Apr 13, 2010

LIBOR Market Models for Smile-Dependent Exotics

Continuing our series on the use of advanced models (see The Bates Model), today we address the issue of pricing complex trades that are heavily dependent on the volatility smile, such as CMS, LIBOR-linked structures and hybrids. For these exotics, the Numerix includes a suite of LIBOR Market Models (LMM), with support for stochastic volatility, which is required to calibrate to the entire volatility surface.

To understand why this model is necessary, let’s look at recent advances in market models for pricing interest-rate derivatives.

Tracing the Evolution of the LMM
In 1997, Brace, Gatarek and Musiela [1] introduced the BGM model—a standard formulation of the LIBOR market model using a lognormal diffusion of the underlying forward LIBORs. By modeling observable rates directly and enabling correlation between observable rates, BGM offered a major advantage over short-rate models that relied on unobservable instantaneous rates.

This multi-factor model allowed de-correlation of LIBOR/CMS rates and provided analytical formulas for European caplets/swaptions, which are necessary for effective calibration. However, lognormal modeling of the underlying LIBORs implies a flat volatility smile. This limitation made BGM ineffective for pricing smile-dependent exotics, such as range accrual notes.

The need for more-complex calibration algorithms led to two key extensions of the classic formulation. First came the shifted BGM, which introduced the ability to reproduce an implied volatility skew [2]. Further, a stochastic volatility process was added to the LIBORs’ evolution to calibrate both the skew and curvature of the smile [3].

The general name “LIBOR Market Model” emerged from these extensions to the classical BGM model. In its general formulation, LMM supports a shift and stochastic volatility process, as well as a wide spectrum of volatility parameterization, enabling fine-tuned calibration to the market-observed volatility smile.

LIBOR Market Models in Numerix Analytics
While the LMM has traditionally been considered “computationally heavy,” we’ve developed highly efficient proprietary numerical methods, including an accelerated Monte Carlo, that enable extremely rapid calculations for complex callable derivatives. By providing a unified framework for both shift and stochastic volatility, Numerix allows the user to capture the term structure of volatility smiles across the swaption grid, enabling the most accurate calibrations.

The user also has the option to turn shift and stochastic volatility on or off in an arbitrary order and set—for example, switching them both off to replicate the classical log-normal BGM model for some vanilla deals.

Numerix has developed advanced analytical methods that permit calibration to:

  • European caps and swaptions
  • CMS European products, such as CMS swaps, CMS caps/floors and CMS spread options
  • Various correlations, including those between Libor and CMS rates

The LMM also features unique characteristics that allow it to be applied to interest rates within a broader context of equity, FX and commodity modeling for use in pricing hybrid products. We’ve incorporated this feature to enable the use of the LMM within our unique hybrid model framework.

Why Should I Use the Numerix LMM?

  • One of most complete and rich dynamic models for interest rates, permitting calibration to options’ smile and skew—indispensable for pricing complex structured products
  • Applicable to both single- and multi-currency fixed-income instruments—as well as the domestic IR component of hybrid instruments
  • Offers a wide range of “flavors,” such as shifted LMM with stochastic volatility that follows CIR process with time-dependent volatility of volatility
  • Includes a large set of calibration options, including user-defined sets or arbitrary sets of caplets/swaptions, European CMS products, and correlations between arbitrary LIBOR and CMS rates
  • Enables the definition of arbitrary payoffs using Numerix’s unique payoff structuring interface

For more quantitative research from Numerix, visit our online library, or contact our sales team to learn how to integrate the Numerix LMM in your systems.


Follow-up: See the April 14 article on comparing market model accuracy to the volatility smile, with sample calibrations of BGM, SBGM and SV-LMM.


[1] A. Brace, D. Gatarek, M. Musiela, The Market Model of Interest Rate Dynamics, Mathematical Finance, 1997, Vol. 7, No. 2, pp. 127-155.

[2] L. Andersen and J. Andreasen, Volatility Skews and Extensions of the LIBOR Market Model, Applied Mathematical Finance, 2000, Vol. 7, No. 1, pp. 1–32.

[3] V. Piterbarg, Stochastic Volatility Model with Time-Dependent Skew, Applied Mathematical Finance, June 2005, Vol. 12, No. 2, pp. 147–185.

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