arXiv subjects

Daniel Sevcovic

Publications and source records attributed to Daniel Sevcovic.

At least 37 records · Page 2Linked to original sources

arXiv2018

Expected Utility Maximization and Conditional Value-at-Risk Deviation-based Sharpe Ratio in Dynamic Stochastic Portfolio Optimization

In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation ($CVaRD$) based Sharpe ratio for measuring risk-adjusted performance of a dynamic portfolio. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index and we evaluate and analyze the dependence of the $CVaRD$-based Sharpe ratio on the utility function and the associated risk aversion level.

q-fin.PM↗

arXiv2018

Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function

In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for 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 relaxation 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.

q-fin.CP↗

arXiv2018

On Construction of Upper and Lower Bounds for the HOMO-LUMO Spectral Gap

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.

math.OC↗

arXiv2017

Pricing Perpetual Put Options by the Black-Scholes Equation with a Nonlinear Volatility Function

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.

q-fin.MF↗

arXiv2017

On a Construction of Integrally Invertible Graphs and their Spectral Properties

Godsil (1985) defined a graph to be invertible if it has a non-singular adjacency matrix whose inverse is diagonally similar to a nonnegative integral matrix; the graph defined by the last matrix is then the inverse of the original graph. In this paper we call such graphs positively invertible and introduce a new concept of a negatively invertible graph by replacing the adjective `nonnegative' by `nonpositive in Godsil's definition; the graph defined by the negative of the resulting matrix is then the negative inverse of the original graph. We propose new constructions of integrally invertible graphs (those with non-singular adjacency matrix whose inverse is integral) based on an operation of `bridging' a pair of integrally invertible graphs over subsets of their vertices, with sufficient conditions for their positive and negative invertibility. We also analyze spectral properties of graphs arising from bridging and derive lower bounds for their least positive eigenvalue. As an illustration we present a census of graphs with a unique 1-factor on $m\le 6$ vertices and determine their positive and negative invertibility.

math.CO↗

arXiv2017

Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper 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.

q-fin.PR↗

arXiv2017

Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

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.

q-fin.CP↗

arXiv2016

Invertibility of graphs with a unique perfect matching

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 graphs exhibit properties like selfinvertibility which cannot be observed within the class of positively invertible non-bipartite graphs with a unique perfect matching.

math.CO↗

arXiv2016

Maximization of the Spectral Gap for Chemical Graphs by means of a Solution to a Mixed Integer Semidefinite Program

In this paper we analyze the spectral gap of a weighted graph which is the difference between the smallest positive and largest negative eigenvalue of its adjacency matrix. Such a graph can represent e.g. a chemical organic molecule. Our goal is to construct a new graph by bridging two given weighted graphs over a bipartite graph. The aim is to maximize the spectral gap with respect to a bridging graph. To this end, we construct a mixed integer semidefinite program for maximization of the spectral gap and compute it numerically.

math.OC↗

arXiv2016

Numerical and analytical methods for bond pricing in short rate convergence models of interest rates

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 stochastic European short rate. In short rate models, the bond prices, which determine the term structure of interest rate, are obtained as solutions to partial differential equations. Analytical solutions are available only in special cases; therefore we consider the question of obtaining their approximations. We use both analytical and numerical methods to get an approximate solution to the partial differential equation for bond prices.

q-fin.MF↗

arXiv2016

Analysis of the nonlinear option pricing model under variable transaction costs

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 of the option price. We show existence of a classical smooth solution and prove useful bounds on the option prices. Furthermore, we construct an effective numerical scheme for approximation of the solution. The solutions are obtained by means of the efficient numerical discretization scheme of the Gamma equation. Several computational examples are presented.

q-fin.PR↗

arXiv2015

Comparison of the analytical approximation formula and Newton's method for solving a class of nonlinear Black-Scholes parabolic equations

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 equations including, in particular, the market illiquidity model and the risk-adjusted pricing model. Accuracy and time complexity of both numerical methods are compared. Furthermore, market quotes data was used to calibrate model parameters.

q-fin.PR↗

arXiv2015

Application of the Enhanced Semidefinite Relaxation Method to Construction of the Optimal Anisotropy Function

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 anisotropy function. The idea is to construct an optimal anisotropy function as a minimizer for the anisotropic interface energy functional for a given Jordan curve in the plane. We present computational examples of resolving the optimal anisotropy function. The examples include boundaries of real snowflakes.

math.OC↗

arXiv2014

Solution to the Inverse Wulff Problem by Means of the Enhanced Semidefinite Relaxation Method

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 obtained by a semidefinite relaxation of the original problem augmented by quadratic-linear constraints. We show that the sequence of finite dimensional approximations of the optimal anisoperimetric ratio converges to the optimal anisoperimetric ratio which is a solution to the inverse Wulff problem. Several computational examples, including those corresponding to boundaries of real snowflakes and discussion on the rate of convergence of numerical method are also presented in this paper.

math.OC↗

arXiv2014

Computational studies of conserved mean-curvature flow

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 are presented as well.

math.NA↗

arXiv2013

Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical Hölder smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method.

q-fin.PM↗

arXiv2013

On a gradient flow of plane curves minimizing the anisoperimetric ratio

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 energies corresponding to different anisotropy functions. Finally, we present several computational examples illustrating theoretical results.

math.DG↗

arXiv2012

On traveling wave solutions to Hamilton-Jacobi-Bellman equation with inequality constraints

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.

q-fin.PM↗