Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation

Akrami, Mohammad Hossein; Erjaee, G.H.

doi:10.1515/fca-2015-0004

Cited by 27 publications

(14 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The works of Cartea and Meyer-Brandis [49] and Cartea [50] proposed a stock price model that uses information about the waiting time between trades. In this model the arrival of trades is driven by a counting process, in which the waiting-time between trades processes is described by the Mittag-Leffler survival function (see also [51]). In the paper [50], Cartea proposed that the value of derivatives satisfies the fractional Black-Scholes equation that contains the Caputo fractional derivative with respect to time.…”

Section: Arfima Stage (Approach)mentioning

confidence: 99%

On History of Mathematical Economics: Application of Fractional Calculus

Tarasov

2019

Mathematics

182

View full text Add to dashboard Cite

Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed.

show abstract

Section: Arfima Stage (Approach)mentioning

confidence: 99%

On History of Mathematical Economics: Application of Fractional Calculus

Tarasov

2019

Mathematics

182

View full text Add to dashboard Cite

show abstract

“…In the past few decades, many authors require the construction of variants of the Black-Scholes models in various financial markets, in which the fractional derivatives are defined. In particular, two types of fractional Black-Scholes equations are mainly studied, i.e., timefractional Black-Scholes equation with the Caputo derivatives 7,8 and pricing-fractional Black-Scholes equation with the Weyl fractional derivative 9, 10 .…”

Section: Introductionmentioning

confidence: 99%

Invariant subspace method for fractional Black-Scholes equations

Kittipoom¹

2018

ScienceAsia

View full text Add to dashboard Cite

In this paper, the invariant subspace method is used to solve the time-and pricing-fractional Black-Scholes equations, in which the fractional derivatives are described in the Caputo and Weyl sense, respectively. We introduce invariant subspaces for time and pricing differential operators. Applying an appropriate invariant subspace will reduce the fractional Black-Scholes equation to a system of ordinary fractional differential equations. Finally, using point symmetries of the obtained system, we construct the explicit solutions of the fractional Black-Scholes equations.

show abstract

“…For this purpose, we use the reconstruction of variational iteration method [1] and develop the analytical solution of generalized fractional Black-Scholes equation for European option pricing problems. This equation is described by the following equation…”

Section: Introductionmentioning

confidence: 99%

“…For a generalization of the fractional Black-Scholes model with the Caputo derivative to the fractional Black-Scholes model with the regularized Prabhakar derivative, the initial condition in the following examples is similar to that of the initial condition given in[1].…”

mentioning

confidence: 99%