Browsing by Author "Hozman, Jiří"
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- ItemAnalysis and application of the discontinuous Galerkin method to the RLW equation(Springer International Publishing Ag, 2013) Hozman, Jiří; Lamač, JanIn this work, our main purpose is to develop of a sufficiently robust, accurate and efficient numerical scheme for the solution of the regularized long wave (RLW) equation, an important partial differential equation with quadratic nonlinearity, describing a large number of physical phenomena. The crucial idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semi-discretization and the backward difference formula for the time discretization. Furthermore, a suitable linearization preserves a linear algebraic problem at each time level. We present error analysis of the proposed scheme for the case of nonsymmetric discretization of the dispersive term. The appended numerical experiments confirm theoretical results and investigate the conservative properties of the RLW equation related to mass, momentum and energy. Both procedures illustrate the potency of the scheme consequently.
- ItemDG Method for Pricing European Options under Merton Jump-Diffusion Model(Springer Heidelberg, 2019) Hozman, Jiří; Tichý, Tomáš; Vlasák, MiloslavUnder real market conditions, there exist many cases when it is inevitable to adopt numerical approximations of option prices due to non-existence of analytical formulae. Obviously, any numerical technique should be tested for the cases when the analytical solution is well known. The paper is devoted to the discontinuous Galerkin method applied to European option pricing under the Merton jump-diffusion model, when the evolution of the asset prices is driven by a Lévy process with finite activity. The valuation of options under such a model with lognormally distributed jumps requires solving a parabolic partial integro-differential equation which involves both the integrals and the derivatives of the unknown pricing function. The integral term related to jumps leads to new theoretical and numerical issues regarding the solving of the pricing equation in comparison with the standard approach for the Black-Scholes equation. Here we adopt the idea of the relatively modern technique that the integral terms in Merton-type models can be viewed as solutions of proper differential equations, which can be accurately solved in a simple way. For practical purposes of numerical pricing of options in such models we propose a two-stage implicit-explicit scheme arising from the discontinuous piecewise polynomial approximation, i.e., the discontinuous Galerkin method. This solution procedure is accompanied with theoretical results and discussed within the numerical results on reference benchmarks.
- ItemNumerical Pricing of Options under the Exponential Ornstein-Uhlenbeck Stochastic Volatility Model based on a DG Technique(2018) Hozman, Jiří; Tichý, TomášStochastic volatility models are a variance extension of the classical Black-Scholes model dynamics by introducing another auxiliary processes to model the volatility of the underlying asset returns. Here we study the pricing problem for European-style options under a one-factor stochastic volatility model when the volatility of the underlying price is governed by the exponential Ornstein-Uhlenbeck process. The problem can be formulated as a non-stationary second-order degenerate partial differential equation accompanied by initial and boundary conditions, whose analytical solutions are not available in general. Therefore, the approximate option value is obtained by a numerical procedure based on a discontinuous Galerkin technique that provides promising results. Finally, reference numerical experiments are provided with the emphasis on the behaviour of the option values with respect to the discretization parameters.
- ItemNumerical solution of the mew equation by the semi-implicit numerical scheme(Technická univerzita v Liberci, Česká republika, 2012) Hozman, JiříIn this paper we deal with the development of a numerical method for the solution of the mo- dified equal width wave (MEW) equation – a very important equation with a cubic nonlinearity describing a large number of physical phenomena. The crucial idea of introduced approach is based on the discretization of the MEW equation with the aid of a combination of the discontin- uous Galerkin (DG) method for the space semi-discretization and the backward Euler method for the time discretization. The appended numerical experiments investigate the conservative properties of the MEW equation related to mass, momentum and energy, and illustrate the po- tency of this scheme, consequently.
- ItemOption Pricing under the Kou Jump-Diffusion Model: a DG Approach(AMER INST PHYSICS, 2 HUNTINGTON QUADRANGLE, STE 1NO1, MELVILLE, NY 11747-4501 USA, 2019) Hozman, Jiří; Tichý, TomášMore empiricism in modelling of option contracts is obtained when the jump-diffusion models are employed. Such models extend the standard Black-Scholes framework by adding jumps to the dynamics of underlying asset prices and enable to describe large and sudden changes in the underlying. The paper is devoted to the discontinuous Galerkin method applied to European option pricing under the Kou model where jump sizes are double exponentially distributed. The pricing function satisfies a partial integro-differential equation, which involves both integrals and derivatives of an unknown option value function. With a localization to a bounded spatial domain, the governing equation is discretized by the discontinuous Galerkin method over a finite element mesh and it is integrated in temporal variable by a semi-implicit Euler scheme, where the differential part is treated implicitly while the integral one explicitly by the composite trapezoidal rule. This approach thus leads to a sparse linear algebraic system at each time level. Finally, numerical results demonstrate the capability of the scheme presented within the reference benchmarks.
- ItemThe discontinuous Galerkin method for discretely observed Asian options(John Wiley and Sons Ltd, 2020) Hozman, Jiří; Tichý, TomášAsian options represent an important subclass of the path-dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black-Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.