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A displaced-diffusion stochastic volatility LIBOR market model: motivation, definition and implementation

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  • Mark Joshi
  • Riccardo Rebonato

Abstract

We present an extension of the LIBOR market model which allows for stochastic instantaneous volatilities of the forward rates in a displaced-diffusion setting. We show that virtually all the powerful and important approximations that apply in the deterministic setting can be successfully and naturally extended to the stochastic volatility case. In particular we show that (i) the caplet market can still be efficiently and accurately fit; (ii) that the drift approximations that allow the evolution of the forward rates over time steps as long as several years are still valid; (iii) that in the new setting the European swaption matrix implied by a given choice of volatility parameters can be efficiently approximated with a closed-form expression without having to carry out a Monte Carlo simulation for the forward rate process; and (iv) that it is still possible to calibrate the model virtually perfectly via simply matrix manipulations so that the prices of the co-terminal swaptions underlying a given Bermudan swaption will be exactly recovered, while retaining a desirable behaviour for the evolution of the term structure of volatilities.

Suggested Citation

  • Mark Joshi & Riccardo Rebonato, 2003. "A displaced-diffusion stochastic volatility LIBOR market model: motivation, definition and implementation," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 458-469.
  • Handle: RePEc:taf:quantf:v:3:y:2003:i:6:p:458-469
    DOI: 10.1088/1469-7688/3/6/305
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    References listed on IDEAS

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    1. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    2. Beckers, Stan, 1980. " The Constant Elasticity of Variance Model and Its Implications for Option Pricing," Journal of Finance, American Finance Association, vol. 35(3), pages 661-673, June.
    3. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. " Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-430, March.
    4. Rubinstein, Mark, 1983. " Displaced Diffusion Option Pricing," Journal of Finance, American Finance Association, vol. 38(1), pages 213-217, March.
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    Cited by:

    1. Roger Lee & Dan Wang, 2012. "Displaced lognormal volatility skews: analysis and applications to stochastic volatility simulations," Annals of Finance, Springer, vol. 8(2), pages 159-181, May.
    2. repec:eee:empfin:v:42:y:2017:i:c:p:175-198 is not listed on IDEAS
    3. L. Steinruecke & R. Zagst & A. Swishchuk, 2015. "The Markov-switching jump diffusion LIBOR market model," Quantitative Finance, Taylor & Francis Journals, vol. 15(3), pages 455-476, March.

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