1932

Abstract

I review how the theoretical modeling of the dynamics of forward rates in the context of derivatives pricing has evolved over time. I review the theoretical developments from the short rate models of the 1980s to the stochastic-volatility extensions of the SABR model. I argue that how the theory developed can be understood only by taking into account the institutional setting of derivatives trading and that the modeling choices were motivated to a surprisingly large extent by how the market evolved. I conclude with an assessment of which of these theoretical contributions have had a lasting and meaningful effect on the financial theory of asset pricing.

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2023-11-01
2024-04-25
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Literature Cited

  1. Alexander C, Lvov D. 2003. Statistical properties of forward Libor rates. ISMA Discuss. Pap. Finance 2003-03 ISMA Cent., Univ. Reading, Reading UK:
  2. Andersen LBG, Andreasen A. 2000. Volatility skews and extensions of the LIBOR market model. Appl. Math. Finance 7:11–32
    [Google Scholar]
  3. Bailey S. 2012. Principal component analysis with noisy and/or missing data. Publ. Astron. Soc. Pac. 124:9191015–23
    [Google Scholar]
  4. Bartlett B. 2006. Hedging under SABR model. Wilmott Magazine July/Aug. 2–4
    [Google Scholar]
  5. Bjork T. 2004. Arbitrage Theory in Continuous Time Oxford, UK: Oxford Univ. Press
  6. Black F. 1976. The pricing of commodity contracts. J. Financ. Econ. 3:167–79
    [Google Scholar]
  7. Black F, Derman E, Toy W. 1990. A one-factor model of interest rates and its application to Treasury bond options. Financ. Anal. J. 46:33–39
    [Google Scholar]
  8. Black F, Karasinski P. 1991. Bond and option pricing when short rates are log-normal. Financ. Anal. J. 47:452–59
    [Google Scholar]
  9. Brace A, Gatarek D, Musiela M. 1997. The market model of interest rate dynamics. Math. Finance 7:2127–47
    [Google Scholar]
  10. Brigo D, Mercurio F. 2006. Interest Rate Models: Theory and Practice. Berlin: Springer
  11. Carr P, Wu L. 2003. What type of process underlies options? A simple and robust test. J. Finance 58:62581–610
    [Google Scholar]
  12. Carverhill A. 1994. When is the spot rate Markovian?. Math. Finance 4:305–12
    [Google Scholar]
  13. Chaplin G. 1998. A review of term-structure models and their applications. Br. Actuar. J. 4:2323–83
    [Google Scholar]
  14. Cheng S. 1991. On the feasibility of arbitrage-based option pricing with stochastic bond prices. J. Econ. Theory 52:185–98
    [Google Scholar]
  15. Cheyette O. 1996. Markov representation of the Heath–Jarrow–Morton model. Work. Pap. Loomis Sayles New York:
  16. Cochrane J, Piazzesi M. 2004. Reply to Dai, Singleton and Yang Work. Pap. Univ. Chicago
  17. Cochrane J, Piazzesi M. 2005. Bond risk premia. Am. Econ. Rev. 95:138–60
    [Google Scholar]
  18. Cox J, Ingersoll J, Ross SA. 1985. A theory of the term structure of interest rates. Econometrica 53:385–407
    [Google Scholar]
  19. Crump RK, Gospodinov N. 2022. Deconstructing the yield curve Staff Rep. 884 Fed. Reserve Bank N. Y.
  20. Dai Q, Singleton KJ, Yang W. 2004. Predictability of bond risk premia and affine term structure models. Work. Pap. NYU Stern Sch. Bus. New York:
  21. Davydov D, Linetsky V. 2001. Pricing and hedging path-dependent options under the CEV process. Manag. Sci. 47:7881–1027
    [Google Scholar]
  22. de Boer C. 2001. A Practical Guide to Splines. Berlin: Springer. Revis. ed.
  23. de Guillaume N, Rebonato R, Pogudin A. 2013. The nature of the dependence of the magnitude of rate moves on the rates levels: a universal relationship. Quant. Finance 13:3351–67
    [Google Scholar]
  24. Deisenroth MP, Faisal AA, Ong CS. 2020. Mathematics for Machine Learning Cambridge, UK: Cambridge Univ. Press
  25. Derman E, Kani I. 1994. Riding on a smile. Risk 7:32–39
    [Google Scholar]
  26. Derman E, Kani I. 1998. Stochastic implied trees: arbitrage pricing with stochastic term and strike structure of volatility. Int. J. Theor. Appl. Finance 1:61–110
    [Google Scholar]
  27. Dodds S. 1998. Estimating the instantaneous volatilities of forward rates. Work. Pap. Barclays Capital London:
  28. Doust P. 2007. Modelling discrete probabilities. Work. Pap. R. Bank Scotl. Edinburgh:
  29. Doust P. 2011. Yield curve construction—Bezier interpolation. Work. Pap. R. Bank Scotl. Edinburgh:
  30. Duffie D, Stein JC. 2015. Reforming LIBOR and other financial market benchmarks. J. Econ. Perspect. 29:2191–212
    [Google Scholar]
  31. Dupire B. 1994. Pricing with a smile. Risk 7:18–20
    [Google Scholar]
  32. Feller W. 1951. Two singular diffusion problems. Ann. Math. 54:173–82
    [Google Scholar]
  33. Gatheral J. 2006. The Volatility Surface: A Practitioner's Guide. Hoboken, NJ: Wiley
  34. Gibson R, Lhabitant FS, Talay D. 2010. Modeling the term structure of interest rates: a review of the literature. Found. Trends Finance 5:1/21–156
    [Google Scholar]
  35. Glasserman P, Kou SG. 2003. The term structure of simple forward rates with jump risk. Math. Finance 13:3383–410
    [Google Scholar]
  36. Glasserman P, Merener N. 2001. Numerical solutions of jump-diffusion LIBOR market models. Work. Pap. Columbia Univ. New York:
  37. Glasserman P, Merener N. 2003. Cap and swaption approximations in LIBOR market models with jumps. J. Comput. Finance 7:131–52
    [Google Scholar]
  38. Hagan PS, Kumar D, Lesniewski A, Woodward DE. 2002. Managing smile risk. Wilmott Magazine Novemb. 84–108
    [Google Scholar]
  39. Hagan PS, Lesniewski A. 2008. LIBOR market model with SABR style stochastic volatility. Work. Pap. JPMorgan Chase/Ellington Manag. Group
  40. Hagan PS, West G. 2006. Interpolation methods for curve construction. Appl. Math. Finance 13:289–129
    [Google Scholar]
  41. Halberg JH, Nilson EN, Walsh JH. 1967. Theory of Splines and Their Applications San Diego, CA: Academic
  42. Han XA, Ma Y, Huang X. 2008. A novel generalization of Bezier curve and surface. J. Comput. Appl. Math. 217:1180–93
    [Google Scholar]
  43. Harrison JM, Kreps D. 1979. Martingales and arbitrage in multi-period securities markets. J. Econ. Theory 20:381–408
    [Google Scholar]
  44. Harrison JM, Pliska S. 1981. Martingales and stochastic integrals in the theory of continuous trading. Stoch. Proc. Appl. 11:215–60
    [Google Scholar]
  45. Healy J. 2020. Equivalence between forward rate interpolations and discount factor interpolations for the yield curve construction. arXiv:2005.1389 [cs.DB]
  46. Heath D, Jarrow R, Morton A. 1990. Bond pricing and the term structure of interest rates: a discrete time approximation. J. Financ. Quant. Anal. 25:4419–40
    [Google Scholar]
  47. Heath D, Jarrow R, Morton A. 1992. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60:177–105
    [Google Scholar]
  48. Henrad M. 2014. Interest Rate Modelling in the Multi-Curve Framework. New York: Palgrave Macmillan
  49. Henry-Labordère P. 2006. Unifying the BGM and SABR models: a short ride in hyperbolic geometry. Work. Pap. Qube Res. Technol. London:
  50. Ho T, Lee S. 1986. Term structure movements and pricing interest rate contingemt claims. J. Finance 41:1011–29
    [Google Scholar]
  51. Hughston L. 2000. The New Interest Rate Models. London: Risk:
  52. Hull J, White A. 1990. Pricing interest rate derivative securities. Rev. Financ. Stud. 3:573–92
    [Google Scholar]
  53. Hull J, White A. 1993. Bond option pricing based on a model for the evolution of bond prices. Adv. Futur. Opt. Res. 6:1–13
    [Google Scholar]
  54. Hull J, White A. 2000. White, forward rate volatilities, swap rate volatilities and the implementation of the LIBOR market model. J. Fixed Income 10:40–62
    [Google Scholar]
  55. Hunt P, Kennedy JE. 2000. Financial Derivatives in Theory and Practice Hoboken, NJ: Wiley
  56. Ioannidis C, Miao R, Williams JM 2008. Interest rate models: a review. Handbook of Financial Engineering C Zopounidis, M Doumpos, PM Pardalos 157–200. Berlin: Springer
    [Google Scholar]
  57. Jäckel P, Rebonato R. 2003. The link between caplet and swaption volatilities in a Brace-Gatarek-Musiela-Jamshidian framework: approximate solutions and empirical evidence. J. Comput. Finance 6:35–45
    [Google Scholar]
  58. Jamshidian F. 1997. LIBOR and swap market models and measures. Finance Stoch. 1:293–330
    [Google Scholar]
  59. Johnson RS. 2005. Singular Perturbation Theory: Mathematical and Analytical Techniques. Berlin: Springer
  60. Joshi M, Rebonato R. 2003. A displaced-diffusion stochastic volatility LIBOR market model: motivation, definition and implementation. Quant. Finance 3:6458–69
    [Google Scholar]
  61. Karatzas I, Shreve S. 1991. Brownian Motion and Stochastic Calculus. Berlin: Springer. , 2nd ed..
  62. Laurini MP, Ohashi A. 2014. A noisy principal component analysis for forward rate curves. Work. Pap. Univ. Fed. Paraiba Paraiba, Braz.:
  63. Litterman R, Scheinkman JA. 1991. Common factors affecting bond returns. J. Fixed Income 1:54–61
    [Google Scholar]
  64. Longstaff FA, Santa-Clara P, Schwartz ES. 1999. The relative valuation of caps and swaptions: theory and empirical evidence. Work. Pap. Univ. Calif. Los Angeles:
  65. Longstaff FA, Schwartz ES. 2001. Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14:1113–47
    [Google Scholar]
  66. Lord R, Pelsser A. 2007. Level–slope–curvature—fact or artefact?. Appl. Math. Finance 14:2105–30
    [Google Scholar]
  67. Lyashenko A, Mercurio F. 2019. LIBOR replacement: a modeling framework for in-arrears term rates. Risk June 57–62
    [Google Scholar]
  68. Lyashenko A, Mercurio F. 2020. LIBOR replacement II: completing the generalized forward market model. Risk July 61–66
    [Google Scholar]
  69. Marris D. 1999. Financial option pricing and skewed volatility PhD Thesis Univ. Cambridge Cambridge, UK:
  70. McCulloch J. 1975. The tax-adjusted yield curve. J. Finance 30:811–30
    [Google Scholar]
  71. Merton R. 1976. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3:125–44
    [Google Scholar]
  72. Miltersen KR, Sandmann K, Sondermann D. 1997. Closed-form solutions for term structure derivatives with log-normal interest rates. J. Finance 52:409–30
    [Google Scholar]
  73. Morton KW, Mayers DF. 2005. Numerical Solution of Partial Differential Equations Cambridge, UK: Cambridge Univ. Press
  74. Nawalkha SK, Rebonato R. 2011. What interest rate models to use? Buy side versus sell side. J. Invest. Manag. 9:35–18
    [Google Scholar]
  75. Pienaar R, Chodhry M. 2001. Fitting the term structure of interest rates: the practical implementation of cubic spline methodology Work. Pap. Cent. Math. Trad. Finance, City Univ. Bus. Sch. London:
  76. Pietersz R. 2005. Pricing models for Bermudan-style interest rate derivatives. PhD Thesis Res. Inst. Manag., Erasmus Univ. Rotterdam, Neth.:
  77. Pirjol D, Zhu L. 2017. Small-noise limit of the quasi-Gaussian log-normal HJM model. Oper. Res. Lett. 45:16–11
    [Google Scholar]
  78. Piterbarg V. 2003. A stochastic volatility forward LIBOR model with a term structure of volatility smiles. Work. Pap. Imperial College London
  79. Piterbarg V, Renedo MA. 2004. Eurodollar futures convexity adjustment in stochastic volatility models. Work. Pap. Imperial College London/Bank Am.
  80. Rebonato R. 1999a. On the pricing implications of the joint lognormal assumption for the swaption and cap markets. J. Comput. Finance 2:357–76
    [Google Scholar]
  81. Rebonato R. 1999b. On the simultaneous calibration of multi-factor interest models to black volatilities and to the correlation matrix. J. Comput. Finance 2:45–27
    [Google Scholar]
  82. Rebonato R. 2002. Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond Princeton, NJ: Princeton Univ. Press
  83. Rebonato R. 2003. What process gives rise to the observed dependence of swaption implied volatility on the underlying?. Int. J. Theor. Appl. Finance 6:4419–42
    [Google Scholar]
  84. Rebonato R. 2004a. Interest-rate term-structure pricing models: a review. Proc. R. Soc. A 460:667–728
    [Google Scholar]
  85. Rebonato R. 2004b. Volatility and Correlation: The Perfect Hedger and the Fox Hoboken, NJ: Wiley
  86. Rebonato R, Aouadi AE. 2021. How do the volatilities of rates depend on their level? The ‘universal relationship’ revisited. J. Fixed Income 30:417–31
    [Google Scholar]
  87. Rebonato R, Cooper I. 1995. The limitations of simple two-factor interest-rate models. J. Financ. Eng. 5:1–16
    [Google Scholar]
  88. Rebonato R, Jäckel P. 2000. The most general methodology to create a valid correlation matrix for risk management and option pricing purposes. J. Risk 2:217–27
    [Google Scholar]
  89. Rebonato R, McKay K, White R. 2009. The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives Hoboken, NJ: Wiley
  90. Rebonato R, Pogudin A. 2011. Is it possible to reconcile the caplet and swaption markets? Evidence from the U.S. dollar market. J. Deriv. 19:28–31
    [Google Scholar]
  91. Rebonato R, Pogudin A, White A. 2008. Delta and vega hedging in the SABR and LMM-SABR models. Risk Dec. 94–99
    [Google Scholar]
  92. Rebonato R, Ronzani R. 2022. Is convexity efficiently priced? Evidence from international swap markets. J. Empir. Finance 63:C392–412
    [Google Scholar]
  93. Rebonato R, White R. 2008. A swaption volatility model using Markov regime switching. J. Comput. Finance 12:179–114
    [Google Scholar]
  94. Ritchken P, Sankarasubramanian L. 1995. Volatility structure of forward rates and the dynamics of the term structure. Math. Finance 5:55–72
    [Google Scholar]
  95. Rubinstein M. 1983. Displaced diffusion option pricing. J. Finance 38:1213–17
    [Google Scholar]
  96. Sandmann S, Sondermann D. 1995. On the stability of log-normal interest rate models. Work. Pap. Univ. Bonn Bonn, Ger.:
  97. Schoenmakers JGM, Coffey B. 2000. Stable implied correlation of a multi-factor LIBOR model via a semiparametric correlation structure. Work. Pap. Weierstr. Inst. Angew. Anal. Stoch. Berlin:
  98. Schrimpf A, Shusko V. 2019. Beyond LIBOR: a primer on the new benchmark rates. BIS Q. Rev. March 29–52
    [Google Scholar]
  99. Sidenius J. 2000. LIBOR market models in practice. J. Comput. Finance 3:375–99
    [Google Scholar]
  100. Spears TC. 2014. Engineering value, engineering risk: what derivatives quants know and what their models do PhD Thesis Univ. Edinburgh Edinburgh, UK:
  101. Švàbovà L, Duriča M. 2014. The relationship between the finite difference method and trinomial trees. Proceedings of the 17th International Scientific Conference on Applications of Mathematics and Statistics in Economics259–69. Red Hook, NY: Curran
    [Google Scholar]
  102. Vaillant N. 1995. Convexity adjustment between futures and forward rates using a martingale approach. Work. Pap. BZW London:
  103. Vasicek O. 1977. An equilibrium characterization of the term structure. J. Financ. Econ. 5:177–88
    [Google Scholar]
  104. Wu L, Zhang F. 2002. LIBOR market model: from deterministic to stochastic volatility. Work. Pap. Claremont Grad. Univ./Hong Kong Univ. Sci. Technol.
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