Valuing barrier options using the adaptive discontinuous Galerkin method

summary:This paper is devoted to barrier options and the main objective is to develop a sufficiently robust, accurate and efficient method for computation of their values driven according to the well-known Black-Scholes equation. The main idea is based on the discontinuous Galerkin method together with a spatial adaptive approach. This combination seems to be a promising technique for the solving of such problems with discontinuous solutions as well as for consequent optimization of the number of degrees of freedom and computational cost. The appended numerical experiment illustrates the potency of the proposed numerical scheme

DGM for real options valuation: Options to change operating scale

summary:The real options approach interprets a flexibility value, embedded in a project, as an option premium. The object of interest is to valuate real options to change operating scale, typical for natural resources industry. The evolution of the project as well as option prices is decribed by partial differential equations of the Black-Scholes type, linked through a payoff function given by a type of the flexibility provided. The governing equations are discretized by the discontinuous Galerkin method over a finite element mesh and they are integrated in temporal variable by an implicit Euler scheme. The special attention is paid to the treatment of early exercise feature that is handled by additional penalty term. The capabilities of the approach presented are documented on the selected individual real options from the reference experiments using real market data

Analysis and application of the discontinuous Galerkin method to the RLW equation

In 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.ESF [CZ.1.07/2.3.00/09.0155]; SGS Project 'Modern numerical methods'; TU Libere

Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

SEMIAUTOMATIC DETECTION OF STENOSIS AND OCCLUSION OF PULMONARY ARTERIES FOR PATIENTS WITH CHRONIC THROMBOEMBOLIC PULMONARY HYPERTENSION

Chronic thromboembolic pulmonary hypertension (CTEPH) is a severe lung disease defined by the presence of chronic blood clots in the pulmonary arteries accompanied by severe health complications. It is necessary to go through a large set of axial sections from Computed tomography pulmonary angiogram (CTPA) for diagnosing the disease, which is difficult and time consuming for the radiologist. The radiologist's experience plays a significant role, same as subjective factors such as attention and fatigue. In this work we pursued the design and development of the algorithm for semiautomatic detection of pulmonary artery stenoses and clots for diagnosing CTEPH, which is based on the implementation of semantic segmentation using deep convolutional neural networks. Specifically, it is about the use of the DeepLab V3 + model embedded in the Xception architecture. Within this work we focused on stenoses and clots located in larger pulmonary arteries. Anonymized data of patients diagnosed with CTEPH and one healthy patient in the term of the presence of the disease were used for realization of this work. Statistical analysis of the results is divided into two parts: analysis of the created algorithm based on comparison of outputs with ground truth data (manually marked references) and analysis of pathology detection on new data based on comparison of predictions with reference images from the radiologist. The proposed algorithm correctly detects present vascular pathology in 83% of cases (sensitivity) and precisely selects cases where the investigated pathology does not occur in 72% of cases (specificity). The calculated Matthews correlation coefficient is 0.53. This means that the predictive ability of the algorithm is moderate positive. The designed and developed image analysis algorithm offers the radiologist a "second opinion" and it also could enable to increase the sensitivity of CTEPH diagnostics in cooperation with a radiologist.

Numerical pricing of American options on extrema with continuous sampling

One of the typical option classes is formed by lookback options whose values depend also on the extrema of the underlying asset over a certain period of time. Moreover, incorporating the American constraint, which admits early exercise, has increased the popularity of these hedging and speculation instruments over recent years. In this paper, we consider the problem of pricing continuously observed American-style lookback options with fixed strike. Since no analytic formulae exist for this case, we follow an approach that formulates the corresponding option pricing problem as the parabolic partial differential inequality subject to a constraint, handled by a penalty technique. As a result, we obtain the pricing equation restricted to a triangular domain, where the path-dependent variable appears as a parameter only in the initial and boundary conditions. The contribution of the paper lies in the proposal of a numerical scheme that solves this option pricing problem. The numerical technique proposed arises from the dis- continuous Galerkin that enables easy implementation of penalties and weak enforcement of boundary conditions. Finally, the capabilities of the numerical scheme are demonstrated within a simple empirical study on the reference experiments

SYSTEM FOR MEASURING KINEMATICS OF VESTIBULAR SYSTEM MOVEMENTS IN NEUROLOGICAL PRACTICE

The article deals with the design of a system for studying kinematics of movement of the vestibular system. Up to now there has not existed a system which would enable to measure the kinematic quantities of movement of the individual parts of the vestibular system within its coordinate system. The proposed system removes these deficiencies by suitable positioning of five gyro-accelerometric units on the helmet. The testing of the system took place under two conditions, during Unilateral Rotation on Barany Chair and Head Impulse Test. During the testing, the system justified its application because the results show that the kinematic quantities of the movement of the left and right labyrinths of the vestibular system differ. The introduced device is mainly intended for application in clinical neurology with the aim to enable the physician to measure all linear and angular accelerations of the vestibular system during medical examinations