scholarly journals Pricing Path-Independent Payoffs with Exotic Features in the Fractional Diffusion Model

2020 ◽  
Vol 4 (2) ◽  
pp. 16 ◽  
Author(s):  
Jean-Philippe Aguilar
Keyword(s):  
Fractional Derivative ◽  
Option Pricing ◽  
Diffusion Model ◽  
Financial Models ◽  
Convergent Series ◽  
Fractional Diffusion ◽  
Volatility Modeling ◽  
Time Fractional Derivative ◽  
Digital Options ◽  
Residue Theory

We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs …) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models.

scholarly journals Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 433 ◽  
Author(s):  
Bohdan Datsko ◽  
Igor Podlubny ◽  
Yuriy Povstenko
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Fractional Derivative ◽  
Graphical Representation ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Mass Absorption ◽  
The Laplace Transform ◽  
Time Fractional Diffusion Equation ◽  
Time Fractional Derivative

The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given.

scholarly journals Analysis of two Legendre spectral approximations for the variable-coefficient fractional diffusion-wave equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenping Chen ◽  
Shujuan Lü ◽  
Hu Chen ◽  
Lihua Jiang
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Variable Coefficient ◽  
Equivalent Form ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Diffusion Wave ◽  
Fully Discrete ◽  
Spectral Approximations ◽  
Time Fractional Derivative

Abstract In this paper, we solve the variable-coefficient fractional diffusion-wave equation in a bounded domain by the Legendre spectral method. The time fractional derivative is in the Caputo sense of order $\gamma \in (1,2)$ γ ∈ ( 1 , 2 ) . We propose two fully discrete schemes based on finite difference in temporal and Legendre spectral approximations in spatial discretization. For the first scheme, we discretize the time fractional derivative directly by the $L_{1}$ L 1 approximation coupled with the Crank–Nicolson technique. For the second scheme, we transform the equation into an equivalent form with respect to the Riemann–Liouville fractional integral operator. We give a rigorous analysis of the stability and convergence of the two fully discrete schemes. Numerical examples are carried out to verify the theoretical results.

scholarly journals An Efficient Method for Systems of Variable Coefficient Coupled Burgers’ Equation with Time-Fractional Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hossein Aminikhah ◽  
Nasrin Malekzadeh
Keyword(s):  
Fractional Derivative ◽  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Burgers Equation ◽  
Power Series Expansion ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Time Fractional Derivative ◽  
Coupled Burgers Equation

A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. Results indicate that the introduced method is promising for solving other types of systems of nonlinear fractional-order partial differential equations.

Duality theory of fractional resolvents and applications to backward fractional control systems

2021 ◽  
Vol 24 (2) ◽  
pp. 541-558
Author(s):  
Shouguo Zhu ◽  
Gang Li
Keyword(s):  
Control System ◽  
Fractional Derivative ◽  
Control Systems ◽  
Diffusion Model ◽  
Duality Theory ◽  
Approximate Controllability ◽  
Fractional Diffusion ◽  
Variational Technique ◽  
Liouville Fractional Derivative ◽  
Fractional Control

Abstract We study the duality theory for fractional resolvents, extending and improving some corresponding theorems on semigroups. As applications, we develop the variational technique to analyze the finite-approximate controllability of a backward fractional control system with a right-sided Riemann-Liouville fractional derivative. Moreover, validity of our theoretical findings is given by a fractional diffusion model.

scholarly journals Analytical Solutions of the Diffusion–Wave Equation of Groundwater Flow with Distributed-Order of Atangana–Baleanu Fractional Derivative

2021 ◽  
Vol 11 (9) ◽  
pp. 4142
Author(s):  
Nehad Ali Shah ◽  
Abdul Rauf ◽  
Dumitru Vieru ◽  
Kanokwan Sitthithakerngkiet ◽  
Poom Kumam
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Groundwater Flow ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Integral Transforms ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Time Fractional Derivative ◽  
Distributed Order

A generalized mathematical model of the radial groundwater flow to or from a well is studied using the time-fractional derivative with Mittag-Lefler kernel. Two temporal orders of fractional derivatives which characterize small and large pores are considered in the fractional diffusion–wave equation. New analytical solutions to the distributed-order fractional diffusion–wave equation are determined using the Laplace and Dirichlet-Weber integral transforms. The influence of the fractional parameters on the radial groundwater flow is analyzed by numerical calculations and graphical illustrations are obtained with the software Mathcad.

scholarly journals Determination of the order of fractional derivative for subdiffusion equations

2020 ◽  
Vol 23 (6) ◽  
pp. 1647-1662
Author(s):  
Ravshan Ashurov ◽  
Sabir Umarov
Keyword(s):  
Fractional Derivative ◽  
Fixed Time ◽  
Time Instant ◽  
Observation Data ◽  
Additional Information ◽  
Fractional Modeling ◽  
Time Fractional Derivative ◽  
The Right ◽  
Right Order

Abstract The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring location, as “the observation data”, identifies uniquely the order of the fractional derivative.

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