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Our goal is to analyze the system of Hamilton-Jacobi-Bellman equations arising in derivative securities pricing models. The European style of an option price is constructed as a difference of the certainty equivalents to the value functions solving the system of HJB equations. We introduce the transformation method for solving the penalized nonlinear partial differential equation. The transformed equation involves possibly non-constant the risk aversion function containing the negative ratio between the second and first derivatives of the utility function. Using comparison principles we derive useful bounds on the option price. We also propose a finite difference numerical discretization scheme with some computational examples.
Pedro Pólvora; Daniel Ševčovič. Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation. Journal of Risk and Financial Management 2021, 14, 399 .
AMA StylePedro Pólvora, Daniel Ševčovič. Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation. Journal of Risk and Financial Management. 2021; 14 (9):399.
Chicago/Turabian StylePedro Pólvora; Daniel Ševčovič. 2021. "Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation." Journal of Risk and Financial Management 14, no. 9: 399.
The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.
Daniel Ševčovič; Cyril Udeani. Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing. Mathematics 2021, 9, 1463 .
AMA StyleDaniel Ševčovič, Cyril Udeani. Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing. Mathematics. 2021; 9 (13):1463.
Chicago/Turabian StyleDaniel Ševčovič; Cyril Udeani. 2021. "Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing." Mathematics 9, no. 13: 1463.
In this paper, we investigate a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati transformation method. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach’s fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space in an abstract setting. Some financial applications of the proposed result are presented in one-dimensional space.
Cyril Izuchukwu Udeani; Daniel Ševčovič. Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem. Japan Journal of Industrial and Applied Mathematics 2021, 38, 693 -713.
AMA StyleCyril Izuchukwu Udeani, Daniel Ševčovič. Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem. Japan Journal of Industrial and Applied Mathematics. 2021; 38 (3):693-713.
Chicago/Turabian StyleCyril Izuchukwu Udeani; Daniel Ševčovič. 2021. "Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem." Japan Journal of Industrial and Applied Mathematics 38, no. 3: 693-713.
As the American early exercise results in a free boundary problem, in this article we add a penalty term to obtain a partial differential equation, and we also focus on an improved definition of the penalty term for American options. We replace the constant penalty parameter with a time-dependent function. The novelty and advantage of our approach consists in introducing a bounded, time-dependent penalty function, enabling us to construct an efficient, stable, and adaptive numerical approximation scheme, while in contrast, the existing standard approach to the penalisation of the American put option-free boundary problem involves a constant penalty parameter. To gain insight into the accuracy of our proposed extension, we compare the solution of the extension to standard reference solutions from the literature. This illustrates the improvement of using a penalty function instead of a penalising constant.
Anna Clevenhaus; Matthias Ehrhardt; Michael Günther; Daniel Ševčovič. Pricing American Options with a Non-Constant Penalty Parameter. Journal of Risk and Financial Management 2020, 13, 1 .
AMA StyleAnna Clevenhaus, Matthias Ehrhardt, Michael Günther, Daniel Ševčovič. Pricing American Options with a Non-Constant Penalty Parameter. Journal of Risk and Financial Management. 2020; 13 (6):1.
Chicago/Turabian StyleAnna Clevenhaus; Matthias Ehrhardt; Michael Günther; Daniel Ševčovič. 2020. "Pricing American Options with a Non-Constant Penalty Parameter." Journal of Risk and Financial Management 13, no. 6: 1.
In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black–Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Lévy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Lévy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Lévy stochastic processes. We also present numerical results and comparison of option prices for various Lévy stochastic processes modelling underlying asset dynamics.
José M. T. S. Cruz; Daniel Ševčovič. On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models. Japan Journal of Industrial and Applied Mathematics 2020, 37, 697 -721.
AMA StyleJosé M. T. S. Cruz, Daniel Ševčovič. On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models. Japan Journal of Industrial and Applied Mathematics. 2020; 37 (3):697-721.
Chicago/Turabian StyleJosé M. T. S. Cruz; Daniel Ševčovič. 2020. "On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models." Japan Journal of Industrial and Applied Mathematics 37, no. 3: 697-721.
In this paper, we investigate a system of geometric evolution equations describing a curvature‐driven motion of a family of planar curves with mutual interactions that can have local as well as nonlocal character, and the entire curve may influence evolution of other curves. We propose a direct Lagrangian approach for solving such a geometric flow of interacting curves. We prove local existence, uniqueness, and continuation of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. A numerical solution to the governing system has been constructed by means of the method of flowing finite volumes. We also discuss various applications of the motion of interacting curves arising in nonlocal geometric flows of curves as well as an interesting physical problem of motion of two interacting dislocation loops in the material science.
Michal Beneš; Miroslav Kolář; Daniel Ševčovič. Curvature driven flow of a family of interacting curves with applications. Mathematical Methods in the Applied Sciences 2020, 1 .
AMA StyleMichal Beneš, Miroslav Kolář, Daniel Ševčovič. Curvature driven flow of a family of interacting curves with applications. Mathematical Methods in the Applied Sciences. 2020; ():1.
Chicago/Turabian StyleMichal Beneš; Miroslav Kolář; Daniel Ševčovič. 2020. "Curvature driven flow of a family of interacting curves with applications." Mathematical Methods in the Applied Sciences , no. : 1.
In this article we propose new linear and nonlinear partial differential equations (PDEs) models for pricing American options and total value adjustment in the presence of counterparty risk. An innovative aspect comes from the consideration of stochastic spreads, which increases the dimension of the problem. In this setting, we pose new complementarity problems associated to linear and nonlinear PDEs. Moreover, using the mathematical tools of semilinear variational inequalities for parabolic equations, we prove the existence and uniqueness of a solution for these models. For the numerical solution, we mainly combine a semi-Lagrangian time discretization scheme, a fixed point method to cope with nonlinear terms and a finite element method for the spatial discretization, jointly with an augmented Lagrangian active set method to solve the fully discretized complementarity problem. Finally, numerical examples illustrate the expected behaviour of the option prices and the corresponding total value adjustment, as well as the performance of the proposed numerical techniques. Moreover, we compare the numerical results from the PDEs approach with those obtained by applying Monte Carlo techniques.
Iñigo Arregui; Beatriz Salvador; Daniel Ševčovič; Carlos Vázquez. PDE models for American options with counterparty risk and two stochastic factors: Mathematical analysis and numerical solution. Computers & Mathematics with Applications 2019, 79, 1525 -1542.
AMA StyleIñigo Arregui, Beatriz Salvador, Daniel Ševčovič, Carlos Vázquez. PDE models for American options with counterparty risk and two stochastic factors: Mathematical analysis and numerical solution. Computers & Mathematics with Applications. 2019; 79 (5):1525-1542.
Chicago/Turabian StyleIñigo Arregui; Beatriz Salvador; Daniel Ševčovič; Carlos Vázquez. 2019. "PDE models for American options with counterparty risk and two stochastic factors: Mathematical analysis and numerical solution." Computers & Mathematics with Applications 79, no. 5: 1525-1542.
In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) equation. We propose the so-called Riccati method for transformation of the fully nonlinear HJB equation into a quasi-linear parabolic equation with non-local terms involving the intertemporal utility function. As a numerical method we propose a semi-implicit scheme in time based on a finite volume approximation in the spatial variable. By analyzing an explicit traveling wave solution we show that the numerical method is of the second experimental order of convergence. As a practical application we compute optimal strategies for a portfolio investment problem motivated by market financial data of German DAX 30 Index and show the effect of considering intertemporal utility on optimal portfolio selection.
Soňa Kilianová; Daniel Ševčovič. Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton–Jacobi–Bellman equation. Japan Journal of Industrial and Applied Mathematics 2019, 36, 497 -519.
AMA StyleSoňa Kilianová, Daniel Ševčovič. Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton–Jacobi–Bellman equation. Japan Journal of Industrial and Applied Mathematics. 2019; 36 (2):497-519.
Chicago/Turabian StyleSoňa Kilianová; Daniel Ševčovič. 2019. "Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton–Jacobi–Bellman equation." Japan Journal of Industrial and Applied Mathematics 36, no. 2: 497-519.
In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) equation. We propose the so-called Riccati method for transformation of the fully nonlinear HJB equation into a quasi-linear parabolic equation with non-local terms involving the intertemporal utility function. As a numerical method we propose a semi-implicit scheme in time based on a finite volume approximation in the spatial variable. By analyzing an explicit traveling wave solution we show that the numerical method is of the second experimental order of convergence. As a practical application we compute optimal strategies for a portfolio investment problem motivated by market financial data of German DAX 30 Index and show the effect of considering intertemporal utility on optimal portfolio selection.
Sona Kilianova; Daniel Sevcovic. Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation. 2019, 1 .
AMA StyleSona Kilianova, Daniel Sevcovic. Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation. . 2019; ():1.
Chicago/Turabian StyleSona Kilianova; Daniel Sevcovic. 2019. "Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation." , no. : 1.
In this work, we analyze a nonlinear partial differential equation (PDE) model for the total value adjustment on European options in the presence of a counterparty risk. We transform the nonlinear PDE into an equivalent one, involving a sectorial operator, and prove the existence and uniqueness of a solution. Dans ce travail, nous analysons un modèle d'équations aux dérivées partielles (EDP) non linéaires pour l'ajustement XVA d'options européennes en présence d'un risque de contrepartie. Nous transformons l'EDP non linéaire en une équation équivalente, impliquant un opérateur sectoriel, et prouvons l'existence et l'unicité de la solution.
Iñigo Arregui; Beatriz Salvador; Daniel Ševčovič; Carlos Vázquez. Mathematical analysis of a nonlinear PDE model for European options with counterparty risk. Comptes Rendus Mathematique 2019, 357, 252 -257.
AMA StyleIñigo Arregui, Beatriz Salvador, Daniel Ševčovič, Carlos Vázquez. Mathematical analysis of a nonlinear PDE model for European options with counterparty risk. Comptes Rendus Mathematique. 2019; 357 (3):252-257.
Chicago/Turabian StyleIñigo Arregui; Beatriz Salvador; Daniel Ševčovič; Carlos Vázquez. 2019. "Mathematical analysis of a nonlinear PDE model for European options with counterparty risk." Comptes Rendus Mathematique 357, no. 3: 252-257.
In this paper we investigate two non-local geometric geodesic curvature driven flows of closed curves preserving either their enclosed surface area or their total length on a given two-dimensional surface. The method is based on projection of evolved curves on a surface to the underlying plane. For such a projected flow we construct the normal velocity and the external nonlocal force. The evolving family of curves is parametrized by a solution to the fully nonlinear parabolic equation for which we derive a flowing finite volume approximation numerical scheme. Finally, we present various computational examples of evolution of the surface area and length preserving flows of surface curves. We furthermore analyse the experimental order of convergence. It turns out that the numerical scheme is of the second order of convergence.
Miroslav Kolář; Michal Beneš; Daniel Ševčovič. On Surface Area and Length Preserving Flows of Closed Curves on a Given Surface. Lecture Notes in Computational Science and Engineering 2019, 279 -287.
AMA StyleMiroslav Kolář, Michal Beneš, Daniel Ševčovič. On Surface Area and Length Preserving Flows of Closed Curves on a Given Surface. Lecture Notes in Computational Science and Engineering. 2019; ():279-287.
Chicago/Turabian StyleMiroslav Kolář; Michal Beneš; Daniel Ševčovič. 2019. "On Surface Area and Length Preserving Flows of Closed Curves on a Given Surface." Lecture Notes in Computational Science and Engineering , no. : 279-287.
Soña Pavlíková; Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control & Optimization 2019, 9, 53 -69.
AMA StyleSoña Pavlíková, Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control & Optimization. 2019; 9 (1):53-69.
Chicago/Turabian StyleSoña Pavlíková; Daniel Ševčovič. 2019. "On construction of upper and lower bounds for the HOMO-LUMO spectral gap." Numerical Algebra, Control & Optimization 9, no. 1: 53-69.
Iñigo Arregui; Beatriz Salvador; Daniel Ševčovič; Carlos Vázquez. Total value adjustment for European options with two stochastic factors. Mathematical model, analysis and numerical simulation. Computers & Mathematics with Applications 2018, 76, 725 -740.
AMA StyleIñigo Arregui, Beatriz Salvador, Daniel Ševčovič, Carlos Vázquez. Total value adjustment for European options with two stochastic factors. Mathematical model, analysis and numerical simulation. Computers & Mathematics with Applications. 2018; 76 (4):725-740.
Chicago/Turabian StyleIñigo Arregui; Beatriz Salvador; Daniel Ševčovič; Carlos Vázquez. 2018. "Total value adjustment for European options with two stochastic factors. Mathematical model, analysis and numerical simulation." Computers & Mathematics with Applications 76, no. 4: 725-740.
In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.
Sona Pavlikova; Daniel Ševčovič. On Construction of Upper and Lower Bounds for the HOMO-LUMO Spectral Gap. 2018, 1 .
AMA StyleSona Pavlikova, Daniel Ševčovič. On Construction of Upper and Lower Bounds for the HOMO-LUMO Spectral Gap. . 2018; ():1.
Chicago/Turabian StyleSona Pavlikova; Daniel Ševčovič. 2018. "On Construction of Upper and Lower Bounds for the HOMO-LUMO Spectral Gap." , no. : 1.
We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black–Scholes equation in which the volatility function may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.
Maria Do Rosário Grossinho; Yaser Kord Faghan; Daniel Ševčovič. Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function. Asia-Pacific Financial Markets 2017, 24, 291 -308.
AMA StyleMaria Do Rosário Grossinho, Yaser Kord Faghan, Daniel Ševčovič. Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function. Asia-Pacific Financial Markets. 2017; 24 (4):291-308.
Chicago/Turabian StyleMaria Do Rosário Grossinho; Yaser Kord Faghan; Daniel Ševčovič. 2017. "Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function." Asia-Pacific Financial Markets 24, no. 4: 291-308.
Soňa Pavlíková; Daniel Ševčovič. On a construction of integrally invertible graphs and their spectral properties. Linear Algebra and its Applications 2017, 532, 512 -533.
AMA StyleSoňa Pavlíková, Daniel Ševčovič. On a construction of integrally invertible graphs and their spectral properties. Linear Algebra and its Applications. 2017; 532 ():512-533.
Chicago/Turabian StyleSoňa Pavlíková; Daniel Ševčovič. 2017. "On a construction of integrally invertible graphs and their spectral properties." Linear Algebra and its Applications 532, no. : 512-533.
This survey chapter is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.
Daniel Ševčovič. Nonlinear Parabolic Equations Arising in Mathematical Finance. Novel Methods in Computational Finance 2017, 3 -15.
AMA StyleDaniel Ševčovič. Nonlinear Parabolic Equations Arising in Mathematical Finance. Novel Methods in Computational Finance. 2017; ():3-15.
Chicago/Turabian StyleDaniel Ševčovič. 2017. "Nonlinear Parabolic Equations Arising in Mathematical Finance." Novel Methods in Computational Finance , no. : 3-15.
Miroslav Kolář; Michal BeneŠ; Daniel Ševčovič. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B 2017, 22, 3671 -3689.
AMA StyleMiroslav Kolář, Michal BeneŠ, Daniel Ševčovič. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B. 2017; 22 (10):3671-3689.
Chicago/Turabian StyleMiroslav Kolář; Michal BeneŠ; Daniel Ševčovič. 2017. "Area preserving geodesic curvature driven flow of closed curves on a surface." Discrete & Continuous Dynamical Systems - B 22, no. 10: 3671-3689.
We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on 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 including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary differential equation and implicit equation for the free boundary position. We finally present results of numerical approximation of the early exercise boundary, option price and their dependence on model parameters.
Maria Do Rosário Grossinho; Yaser Faghan Kord; Daniel Ševčovič. Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations. 2017, 1 .
AMA StyleMaria Do Rosário Grossinho, Yaser Faghan Kord, Daniel Ševčovič. Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations. . 2017; ():1.
Chicago/Turabian StyleMaria Do Rosário Grossinho; Yaser Faghan Kord; Daniel Ševčovič. 2017. "Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations." , no. : 1.
In this paper we analyze a nonlinear generalization of the Black-Scholes equation for pricing American style call option in which the volatility may depend on the underlying asset price and the Gamma of the option. We propose a novel method of pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over-relation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, we present several computational examples for the nonlinear Black-Scholes equation for pricing American style call option under presence of variable transaction costs.
Maria Do Rosário Grossinho; Yaser Faghan Kord; Daniel Ševčovič. Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function. 2017, 1 .
AMA StyleMaria Do Rosário Grossinho, Yaser Faghan Kord, Daniel Ševčovič. Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function. . 2017; ():1.
Chicago/Turabian StyleMaria Do Rosário Grossinho; Yaser Faghan Kord; Daniel Ševčovič. 2017. "Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function." , no. : 1.