Fair prices under a unified lattice approach for interest rate derivatives

  • Giacomo MorelliEmail author
S.I. : Recent Developments in Financial Modeling and Risk Management


An open question in interest rates derivative pricing is whether the price of the contracts should be computed by means of a multi-curve approach (different yield curves for discounting and forwarding) or by using a single curve (just one yield curve both for discounting and forwarding). The answer is of primary importance for financial markets as it allows to define a class of fair contracts. This paper calculates and compares the price of a simple swap within both multi-curve and single curve approaches and proposes a generalization of the lattice approach, which is usually used to approximate short interest rate models in the multi-curve framework. As an example, I show how to use the Black et al. (Financ Anal J 46(1):33–39, 1990) interest rate model on binomial lattice in multi-curve framework and calculate the price of the 2–8 period swaption with a single (LIBOR) curve and two-curve (OIS+LIBOR) approaches. Such technique can be used for pricing any interest rate based contract.


Interest rates Single curve Multiple curve Derivative pricing Fair contracts 

JEL Classification

C02 C60 C63 



  1. Ametrano, F., & Bianchetti, M. (2009). Bootstrapping the illiquidity. Modelling Interest Rates: Advances for Derivatives Pricing. Risk Books.Google Scholar
  2. Arrow, K. J., & Debreu, G. (1954). Existence of an equilibrium for a competitive economy. Econometrica, 22, 265–290.CrossRefGoogle Scholar
  3. Ball, C., & Torous, W. (1999). The stochastic volatility of short-term interest rates: Some international evidence. The Journal of Finance, 54, 2339–2359.CrossRefGoogle Scholar
  4. Bianchetti, M. (2008). Two curves, one price: Pricing & hedging interest rate derivatives decoupling forwarding and discounting yield curves.
  5. Bianchetti, M. (2010). Multiple curves, one price. Paris: Global derivatives.Google Scholar
  6. Bianchetti, M., & Carlicchi, M. (2011). Interest rates after the credit crunch: Multiple curve vanilla derivatives and SABR.
  7. Black, F., E, D., & Toy, W. (1990). A one-factor model of interest rates and its application to treasury bond options. Financial Analysts Journal, 46(1), 33–39.CrossRefGoogle Scholar
  8. Black, F., & Karasinski, P. (1991). Bond and option pricing when short rates are log-normal. Financial Analysts Journal, 47, 52–59.CrossRefGoogle Scholar
  9. Boenkost, W., & Schmidt, W. (1991). Cross currency swap valuation. Financial Analysts Journal, 47, 52–59.CrossRefGoogle Scholar
  10. Brigo, D. (2008). CDS options through candidate market models and the CDS-calibrated CIR++ stochastic intensity model. Milton Park: Taylor & Francis.Google Scholar
  11. Brigo, D., & Mercurio, F. (2006). Interest rate models -PAYPAL-ENV theory and practice with smile, inflation and credit (2nd ed.). Berlin: Springer.Google Scholar
  12. Chibane, M., & Sheldon, G. (2009). Building curves on a good basis, mimeo. Technical report.Google Scholar
  13. Clewlow, L., & Strickland, C. (1998). Implementing derivative models. London: Wiley & Sons.Google Scholar
  14. Cont, R., & Minca, A. (2016). Credit default swaps and systemic risk. Annals of Operations Research, 247(2), 523–547.CrossRefGoogle Scholar
  15. Fujii, M., Shimada, Y., & Takahashi, A. (2010). A note on construction of multiple swap curves with and without collateral. FSA Research Review, 6, 139–157.Google Scholar
  16. Fujii, M., Shimada, Y., & Takahashi, A. (2011). A market model of interest rates with dynamic basis spreads in the presence of collateral and multiple currencies. Wilmott, 2011(54), 61–73.CrossRefGoogle Scholar
  17. Heath, D., Jarrow, R., & Morton, A. (1990). Bond pricing and the term structure of interest rates: A discrete time approximation. Journal of Financial and Quantitative Analysis, 25, 419–440.CrossRefGoogle Scholar
  18. Henrard, M. (2010). The irony in derivatives discounting part ii: The crisis. Wilmott Journal, 2, 301–316.CrossRefGoogle Scholar
  19. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), 327–343.CrossRefGoogle Scholar
  20. Ho, T., & Lee, S. B. (1986). Term structure movements and pricing interest rate contingent claims. The Journal of Finance, 41, 1011–29.CrossRefGoogle Scholar
  21. Hull, J. C. (2009). Options, futures and other derivatives. Upper Saddle River: Pearson Prentice Hall.Google Scholar
  22. Hull, J., & White, A. (1990). Pricing interest rate derivative securities. Review of Financial Studies, 3, 573–592.CrossRefGoogle Scholar
  23. Johannes, M., & Sundaresan, S. (2007). The impact of collateralization on swap rates. The Journal of Finance, 62(1), 383–410.CrossRefGoogle Scholar
  24. Kalotay, A., Williams, G., & Fabozzi, F. (1993). A model for the valuation of bonds and embedded options. Financial Analysts Journal, 49(3), 35–46.CrossRefGoogle Scholar
  25. Karouzakis, N., Hatgioannides, J., & Andriosopoulos, K. (2018). Convexity adjustment for constant maturity swaps in a multi-curve framework. Annals of Operations Research, 266(1–2), 159–181.CrossRefGoogle Scholar
  26. Kijima, M., Tanaka, K., & Wong, T. (2009). A multi-quality model of interest rates. Quantitative Finance, 9(2), 133–145.CrossRefGoogle Scholar
  27. Mercurio, F. (2008). Interest rates and the credit crunch: New formulas and market models. Technical report, QFR, Bloomberg.Google Scholar
  28. Ron, U. (2000). A practical guide to swap curve construction. Technical report, Bank of CanadaGoogle Scholar
  29. Sochacki, J., & Buetow, G. (2001). Term-structure models using binomial trees. Technical report, The Research Foundation of AIMR.Google Scholar
  30. Tse, Y. (1995). Some international evidence on the stochastic behavior of interest rates. Journal of International Money and Finance, 14, 721–738.CrossRefGoogle Scholar
  31. Whittal, C. (2010). Clearnet revalues \$218 trillion swap portfolio using ois. Risk Magazine.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Economics and FinanceLUISS UniversityRomeItaly

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