Academic literature on the topic 'Fractal black-scholes model'

In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown.

In this paper an arbitrage strategy is constructed for the modified Black–Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brownian motion. Work has been done previously on this problem for the case with constant "volatility" and without a time change; here these results are extended to the case of stochastic volatility models when the modulator is fractional Brownian motion or a time change of it. (Volatility in fractional Black–Scholes models does not carry the same meaning as in the classic Black–Scholes framework, which is made clear in the text.) Since fractional Brownian motion is not a semi-martingale, the Black–Scholes differential equation is not well-defined sense for arbitrary predictable volatility processes. However, it is shown here that any almost surely continuous and adapted process having zero quadratic variation can act as an integrator over functions of the integrator and over the family of continuous adapted semi-martingales. Moreover it is shown that the integral also has zero quadratic variation, and therefore that the integral itself can be an integrator. This property of the integral is crucial in developing the arbitrage strategy. Since fractional Brownian motion and a time change of fractional Brownian motion have zero quadratic variation, these results are applicable to these cases in particular. The appropriateness of fractional Brownian motion as a means of modeling stock price returns is discussed as well.

The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

We study the fractional Black–Scholes model (FBSM) of option pricing in the fractal transmission system. In this work, we develop a full-discrete numerical scheme to investigate the dynamic behavior of FBSM. The proposed scheme implements a known L1 formula for the α-order fractional derivative and Fourier-spectral method for the discretization of spatial direction. Energy analysis indicates that the constructed discrete method is unconditionally stable. Error estimate indicates that the 2−α-order formula in time and the spectral approximation in space is convergent with order OΔt2−α+N1−m, where m is the regularity of u and Δt and N are step size of time and degree, respectively. Several numerical results are proposed to confirm the accuracy and stability of the numerical scheme. At last, the present method is used to investigate the dynamic behavior of FBSM as well as the impact of different parameters.

Black & Scholes (1973) desenvolveram um modelo para a avaliação de opções de compra e venda do tipo Europeu. Merton (1973) estendeu o modelo para ações que pagam dividendos. Muitos outros desenvolvimentos foram feitos acerca dos dois trabalhos citados, mas talvez um dos mais importantes foi proposto por Cox, Ross & Rubinstein (1979), onde o processo estocástico (para o preço da ação objeto) em tempo e estado contínuo (Movimento Geométrico Browniano) proposto por Black & Scholes foi aproximado por um processo de tempo e estado discreto (Random Walk). O modelo de Cox, Ross & Rubinstein, hoje conhecido como Modelo Binomial, tornou-se um dos métodos mais utilizados para calcular o valor de opções, principalmente opções americanas, devido a sua simplicidade e fácil implementação computacional. Mas, o modelo binomial possui uma convergência fraca, em forma oscilatória, para o valor verdadeiro. Este artigo pretende mostrar as principais soluções encontradas na literatura para acelerar a convergência.

A Hipótese de Eficiência do Mercado é uma das bases da moderna teoria de finanças e estabelece que o comportamento aleatório na variação dos preços decorre do fluxo de informações não antecipadas. Um de seus paradigmas define que a distribuição dos retornos dos preços é aleatória e normalmente distribuída. Neste artigo, avaliamos a hipótese de que o processo estocástico gerador dos retornos de diversos mercados emergentes da Ásia e das Américas segue um processo aleatório não-normal alfa-estável. Por meio de estimativas dos parâmetros da distribuição e de simulações, encontramos evidências de que esses retornos realmente seriam mais bem descritos pela distribuição alfa-estável ou distribuição fractal. Estas distribuições acomodam flutuações grandes e freqüentes de preço melhor, pois a probabilidade de perdas substanciais é maior do que a prevista por uma distribuição normal e a assimetria da distribuição é considerada. As estimativas fornecidas para modelos usuais de finanças, como os otimizadores de média e variância de Markowitz e o modelo de apreçamento de opções de Black e Scholes, podem ser mais bem estimadas por meio da distribuição alfa-estável.

La aplicación de la metodología (R/S), de la teoría de fractales, para la determinación del coeficiente Hurst, revela la posibilidad de un comportamiento de memoria larga en alguna de las variables de mercado representativas de México. Aunque bajo ciertas pruebas dichos resultados pueden resultar ser estadísticamente no significativos. A partir del Movimiento Browniano Fraccional (MBF), que es un proceso estocástico más general que el movimiento browniano tradicional, pueden modelarse procesos con persistencia o antipersistencia. Con base en este proceso, y utilizando bases matemáticas más generales, se deduce una forma más amplia de valuación de opciones europeas y la ecuación Black-Scholes, así como la ecuación general de bonos y la estructura de plazos del modelo de tasas de Vasicek, útiles en los casos en donde las series financieras muestran comportamientos de persistencia. Dichas modelaciones se aplican al caso de una variable de mercado mexicano y se obtienen resultados interesantes.

Магістерська дисертація: 94 с., 10 рис., 31 табл., 3 додатки, 21 джерело. В роботі розглядається процеси побудови математичних моделей для прогнозування ціни опціонів на економічні індекси. У роботі обговорюються три типи моделей – фрактальна модель Блека - Шоулза, класична модель Блека - Шоулза, модель Stochastic Alpha, Betha, Rho. Актуальність даної дисертації полягає у висвітленні нового підходу до моделювання опціонного ціноутворення, який ще недостатньо досліджений на практиці. Об’єктом дослідження є фрактальні моделі економічних процесів – моделювання фінансових процесів,що описуються часовими рядами. Предметом дослідження є нестаціонарні часові ряди з кореляцією, що повільно змінюються з часом та відповідають характеристикам фрактальних рядів, а також економіко-математичні моделі, побудовані на основі таких рядів. Мета роботи – побудова фрактальної математичної моделі для оцінки справедливої вартості фінансових деривативів (опціонів); порівняння фрактальної моделі з існуючими класичними та сучасними моделями; порівняння фрактальної моделі з існуючими класичними та сучасними моделями. Методологія реалізована на основі уже відомих алгоритмівб що були детально досліджені та пристосовані до специфіки завдань роботи. За результатами дослідження методології, створено програмний продукт, що автоматизує процес обчислення, на основі мови R.
Master thesis: 94p., 10 pictures, 31 tables, 3 appendices, 21 citations. Current work describes the construction methodology for the fractal model of option pricing of the index-based underlying assets, that are subject to trade on stock exchanges. Three types of models are discussed: fractional Black – Scholes model, classical Black – Scholes model, Stochastic Alpha, Betha, Rho model. The relevance of master thesis is in the explanation of the fractal approach to the modeling of options’ price, that is not sufficiently studied for the practical applicability by the researchers. The aim of the study: build a model of fractal analysis; compare the model in terms of accuracy of prediction to the available classical andmodern option pricing models; develop software that implements algorithms of fractal analysis. The object of the research is fractional models of financial processes, modeling of objects which are described by distributed time series data. The subject of the study is non-stationary time series with correlations slowly changing with the time and match the characteristics of fractal time series, as well as mathematical and economic models that are build on top of that time series. Theoretical and methodological basis of the study are works of domestic and foreign scholars in the field of economic theory, mathematical modeling, predictive models and fractal theory market. The methodology is implemented on the basis of already known algorithms and using own development. The software for the automatization of modek estimation is implemented using the programming language R. The recommendations for further research are given.