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- ItemDG Method for Pricing European Options under Merton Jump-Diffusion Model(Springer Heidelberg, 2019) Hozman, Jiří; Tichý, Tomáš; Vlasák, Miloslav
Show more Under 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.Show more - ItemDual Focus on Systemic Risk in Portfolio Management(Technická Univerzita v Liberci, ) Neděla, David; Tichý, Tomáš; Ekonomická fakulta
Show more In this paper, we examine a complex portfolio selection strategy with a dual emphasis on systemic risk. This strategy or only its elements are advisable for both portfolio managers as well as macroprudential regulators. In particular, first, we present the concept of an early warning system (alarm) employing selected entropy measures, which allow us to detect systemic risk in financial markets. Secondly, we apply the two-phase optimization framework to determine the optimal composition of the portfolio. Essentially, the first phase of this strategy includes the reward‒risk ratio maximization part and the following phase aims at systematic risk minimization. Furthermore, we approximate the returns using a dynamic set of components obtained from the principal component analysis and the classical ordinary least squares regression. In the empirical analysis using US market data, the wealth paths and statistics of different portfolio strategies are compared with each other. Ex-post results confirm higher profitability of the early warning system with double optimization, even if the transaction costs are taken into account. However, the main benefit lies in the significantly better risk properties of the proposed strategy.Show more - ItemNumerical Pricing of Options under the Exponential Ornstein-Uhlenbeck Stochastic Volatility Model based on a DG Technique(2018) Hozman, Jiří; Tichý, Tomáš
Show more Stochastic 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.Show more - 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, Tomáš
Show more More 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.Show more - ItemOption pricing with simulation of fuzzy stochastic variables(Technická Univerzita v Liberci, 2013-08) Holčapek, Michal; Tichý, Tomáš; Ekonomická fakulta
Show more During last decades the stochastic simulation approach, both via MC and QMC has been vastly applied and subsequently analyzed in almost all branches of science. Very nice applications can be found in areas that rely on modeling via stochastic processes, such as finance. However, since financial quantities as opposed to natural processes depend on human activity, their modeling is often very challenging. Many scholars therefor suggest to specify some parts of financial models by means of fuzzy set theory. Many financial problems, such as pricing and hedging of specific financial derivatives, are too complex to be solved analytically even in a crisp case, it can be efficient to apply (Quasi) Monte Carlo simulation. In this contribution a recent knowledge of fuzzy numbers and their approximation is utilized in order to suggest fuzzy-MC simulation to option price modeling in terms of fuzzy-random variables. In particular, we suggest three distinct fuzzy-random processes as an alternative to a standard crisp model. Application possibilities are shown on illustrative examples assuming a plain vanilla European put option under Brownian motion with fuzzy parameter (volatility), Brownian motion with fuzzy subordinator and Brownian motion with fuzzyfied subordinator. In each case the model result into a whole set of prices – thus, since we assume one of the input data as LU fuzzy number, we get the price in terms of the LU fuzzy number as well. The payoff function of analyzed put option can be obviously replaced by more complex payoff structure.Show more - ItemThe discontinuous Galerkin method for discretely observed Asian options(John Wiley and Sons Ltd, 2020) Hozman, Jiří; Tichý, Tomáš
Show more Asian 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.Show more