High order discretization schemes for the CIR process: Application to affine term structure and Heston models

Alfonsi, Aurélien

doi:10.1090/s0025-5718-09-02252-2

Cited by 149 publications

(168 citation statements)

References 22 publications

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“…A couple years later, a high order discretization schemes for the variance process were presented in the paper by Alfonsi [43], but rigorous error analysis for Heston model was regarded as an open issue. Possible extensions of the QE scheme include GammaQE scheme and Double Gamma scheme introduced by Chan and Joshi [44].…”

Section: Qe Schemementioning

confidence: 99%

Calibration and simulation of Heston model

Mrázek

Pospı́šil

2017

Open Mathematics

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Abstract:We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods.We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.

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Section: Qe Schemementioning

confidence: 99%

Calibration and simulation of Heston model

Mrázek

Pospı́šil

2017

Open Mathematics

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“…An alternative approach can be found in Kusuoka [21,22]. Its implementation as a splitting method is given in Ninomiya and Victoir [26], see also Alfonsi [1], Ninomiya and Ninomiya [25], and Tanaka and Kohatsu-Higa [33]. Our strategy is as follows.…”

Section: Stability Of Cubature Schemesmentioning

confidence: 99%

“…This is also a typical assumption in other approximation methods for stochastic differential equations, e.g., in Talay and Tubaro [32]. Some success in relaxing these assumptions, which are rarely satisfied in practical problems, was achieved for approximations of the splitting type in Alfonsi [1] and Tanaka and Kohatsu-Higa [33]. While the first one focuses on the CIR process and the second one on Lévy driving noise, it was recognised in both works that polynomially bounded test functions are the correct context for problems with Lipschitz continuous vector fields.…”

Section: Introductionmentioning

confidence: 96%

Cubature methods for stochastic (partial) differential equations in weighted spaces

Dörsek

Teichmann

Velušček

2013

Stoch PDE: Anal Comp

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The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded test functions. We first describe a recently introduced flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da PratoZabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced technical assumption on weak symmetry of the cubature formula. Computational results for a cubature discretization of a spatially extended stochastic FitzHugh-Nagumo model, an SPDE model from mathematical biology, are shown, illustrating the applicability of our theory.

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“…For an extensive list of articles on this subject we refer to [3] and [7]. Besides [3] and [7] we also refer to [1,2,11,12], where a number of discretization schemes for the CIR process can be found. Further we note that in [17] a weakly convergent fully implicit method is implemented for the Heston model.…”

Section: Introductionmentioning

confidence: 99%

Uniform approximation of the Cox-Ingersoll-Ross process

Milstein¹,

Schoenmakers

2015

Adv. Appl. Probab.

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The Doss-Sussmann (DS) approach is used for uniform simulation of the CoxIngersoll-Ross (CIR) process. The DS formalism allows to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near zero. From a conceptual point of view the proposed method gives a better quality of approximation (from a path-wise point of view) than standard, or even exact simulation of the SDE at some discrete time grid.

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High order discretization schemes for the CIR process: Application to affine term structure and Heston models

Cited by 149 publications

References 22 publications

Calibration and simulation of Heston model

Calibration and simulation of Heston model

Cubature methods for stochastic (partial) differential equations in weighted spaces

Uniform approximation of the Cox-Ingersoll-Ross process

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