April  2006, 2(2): 199-227. doi: 10.3934/jimo.2006.2.199

LIBOR market model with stochastic volatility


Department of Mathematics, University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China, China

Received  December 2005 Revised  February 2006 Published  April 2006

In this paper we extend the standard LIBOR market model to accommodate the pronounced phenomenon of implied volatility smiles/skews. We adopt a multiplicative stochastic factor to the volatility functions of all relevant forward rates. The stochastic factor follows a square-root diffusion process, and it can be correlated to the forward rates. For any swap rate, we derive an approximate process under its corresponding forward swap measure. By utilizing the analytical tractability of the approximate process, we develop a closed-form formula for swaptions in term of Fourier transforms. Extensive numerical tests are carried out to support the swaptions formula. The extended model captures the downward volatility skews by taking negative correlations between forward rates and their volatilities, which is consistent with empirical findings.
Citation: Lixin Wu, Fan Zhang. LIBOR market model with stochastic volatility. Journal of Industrial and Management Optimization, 2006, 2 (2) : 199-227. doi: 10.3934/jimo.2006.2.199

Laurent Devineau, Pierre-Edouard Arrouy, Paul Bonnefoy, Alexandre Boumezoued. Fast calibration of the Libor market model with stochastic volatility and displaced diffusion. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1699-1729. doi: 10.3934/jimo.2019025


Partha Sharathi Dutta, Soumitro Banerjee. Period increment cascades in a discontinuous map with square-root singularity. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 961-976. doi: 10.3934/dcdsb.2010.14.961


Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057


Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965


Zhenglin Wang. Fast non-uniform Fourier transform based regularization for sparse-view large-size CT reconstruction. STEM Education, 2022, 2 (2) : 121-139. doi: 10.3934/steme.2022009


Vincenzo Ambrosio, Giovanni Molica Bisci, Dušan Repovš. Nonlinear equations involving the square root of the Laplacian. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 151-170. doi: 10.3934/dcdss.2019011


Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, 2021, 3 (3) : 563-588. doi: 10.3934/fods.2021003


Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli. Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model. Conference Publications, 2007, 2007 (Special) : 354-363. doi: 10.3934/proc.2007.2007.354


Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005


Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579


Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381


Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005


Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023


Michael Music. The nonlinear Fourier transform for two-dimensional subcritical potentials. Inverse Problems and Imaging, 2014, 8 (4) : 1151-1167. doi: 10.3934/ipi.2014.8.1151


Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339


Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems and Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885


Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure and Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007


Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control and Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036


Jia Yue, Nan-Jing Huang. Neutral and indifference pricing with stochastic correlation and volatility. Journal of Industrial and Management Optimization, 2018, 14 (1) : 199-229. doi: 10.3934/jimo.2017043


Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435

2021 Impact Factor: 1.411


  • PDF downloads (235)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]