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Sep 19, 2013
09/13

by
Daniel Sevcovic

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The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. Results of numerical computation of the early exercise boundary for various nonlinear Black--Scholes equations are also presented.

Source: http://arxiv.org/abs/1009.5973v2

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Sep 23, 2013
09/13

by
Daniel Sevcovic

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The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a...

Source: http://arxiv.org/abs/0710.5301v1

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Sep 22, 2013
09/13

by
Daniel Sevcovic

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The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonlinear Black-Scholes equations with a volatility coefficient which can be a nonlinear function of the second...

Source: http://arxiv.org/abs/0805.0611v1

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8.0

Jun 29, 2018
06/18

by
Daniel Sevcovic; Magdalena Zitnanska

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In this paper we analyze a nonlinear Black--Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price $V$ is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the classical Black--Scholes model can be analyzed by means of transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for the second derivative...

Topics: Quantitative Finance, Pricing of Securities

Source: http://arxiv.org/abs/1603.03874

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Jun 30, 2018
06/18

by
Daniel Sevcovic; Maria Trnovska

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We propose a novel method of resolving the optimal anisotropy function. The idea is to construct the optimal anisotropy function as a solution to the inverse Wulff problem, i.e. as a minimizer for the anisoperimetric ratio for a given Jordan curve in the plane. It leads to a nonconvex quadratic optimization problem with linear matrix inequalities. In order to solve it we propose the so-called enhanced semidefinite relaxation method which is based on a solution to a convex semidefinite problem...

Topics: Mathematics, Optimization and Control

Source: http://arxiv.org/abs/1402.5668

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Sep 23, 2013
09/13

by
Karol Mikula; Daniel Sevcovic

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We discuss the role of tangential stabilization in a curvature driven flow of planar curves. The governing system of nonlinear parabolic equations includes a nontrivial tangential velocity functional yielding a uniform redistribution of grid points along the evolving family of curves preventing numerically computed curves from forming various instabilities.

Source: http://arxiv.org/abs/0710.5314v1

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6.0

Jun 29, 2018
06/18

by
Sona Pavlikova; Daniel Sevcovic

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In this paper we investigate invertibility of graphs with a unique perfect matching, i.e. graphs having a unique 1-factor. We recall the new notion of the so-called negatively invertible graphs investigated by the authors in the recent paper. It is an extension of the classical definition of an inverse graph due to Godsil. We characterize all graphs with a unique perfect matching on $m\le 6$ vertices with respect to their positive and negative invertibility. We show that negatively invertible...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1612.02234

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Sep 20, 2013
09/13

by
Beata Stehlikova; Daniel Sevcovic

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We analyze analytic approximation formulae for pricing zero-coupon bonds in the case when the short-term interest rate is driven by a one-factor mean-reverting process with a volatility nonlinearly depending on the interest rate itself. We derive the order of accuracy of the analytical approximation due to Choi and Wirjanto. We furthermore give an explicit formula for a higher order approximation and we test both approximations numerically for a class of one-factor interest rate models.

Source: http://arxiv.org/abs/0802.3039v2

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Sep 19, 2013
09/13

by
Daniel Sevcovic; Shigetoshi Yazaki

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We study evolution of a closed embedded plane curve with the normal velocity depending on the curvature, the orientation and the position of the curve. We propose a new method of tangential redistribution of points by curvature adjusted control in the tangential motion of evolving curves. The tangential velocity distributes grid points along the curve not only uniform but also lead to a suitable concentration and/or dispersion depending on the curvature. Our study is based on solutions to the...

Source: http://arxiv.org/abs/1009.2588v2

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Sep 21, 2013
09/13

by
Zuzana Macova; Daniel Sevcovic

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The main purpose of this paper is to analyze solutions to a fully nonlinear parabolic equation arising from the problem of optimal portfolio construction. We show how the problem of optimal stock to bond proportion in the management of pension fund portfolio can be formulated in terms of the solution to the Hamilton-Jacobi-Bellman equation. We analyze the solution from qualitative as well as quantitative point of view. We construct useful bounds of solution yielding estimates for the optimal...

Source: http://arxiv.org/abs/0905.0155v2

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Sep 17, 2013
09/13

by
Tomas Bokes; Daniel Sevcovic

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In this paper we generalize and analyze the model for pricing American-style Asian options due to (Hansen and Jorgensen 2000) by including a continuous dividend rate $q$ and a general method of averaging of the floating strike. We focus on the qualitative and quantitative analysis of the early exercise boundary. The first order Taylor series expansion of the early exercise boundary close to expiry is constructed. We furthermore propose an efficient numerical algorithm for determining the early...

Source: http://arxiv.org/abs/0912.1321v1

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Jul 20, 2013
07/13

by
Daniel Sevcovic; Shigetoshi Yazaki

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We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of the curve. In contrast to the gradient flow for the isoperimetric ratio, we show there exist initial curves for which the enclosed area is decreasing with respect to time. We also derive a mixed anisoperimetric inequality for the product of total interfacial...

Source: http://arxiv.org/abs/1203.2238v2

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Sep 22, 2013
09/13

by
Daniel Sevcovic; Martin Takac

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In this paper we analyze American style of floating strike Asian call options belonging to the class of financial derivatives whose payoff diagram depends not only on the underlying asset price but also on the path average of underlying asset prices over some predetermined time interval. The mathematical model for the option price leads to a free boundary problem for a parabolic partial differential equation. Applying fixed domain transformation and transformation of variables we develop an...

Source: http://arxiv.org/abs/1101.3071v1

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Sep 20, 2013
09/13

by
Martin Lauko; Daniel Sevcovic

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In this paper we present qualitative and quantitative comparison of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of the American put option paying zero dividends. First we analyze their asymptotic behavior close to expiration. In the second part of the paper, we introduce a new numerical scheme for computing the entire early exercise boundary. The local iterative numerical scheme is based on a solution to a nonlinear integral...

Source: http://arxiv.org/abs/1002.0979v2

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6.0

Jun 30, 2018
06/18

by
Daniel Sevcovic; Maria Trnovska

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In this paper we propose and apply the enhanced semidefinite relaxation technique for solving a class of non-convex quadratic optimization problems. The approach is based on enhancing the semidefinite relaxation methodology by complementing linear equality constraints by quadratic-linear constrains. We give sufficient conditions guaranteeing that the optimal values of the primal and enhanced semidefinite relaxed problems coincide. We apply this approach to the problem of resolving the optimal...

Topics: Mathematics, Optimization and Control

Source: http://arxiv.org/abs/1405.4382

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45

Sep 21, 2013
09/13

by
Karol Mikula; Daniel Sevcovic

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In this review paper we present a stable Lagrangian numerical method for computing plane curves evolution driven by the fourth order geometric equation. The numerical scheme and computational examples are presented.

Source: http://arxiv.org/abs/0905.1433v1

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Sep 21, 2013
09/13

by
Naoyuki Ishimura; Daniel Sevcovic

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The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct monotone traveling wave solutions and identify parametric regions for which the traveling wave solution has a positive or negative wave speed.

Source: http://arxiv.org/abs/1108.1035v3

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6.0

Jun 30, 2018
06/18

by
Miroslav Kolar; Michal Benes; Daniel Sevcovic

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The paper presents the results of numerical solution of the evolution law for the constrained mean-curvature flow. This law originates in the theory of phase transitions for crystalline materials and describes the evolution of closed embedded curves with constant enclosed area. It is reformulated by means of the direct method into the system of degenerate parabolic partial differential equations for the curve parametrization. This system is solved numerically and several computational studies...

Topics: Mathematics, Numerical Analysis

Source: http://arxiv.org/abs/1402.6917

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Sep 20, 2013
09/13

by
Harald Garcke; Yoshihito Kohsaka; Daniel Sevcovic

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In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove...

Source: http://arxiv.org/abs/0802.3036v1

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Jun 29, 2018
06/18

by
Zuzana Buckova; Beata Stehlikova; Daniel Sevcovic

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In this survey paper we discuss recent advances on short interest rate models which can be formulated in terms of a stochastic differential equation for the instantaneous interest rate (also called short rate) or a system of such equations in case the short rate is assumed to depend also on other stochastic factors. Our focus is on convergence models, which explain the evolution of interest rate in connection with the adoption of Euro currency. Here, the domestic short rate depends on a...

Topics: Mathematical Finance, Quantitative Finance

Source: http://arxiv.org/abs/1607.04968

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56

Sep 23, 2013
09/13

by
Karol Mikula; Daniel Sevcovic; Martin Balazovjech

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A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the...

Source: http://arxiv.org/abs/0810.1745v2

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56

Sep 23, 2013
09/13

by
Vladimír Klement; Tomáš Oberhuber; Daniel Ševčovič

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We propose a new constrained level-set method for semi-automatic image segmentation. The method allows to specify which parts of the image lie inside respectively outside the segmented objects. Such an a-priori information can be expressed in terms of upper and lower constraints prescribed for the level-set function. Constraints have the same meaning as the initial seeds of the graph-cuts based methods for image segmentation. A numerical approximation scheme is based on the complementary-finite...

Source: http://arxiv.org/abs/1105.1429v1

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117

Sep 23, 2013
09/13

by
Michal Benes; Karol Mikula; Tomas Oberhuber; Daniel Sevcovic

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The main goal of this paper is to present results of comparison study for the level set and direct Lagrangian methods for computing evolution of the Willmore flow of embedded planar curves. To perform such a study we construct new numerical approximation schemes for both Lagrangian as well as level set methods based on semi-implicit in time and finite/complementary volume in space discretizations. The Lagrangian scheme is stabilized in tangential direction by the asymptotically uniform grid...

Source: http://arxiv.org/abs/0710.5305v1

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7.0

Jun 29, 2018
06/18

by
Maria do Rosario Grossinho; Yaser Kord Faghan; Daniel Sevcovic

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We investigate qualitative and quantitative behavior of a solution to the problem of pricing American style of perpetual put options. We assume the option priceWe investigate qualitative and quantitative behavior of a solution to the problem of pricing American style of perpetual put options. We assume the option price is a solution to a stationary generalized Black-Scholes equation in which the volatility may depend on the second derivative of the option price itself. We prove existence and...

Topics: Mathematical Finance, Quantitative Finance

Source: http://arxiv.org/abs/1611.00885

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7.0

Jun 28, 2018
06/18

by
Karol Duris; Shih-Hau Tan; Choi-Hong Lai; Daniel Sevcovic

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Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE based option pricing models can be described by solutions to the generalized Black-Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. Different linearization techniques such as Newton's method and analytic asymptotic approximation formula are adopted and compared for a wide class of nonlinear Black-Scholes...

Topics: Pricing of Securities, Quantitative Finance

Source: http://arxiv.org/abs/1511.05661