Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets

Cozma, Andrei; Mariapragassam, Matthieu; Reisinger, Christoph

doi:10.48550/arxiv.1501.06084

Cited by 4 publications

(25 citation statements)

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“…On the other hand, we infer from Figure 1a that both the explosion time for the exact process and the lower bounds with the explicit Euler discretizations approach infinity as ω approaches one, i.e., that lim ω→1 + T * (ω) = ∞. This ensures the uniform boundedness of moments, for ω sufficiently close to one, of the explicit schemes even for very long maturities, an important ingredient in proving the strong convergence of the approximation process (see Cozma and Reisinger 2015). Note that the green and the yellow curves in Figure 1 overlap when ρ ≥ 0.…”

Section: Moment Stability In the Heston Modelmentioning

confidence: 81%

“…However, (3.66) holds for some η ≥ 1 if and only if the second-order polynomial in η on the left-hand side has a real root greater or equal to one. Therefore, we find the necessary and sufficient conditions: The following result is an extension of Proposition 3.3 in Cozma and Reisinger (2015).…”

Section: Explicit Euler Schemes With Absorption Fixesmentioning

confidence: 90%

“…Although weak convergence is important when estimating expectations of payoffs, strong convergence may be required for complex path-dependent derivatives and plays a crucial role in multilevel Monte Carlo methods (Giles 2008). An important step in deriving strong convergence is proving the finiteness of moments of order higher than one of the process and its approximation (Higham et al 2002;Cozma and Reisinger 2015). In addition, for a number of stochastic volatility models, moments of order higher than one can explode in finite time (Andersen and Piterbarg 2007).…”

Section: Introductionmentioning

confidence: 99%

“…Hence, we need to examine the stability of moments of the actual and the approximated processes, a problem directly related to the exponential integrability of the CIR process and its discretization (Cozma and Reisinger 2015). Hutzenthaler et al (2014) proved that a class of stopped increment-tamed Euler approximations for nonlinear systems of SDEs with locally Lipschitz drift and diffusion coefficients retain the exponential integrability of the exact solution under some mild assumptions.…”

Section: Introductionmentioning

confidence: 99%

“…However, the diffusion coefficient in (1.1) is not locally Lipschitz, so their analysis does not apply to the present work. Cozma and Reisinger (2015) derived the exponential integrability of full truncation Euler approximations (Lord et al 2010) for the CIR process up to a critical time. In the present work, we first extend the aforementioned result to more general exponential functionals of the CIR process, and then, we prove that the drift-implicit and a number of explicit Euler discretizations for the CIR process preserve exponential integrability properties.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process

Cozma¹,

Reisinger²

2016

Preprint

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We analyze exponential integrability properties of the Cox-Ingersoll-Ross (CIR) process and its Euler discretizations with various types of truncation and reflection at 0. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of stochastic differential equations arising in finance. We prove that both implicit and explicit Euler-Maruyama discretizations for the CIR process preserve the exponential integrability of the exact solution for a wide range of parameters, and find lower bounds on the explosion time.

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Section: Moment Stability In the Heston Modelmentioning

confidence: 81%

Section: Explicit Euler Schemes With Absorption Fixesmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process

Cozma¹,

Reisinger²

2016

Preprint

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Strong order 1/2 convergence of full truncation Euler approximations to the Cox–Ingersoll–Ross process

Cozma

Reisinger

2018

IMA Journal of Numerical Analysis

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We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller ratio is greater than three), we establish the strong order 1/2 convergence in L p of the scheme to the exact solution. This is consistent with the optimal rate of strong convergence for approximations to the Cox-Ingersoll-Ross process based on sequential evaluations of the driving Brownian motion.

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Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility Models

Cozma¹,

Reisinger²

2018

SIAM J. Numer. Anal.

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We consider a class of stochastic path-dependent volatility models where the stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is multiplied by a (leverage) function of the spot process, its running maximum, and time. We propose a Monte Carlo simulation scheme which combines a log-Euler scheme for the spot process with the full truncation Euler scheme or the backward Euler-Maruyama scheme for the squared stochastic volatility component. Under some mild regularity assumptions and a condition on the Feller ratio, we establish the strong convergence with order 1/2 (up to a logarithmic factor) of the approximation process up to a critical time. The model studied in this paper contains as special cases Heston-type stochasticlocal volatility models, the state-of-the-art in derivative pricing, and a relatively new class of path-dependent volatility models. The present paper is the first to prove the convergence of the popular Euler schemes with a positive rate, which is moreover consistent with that for Lipschitz coefficients and hence optimal.

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Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets

Cited by 4 publications

References 0 publications

Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process

Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process

Strong order 1/2 convergence of full truncation Euler approximations to the Cox–Ingersoll–Ross process

Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility Models

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