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Keywords = fractional CKLS model

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14 pages, 2567 KiB  
Article
Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS Model
by Jiajia Zhao and Zuoliang Xu
Fractal Fract. 2022, 6(7), 344; https://doi.org/10.3390/fractalfract6070344 - 21 Jun 2022
Cited by 3 | Viewed by 1781
Abstract
In this paper, we reconstruct the time-dependent volatility function of the underlying asset and the mean-reverting parameter γ of the interest rate for European options under the fractional Chan–Karolyi–Longstaff–Sanders (CKLS) stochastic interest rate model. Tikhonov regularization is used to solve the ill-posedness of [...] Read more.
In this paper, we reconstruct the time-dependent volatility function of the underlying asset and the mean-reverting parameter γ of the interest rate for European options under the fractional Chan–Karolyi–Longstaff–Sanders (CKLS) stochastic interest rate model. Tikhonov regularization is used to solve the ill-posedness of the inverse problem. The existence and stability of the solution of the regularization problem are given. We employ the alternating direction method of multipliers (ADMM) to iteratively optimize the volatility function and the parameter γ. Finally, numerical simulations and the empirical analysis are presented to illustrate the efficiency of the proposed method. Full article
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14 pages, 1311 KiB  
Article
Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient
by Kęstutis Kubilius and Aidas Medžiūnas
Mathematics 2021, 9(1), 18; https://doi.org/10.3390/math9010018 - 23 Dec 2020
Cited by 9 | Viewed by 2887
Abstract
We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories [...] Read more.
We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories of the fractional CKLS model with β>1 are not necessarily positive. We obtain the almost sure convergence rate of the backward Euler approximation scheme for solutions of the considered SDEs. We also obtain a strongly consistent and asymptotically normal estimator of the Hurst index H>1/2 for positive solutions of FSDEs. Full article
(This article belongs to the Special Issue Applied Probability)
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