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R. Szmytkowski and S. Bielski, “Comment on the Orthogonality of the Macdonald Functions of Imaginary Order,” Journal of Mathematical Analysis and Applications, Vol. 365, No. 1, 2010, pp. 195-197. doi:10.1016/j.jmaa.2009.10.035

has been cited by the following article:

• TITLE: Some Explicitly Solvable SABR and Multiscale SABR Models: Option Pricing and Calibration

  AUTHORS: Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli

  KEYWORDS: Multiscale Stochastic Volatility Models; Option Pricing; Calibration Problem; FX Data

  JOURNAL NAME: Journal of Mathematical Finance, Vol.3 No.1, February 26, 2013

  ABSTRACT: A multiscale SABR model that describes the dynamics of forward prices/rates is presented. New closed form formulae for the transition probability density functions of the normal and lognormal SABR and multiscale SABR models and for the prices of the corresponding European call and put options are deduced. The technique used to obtain these formulae is rather general and can be used to study other stochastic volatility models. A calibration problem for these models is formulated and solved. Numerical experiments with real data are presented.