Fast swaption pricing under the market model with a square-root volatility process

Abstract

In this paper we study a correlation-based LIBOR market model with a square-root volatility process. This model captures downward volatility skews through taking negative correlations between forward rates and the multiplier. An approximate pricing formula is developed for swaptions, and the formula is implemented via fast Fourier transform. Numerical results on pricing accuracy are presented, which strongly support the approximations made in deriving the formula.

‡In the literature both ‘skew’ and ‘smirk’ are used to name a slanted smile.

Keywords:

• LIBOR model
• Stochastic volatility
• Square-root process
• Swaptions
• Fast Fourier transform (FFT)

Acknowledgements

This research is partially supported by RGC Grant HKUST6145/01P. Early versions of this paper were presented in Ecole Polytechnique (June 2002), Stanford University (September 2002), University of Texas at Austin (November 2002), Quantitative Methods in Finance 2002 (December), BNP Paribas (February 2003, New York), The Annual Meeting of Canadian Mathematical Society of 2003, and other occasions. We would like to thank Alan Brace, Rama Cont, Paul Glasserman, George Papanicolaou, Thaleia Zariphopoulou for helpful comments, and Mr. Kalok Chau of HSBC for supplying data. We are responsible for any remaining errors.

Notes

‡In the literature both ‘skew’ and ‘smirk’ are used to name a slanted smile.

†A lognormal process whose volatility follows a square-root process (Cox et al. 1985).

‡Higher peak and fatter tails than that of a normal distribution.

§Such a correspondence in fact also exists in other stochastic volatility models, e.g. Zhou (2003).

†Note that γ _j (t) = 0 for t ≥ T _j since f _j is fixed from the time T _j becomes ‘dead’.

‡The distributional properties of V(t) are well understood (e.g. Avellaneda and Laurence 2000). When 2κθ > ε², in particular, V(t) has a stationary distribution and stays strictly positive.

§The empirical results of Chen and Scott (2001) suggest zero rate–multiplier correlation only for the nearest-term forward rate. In early versions of this paper, we had included plots for implied caplet volatilities of USD for the date of 5 July 2002. While the implied volatility curve of longer maturities appear like downward skews, the implied volatility curve of the six-month caplets is a smile, which is consistent with the finding of zero correlation between the stochastic volatility and the near-term forward rate. The plots are omitted for brevity.

†For convenience we have taken the same ΔT for both caps and swaptions. Note that in reality caps and swaptions can have different intervals between cash flows. In such a case, we may take the smallest interval for ΔT.