Convergence of a finite volume element method for a generalized Black-Scholes equation transformed on finite interval

Valkov, Radoslav L.

doi:10.1007/s11075-014-9838-8

Cited by 12 publications

(4 citation statements)

References 13 publications

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“…The advantage of the weak limit method is that it allows passing to the limits with only uniforme L ∞ estimates on u ε . Problem ( 14)-( 16) and ( 19) can be written as (20) with…”

Section: Viscosity Solutionmentioning

confidence: 99%

“…In this section we prove the convergence of the approximate solutions u ε of the viscosity solution u of ( 14)-( 16) and (19). The idea is that the so-called upper weak limit u and the lower weak limit u are, respectively, a viscosity subsolution and supersolution of (20). On, the one hand, we always have u ≤ u in Q T , and on the other hand, the comparison principle implies that u ≤ u a.e.…”

Section: Convergencementioning

confidence: 99%

“…There are a few papers concerning the calculation of the Delta Greek directly. Radoslav Valkov in [20] presented a convergence analysis of a positivity-preserving fitted finite volume element method (FVEM) for a generalized Black-Scholes equation transformed on finite interval, degenerating on both boundary points. He first formulated the FVEM as a Petrov-Galerkin finite element method using a spatial discretization, previously proposed by the author in [21].…”

Section: Introductionmentioning

confidence: 99%

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Viscosity solution of a Delta Greek nonlinear Black-Scholes equation

Almeida,

Chihaluca,

Duque

2020

Preprint

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In this paper, a class of nonlinear option pricing models involving transaction costs is considered. The diffusion coefficient of the nonlinear parabolic equation for the price V is assumed to be a linear function of the option's underlying asset price and the Gamma Greek Vxx. The main aim of this work is to study the governing PDE of the Delta Greek. The existence of viscosity solutions is proved using the vanishing viscosity method. Regularizing the equation by adding a small perturbation to the initial problem, a sequence of approximate solutions u ε is constructed and then the method of weak limits is applied to prove the convergence of the sequence to the viscosity solution of the Delta equation. The approximate problems constructed are shown to have good regularity, which allows the use of efficient and robust numerical methods.

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“…The advantage of the weak limit method is that it allows passing to the limits with only uniforme L ∞ estimates on u ε . Problem ( 14)-( 16) and ( 19) can be written as (20) with…”

Section: Viscosity Solutionmentioning

confidence: 99%

Section: Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Viscosity solution of a Delta Greek nonlinear Black-Scholes equation

Almeida,

Chihaluca,

Duque

2020

Preprint

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“…At the same time, the presence of four spatial variables for this time-dependent problem as well as the existence of six mixed derivative terms implies that any numerical method is supposed to be expensive. For instance, one can consider spatial discretization of (2.4) by using a FD approach (though this has not yet been studied in the literature) or a FE method, for example [55]. But for four-dimensional problems these methods already suffer from the high computational burden (e.g., due to curse of dimensionality).…”

mentioning

confidence: 99%

Pricing foreign exchange options under stochastic volatility and interest rates using an RBF–FD method

Soleymani

Itkin

2019

Journal of Computational Science

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This paper proposes a numerical method for pricing foreign exchange (FX) options in a model which deals with stochastic interest rates and stochastic volatility of the FX rate. The model considers four stochastic drivers, each represented by an Itô's diffusion with time-dependent drift, and with a full matrix of correlations. It is known that prices of FX options in this model can be found by solving an associated backward partial differential equation (PDE). However, it contains non-affine terms, which makes its difficult to solve it analytically. Also, a standard approach of solving it numerically by using traditional finite-difference (FD) or finite elements (FE) methods suffers from the high computational burden. Therefore, in this paper a flavor of a localized radial basis functions (RBFs) method, RBF-FD, is developed which allows for a good accuracy at a relatively low computational cost. Results of numerical simulations are presented which demonstrate efficiency of such an approach in terms of both performance and accuracy for pricing FX options and computation of the associated Greeks.

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A primitive model for stochastic regular‐impulse population control and its application to ecological problems

et al. 2019

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A primitive mathematical model for population control in natural environment is formulated under a unified regular‐impulse stochastic control formalism. This kind of mathematical models, although they are candidates of the models of population control, seem not to be well studied. In this paper, the impulse control uses deteriorating items and is scheduled only at predetermined times as in many management problems. Finding the optimal control reduces to solving a system of recursive Hamilton‐Jacobi‐Bellman equations. Mathematical analysis on the system provides a viscosity characterization of its solution. The model is applied to management problems of the predatory bird and harmful benthic alga, both of which are concerns in recent inland fisheries. We show that the optimal regular controls for the two problems have qualitatively different behavior with each other, which is due to different interactions between the regular and the impulse controls.

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Convergence of a finite volume element method for a generalized Black-Scholes equation transformed on finite interval

Cited by 12 publications

References 13 publications

Viscosity solution of a Delta Greek nonlinear Black-Scholes equation

Viscosity solution of a Delta Greek nonlinear Black-Scholes equation

Pricing foreign exchange options under stochastic volatility and interest rates using an RBF–FD method

A primitive model for stochastic regular‐impulse population control and its application to ecological problems

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