Fractional Functional with two Occurrences of Integrals and Asymptotic Optimal Change of Drift in the Black-Scholes Mode
- PDF / 537,184 Bytes
- 15 Pages / 439.642 x 666.49 pts Page_size
- 2 Downloads / 145 Views
Fractional Functional with two Occurrences of Integrals and Asymptotic Optimal Change of Drift in the Black-Scholes Model R. A. El-Nabulsi
Received: 5 April 2013 / Accepted: 7 November 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract The fractional action-like variational approach for the case of functional with two occurrences of fractional action integrals is constructed. Our motivation is based on the fact that a functional with two occurrences of integrals has many advantages in optimization problems and financial engineering issues. After deriving the corresponding fractional Euler-Lagrange equation and discussing some examples, we addressed the problem of Asian call option. We derived the asymptotically optimal change of fractional drift for the geometric average Asian call option and gave some new consequences. Keywords Fractional action-like variational approach · Functional with two occurrences of fractional action integrals · Geometric average Asian call option · Optimal change Mathematics Subject Classification (2010) 91G80 · 26A33 · 60H07
1 Introduction The fractional calculus of variations (FCV) is a new branch of applied mathematics with many applications in sciences and engineering. For a good introduction of the topic, we refer the reader to [26, 32]. FCV is considered as a successful mathematical tool to describe dissipative and non-conservative dynamical systems at both classical and quantum levels [1, 2, 12–14, 27, 33, 37]. FCV and the corresponding fractional Euler-Lagrange equations are based on fractional differential and integral operators or simply fractional calculus which is a subject of mathematics dating back to the late part of seventeenth century. For a historical survey in this area, the reader is referred to the bibliography prepared by Ross and reprinted in the monograph by Oldham and Spanier [36]. In reality, FCV was introduced in different forms in literature depending on the nature of the problem under consideration. It is noteworthy that FCV started in 1996 when Riewe replaced in his Lagrangian R. A. El-Nabulsi () Department of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641112, China e-mail: [email protected]
R. A. EL-NABULSI
approach integer derivatives by fractional derivatives and obtained the respective fractional Euler-Lagrange equations, combining accordingly both conservative and nonconservative cases [39, 40]. The main result of Riewe’s approach is that non-conservative forces can be computed directly from potentials represented by fractional derivatives. However, it was observed that due to the complexity of fractional derivative operators, Riewe’s approach is somewhat encountered by mathematical limitations [10, 11, 38] and that the consequential equations are causal and that the procedure to convert these equations into causal ones is not well defined and besides that, the causality principle is violated [8]. The physical
Data Loading...
