoogle Scholar - 87.E. Eberlein, F. Özkan, The Lévy LIBOR model. Finance Stochast.
**9**(3), 327–348 (2005)zbMATHCrossRefGoogle Scholar - 89.E. Eberlein, S. Raible, Term structure models driven by general Lévy processes. Math. Finance
**9**(1), 31–53 (1999)MathSciNetzbMATHCrossRefGoogle Scholar - 94.E. Eberlein, J. Jacod, S. Raible, Lévy term structure models: No-arbitrage and completeness. Finance Stochast.
**9**(1), 67–88 (2005)zbMATHCrossRefGoogle Scholar - 103.D. Filipović,
*Consistency Problems for Heath-Jarrow-Morton Interest Rate Models*, volume 1760 of*Lecture Notes in Mathematics*(Springer, Berlin, 2001)zbMATHCrossRefGoogle Scholar - 105.D. Filipović,
*Term-Structure Models*(Springer, Berlin, 2009)zbMATHCrossRefGoogle Scholar - 108.D. Filipović, S. Tappe, Existence of Lévy term structure models. Finance Stochast.
**12**(1), 83–115 (2008)zbMATHCrossRefGoogle Scholar - 109.D. Filipović, J. Teichmann, Existence of invariant manifolds for stochastic equations in infinite dimension. J. Funct. Anal.
**197**(2), 398–432 (2003)MathSciNetzbMATHCrossRefGoogle Scholar - 110.D. Filipović, M. Larsson, A. Trolle, Linear-rational term structure models. J. Finance
**72**(2), 655–704 (2017)CrossRefGoogle Scholar - 111.D. Filipović, M. Larsson, A. Trolle, On the relation between linearity-generating processes and linear-rational models. arXiv preprint arXiv:1806.03153 (2018)Google Scholar
- 113.B. Flesaker, L. Hughston, Positive interest. Risk
**9**(1), 46–49 (1996)Google Scholar - 124.K. Glau, Z. Grbac, A. Papapantoleon, A unified view of LIBOR models, in
*Advanced Modelling in Mathematical Finance*, vol. 189, ed. by J. Kallsen, A. Papapantoleon (Springer, Cham, 2016), pp. 423–452CrossRefGoogle Scholar - 129.Z. Grbac, W. Runggaldier,
*Interest Rate Modeling: Post-Crisis Challenges and Approaches*(Springer, Cham, 2015)zbMATHCrossRefGoogle Scholar - 131.P. Harms, D. Stefanovits, J. Teichmann, M. Wüthrich, Consistent recalibration of yield curve models. Math. Finance
**28**(3), 757–799 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 140.D. Heath, R. Jarrow, A. Morton, Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica
**60**, 77–105 (1992)zbMATHCrossRefGoogle Scholar - 156.F. Jamshidian, An exact bond option formula. J. Finance
**44**(1), 205–209 (1989)CrossRefGoogle Scholar - 157.F. Jamshidian, Libor and swap market models and measures. Finance Stochast.
**1**(4), 293–330 (1997)zbMATHCrossRefGoogle Scholar - 158.F. Jamshidian, Libor market model with semimartingales. Technical report, Working Paper, NetAnalytic Ltd, 1999Google Scholar
- 159.R. Jarrow, D. Madan, Option pricing using the term structure of interest rates to hedge systematic discontinuities in asset returns. Math. Finance
**5**(4), 311–336 (1995)MathSciNetzbMATHCrossRefGoogle Scholar - 190.M. Keller-Ressel, A. Papapantoleon, J. Teichmann, The affine LIBOR models. Math. Finance
**23**(4), 627–658 (2013)MathSciNetzbMATHCrossRefGoogle Scholar - 193.W. Kluge,
*Time-Inhomogeneous Lévy Processes in Interest Rate and Credit Risk Models*. PhD thesis, University of Freiburg, 2005Google Scholar - 221.K. Miltersen, K. Sandmann, D. Sondermann, Closed form solutions for term structure derivatives with log-normal interest rates. J. Finance
**52**(1), 409–430 (1997)CrossRefGoogle Scholar - 223.M. Musiela, M. Rutkowski,
*Martingale Methods in Financial Modelling*, 2nd edn. Stochastic Modelling and Applied Probability (Springer, Berlin, 2005)zbMATHCrossRefGoogle Scholar - 224.T. Nguyen, F. Seifried, The affine rational potential model (2015). PreprintGoogle Scholar
- 229.A. Papapantoleon, Old and new approaches to LIBOR modeling. Statistica Neerlandica
**64**(3), 257–275 (2010)MathSciNetCrossRefGoogle Scholar - 247.C. Rogers, The potential approach to the term structure of interest rates and foreign exchange rates. Math. Finance
**7**(2), 157–176 (1997)MathSciNetzbMATHCrossRefGoogle Scholar - 255.M. Rutkowski, A note on the Flesaker-Hughston model of the term structure of interest rates. Appl. Math. Finance
**4**(3), 151–163 (1997)zbMATHCrossRefGoogle Scholar - 262.P. Schönbucher,
*Credit Derivatives Pricing Models: Models, Pricing and Implementation*(Wiley, New York, 2003)Google Scholar - 274.H. Shirakawa, Interest rate option pricing with Poisson-Gaussian forward rate curve processes. Math. Finance
**1**(4), 77–94 (1991)zbMATHCrossRefGoogle Scholar - 279.S. Shreve,
*Stochastic Calculus for Finance II: Continuous-Time Models*(Springer, New York, 2004)zbMATHCrossRefGoogle Scholar - 296.R. Zagst,
*Interest-Rate Management*(Springer, Berlin, 2002)zbMATHCrossRefGoogle Scholar

## Copyright information

© Springer Nature Switzerland AG 2019