This paper discusses how to obtain the Black-Scholes equation to evaluate options and how to obtain explicit solutions for Call and Put. The Black-Scholes equation, which is the basis for determining explicit solutions for Call and Put, is a rather sophisticated equation. It is a partial differential equation of the second order, parabolic, similar to the heat equation. The terms of the equation express diffusion in a homogeneous environment, convection and reaction. The main objective of the paper is to present the Black-Scholes methodology and apply this methodology on the underlying asset of the nature of the listed stock on the Bucharest Stock Exchange. Also, a secondary objective is to compare the results obtained in this paper with our results in an article where we determined the values for Call and Put by Monte Carlo simulation.
Category - Vasile BRĂTIAN
Lucian Blaga University Sibiu, Romania
In the present paper, there are presented, theoretical and applicative, two issues: the evaluation of the European options using the Monte Carlo method and the measurement of the entropy of information for the price of the underlying asset of the option. The underlying asset used in our analyses is the share of Compa SA. Through Monte Carlo simulations, scenarios are created on the random evolution of the underlying asset, and the valuation of the option on the underlying asset is made using the Feynman-Kač theorem. The distribution we use is lognormal. Also, in the paper is measured the entropy of information of Shannon type. The measurement of the entropy of information of the stock market price of the underlying asset is calculated annually, considering the stock market price in this case as a discreet random variable.
Portfolio Optimization. Application of the Markowitz Model Using Lagrange and Profitability Forecast
This paper presents the theoretical and applicative model elaborated by Harry Markowitz on the determination of the structure of the efficient securities portfolio. In this sense, in order to determine the structure of the efficient Markowitz portfolio (PE), a Lagrange function is built and minimized. Also, on the basis of the results obtained from the analysis, the profitability of the portfolio is modeled continuous time and determines the range of values in which it can be found over one year after the analysis period. The data used in our analysis are shares of financial investment companies (SIF), traded on the Bucharest Stock Exchange, and the distribution used in the analysis is lognormal. The structure of the portfolio obtained through the Markowitz model can be compared to the structure of the portfolio obtained through the Sharpe model from a previous article titled ”Portfolio optimization - application of Sharpe model using Lagrange” (Brătian, 2017).