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Analytical pricing of the smile in a forward LIBOR market model

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  • D. Brigo
  • F. Mercurio

Abstract

We introduce a general class of analytically tractable diffusions for modelling forward LIBOR rates under their canonical measure. The class, which is based on assuming a smooth functional dependence at expiry between a forward rate and an associated Brownian motion, is highly tractable. It implies explicit dynamics, known marginal and transition densities and explicit caplet prices at any time. As an example, we analyse the dynamics given by a linear combination of geometric Brownian motions with perfectly correlated (decorrelated) returns. We finally construct a specific model in the class that reproduces exactly the market caplet volatilities given in input. Examples of the implied-volatility curves produced by the considered models are also shown.

Suggested Citation

  • D. Brigo & F. Mercurio, 2003. "Analytical pricing of the smile in a forward LIBOR market model," Quantitative Finance, Taylor & Francis Journals, vol. 3(1), pages 15-27.
    • Handle: RePEc:taf:quantf:v:3:y:2003:i:1:p:15-27
    DOI: 10.1080/713666156
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    Citations

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    Cited by:

    1. Da Fonseca, José & Gnoatto, Alessandro & Grasselli, Martino, 2013. "A flexible matrix Libor model with smiles," Journal of Economic Dynamics and Control, Elsevier, vol. 37(4), pages 774-793.
    2. Koichiro Takaoka & Hidenori Futami, 2010. "The Instantaneous Volatility and the Implied Volatility Surface for a Generalized Black–Scholes Model," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 17(4), pages 391-436, December.
    3. Arismendi-Zambrano, Juan & Belitsky, Vladimir & Sobreiro, Vinicius Amorim & Kimura, Herbert, 2022. "The implications of dependence, tail dependence, and bounds’ measures for counterparty credit risk pricing," Journal of Financial Stability, Elsevier, vol. 58(C).
    4. J. C. Arismendi-Zambrano & Vladimir Belitsky & Vinicius Amorim Sobreiro & Herbert Kimura, 2020. "The Implications of Tail Dependency Measures for Counterparty Credit Risk Pricing," Economics Department Working Paper Series n306-20.pdf, Department of Economics, National University of Ireland - Maynooth.
    5. Fries, Christian P. & Nigbur, Tobias & Seeger, Norman, 2017. "Displaced relative changes in historical simulation: Application to risk measures of interest rates with phases of negative rates," Journal of Empirical Finance, Elsevier, vol. 42(C), pages 175-198.
    6. Lixin Wu & Fan Zhang, 2008. "Fast swaption pricing under the market model with a square-root volatility process," Quantitative Finance, Taylor & Francis Journals, vol. 8(2), pages 163-180.
    7. Dariusz Gatarek & Juliusz Jabłecki, 2021. "Between Scylla and Charybdis: The Bermudan Swaptions Pricing Odyssey," Mathematics, MDPI, vol. 9(2), pages 1-32, January.

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