Mariyan Milev

Radial basis functions with application to finance: American put option under jump diffusion

Abstract In this paper, we consider a partial integro-differential equation (PIDE) problem with a free boundary, arising in an American option model when the stock price follows a diffusion process with jump components. We use a front-fixing transformation of the underlying asset variable to fix the free boundary conditions and approximate the integral term by the Laguerre polynomials. We use the Radial basis functions (RBF) method to achieve an implicit nonlinear system of first order equations and apply the Crank–Nicholson scheme. We apply the Predictor–Corrector method, to deal with the system of nonlinear equations. The proposed method is stable and the results are in agreement with those obtained by other numerical methods in literature.

Valuation of exotic options in the framework of Levy processes

In this paper we explore a straightforward procedure to price derivatives by using the Monte Carlo approach when the underlying process is a jump-diffusion. We have compared the Black-Scholes model with one of its extensions that is the Merton model. The latter model is better in capturing the market’s phenomena and is comparative to stochastic volatility models in terms of pricing accuracy. We have presented simulations of asset paths and pricing of barrier options for both Geometric Brownian motion and exponential Levy processes as it is the concrete case of the Merton model. A desired level of accuracy is obtained with simple computer operations in MATLAB for efficient computational time.

The 'core' of symmetric homogeneous polynomial inequalities of degree four of three real variables

Abstract In this paper we explore inequalities between symmetric homogeneous polynomials of degree four of three real variables and three nonnegative real variables. The main theorems describe the cases in which the smallest possible coefficient is not expressed by the other coefficients. The problem is resolved by introducing a parametric representation.

Exponential mean-square stability of two classes of theta Milstein methods for stochastic delay differential equations

This paper establishes exponential mean square stability of two classes of theta Milstein methods, namely split-step theta Milstein (SSTM) method and stochastic theta Milstein (STM) method, for stochastic differential delay equations (SDDEs). We consider the SDDEs problem under a coupled monotone condition on drift and diffusion coefficients, as well as a necessary linear growth condition on the last term of theta Milstein method. It is proved that the SSTM method with θ ∈ [0, ½] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize, but for θ ∈ (½, 1], we proved that the stability results hold for any stepsize. Then, based on the stability results of SSTM method, we examine the exponential mean square stability of the STM method and obtain the similar stability results to that of the SSTM method. In the numerical section the figures show thevalidity of our claims.This paper establishes exponential mean square stability of two classes of theta Milstein methods, namely split-step theta Milstein (SSTM) method and stochastic theta Milstein (STM) method, for stochastic differential delay equations (SDDEs). We consider the SDDEs problem under a coupled monotone condition on drift and diffusion coefficients, as well as a necessary linear growth condition on the last term of theta Milstein method. It is proved that the SSTM method with θ ∈ [0, ½] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize, but for θ ∈ (½, 1], we proved that the stability results hold for any stepsize. Then, based on the stability results of SSTM method, we examine the exponential mean square stability of the STM method and obtain the similar stability results to that of the SSTM method. In the numerical section the figures show thevalidity of our claims.

Efficient option valuation of single and double barrier options

In this paper we present an implementation of pricing algorithm for single and double barrier options using Mellin transformation with Maximum Entropy Inversion and its suitability for real-world applications. A detailed analysis of the applied algorithm is accompanied by implementation in C++ that is then compared to existing solutions in terms of efficiency and computational power. We then compare the applied method with existing closed-form solutions and well known methods of pricing barrier options that are based on finite differences.

Grouping of Bulgarian wines according to grape variety by using statistical methods

68 different types of Bulgarian wines were studied in accordance with 9 optical parameters as follows: color parameters in XYZ and SIE Lab color systems, lightness, Hue angle, chroma, fluorescence intensity and emission wavelength. The main objective of this research is using hierarchical cluster analysis to evaluate the similarity and the distance between examined different types of Bulgarian wines and their grouping based on physical parameters. We have found that wines are grouped in clusters on the base of the degree of identity between them. There are two main clusters each one with two subclusters. The first one contains white wines and Sira, the second contains red wines and rose. The results from cluster analysis are presented graphically by a dendrogram. The other statistical technique used is factor analysis performed by the Method of Principal Components (PCA). The aim is to reduce the large number of variables to a few factors by grouping the correlated variables into one factor and subdividin...

Usage of K-cluster and factor analysis for grouping and evaluation the quality of olive oil in accordance with physico-chemical parameters

25 olive oils were studied– different in origin and ways of extraction, in accordance with 17 physico-chemical parameters as follows: color parameters – a and b, light, fluorescence peaks, pigments – chlorophyll and β-carotene, fatty-acid content. The goals of the current study were: Conducting correlation analysis to find the inner relation between the studied indices; By applying factor analysis with the help of the method of Principal Components (PCA), to reduce the great number of variables into a few factors, which are of main importance for distinguishing the different types of olive oil;Using K-means cluster to compare and group the tested types olive oils based on their similarity. The inner relation between the studied indices was found by applying correlation analysis. A factor analysis using PCA was applied on the basis of the found correlation matrix. Thus the number of the studied indices was reduced to 4 factors, which explained 79.3% from the entire variation. The first one unified the colo...

Using real options analysis to support strategic management decisions

Decision making is a complex process that requires taking into consideration multiple heterogeneous sources of uncertainty. Standard valuation and financial analysis techniques often fail to properly account for all these sources of risk as well as for all sources of additional flexibility. In this paper we explore applications of a modified binomial tree method for real options analysis (ROA) in an effort to improve decision making process. Usual cases of use of real options are analyzed with elaborate study on the applications and advantages that company management can derive from their application. A numeric results based on extending simple binomial tree approach for multiple sources of uncertainty are provided to demonstrate the improvement effects on management decisions.