A stable collocation approach to solve a neutral delay stochastic differential equation of fractional order

Memory effect is an important phenomenon in financial systems, and a number of research works have been carried out to study the long memory in the financial markets. In recent years, fractional order ordinary differential equation is used as an effective instrument for describing the memory effect in complex systems. In this paper, we establish a fractional order stochastic differential equation (FSDE) model to describe the effect of trend memory in financial pricing. We, then, derive a European option pricing formula based on the FSDE model and prove the existence of the trend memory (i.e., the mean value function) in the option pricing formula when the Hurst index is between 0.5 and 1. In addition, we make a comparison analysis between our proposed model, the classic Black-Scholes model, and the stochastic model with fractional Brownian motion. Numerical results suggest that our model leads to more accurate and lower standard deviation in the empirical study.

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A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

Fractal and Fractional ◽

10.3390/fractalfract5030085 ◽

2021 ◽

Vol 5 (3) ◽

pp. 85

Author(s):

Tayyaba Akram ◽

Zeeshan Ali ◽

Faranak Rabiei ◽

Kamal Shah ◽

Poom Kumam

Keyword(s):

Differential Equation ◽

Fractional Order ◽

Numerical Study ◽

Stability And Convergence ◽

Weakly Singular Kernel ◽

Integro Differential Equation ◽

Suggested Technique ◽

Collocation Approach ◽

The Stability ◽

Efficiency And Reliability

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

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Dynamic Hedging Based on Fractional Order Stochastic Model with Memory Effect

Mathematical Problems in Engineering ◽

10.1155/2016/6817483 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Qing Li ◽

Yanli Zhou ◽

Xinquan Zhao ◽

Xiangyu Ge

Keyword(s):

Differential Equation ◽

Stochastic Differential Equation ◽

Fractional Order ◽

Minimum Variance ◽

Hedge Ratio ◽

Out Of Sample ◽

Sample Test ◽

Optimal Dynamic ◽

Optimal Hedge ◽

With Memory

Many researchers have established various hedge models to get the optimal hedge ratio. However, most of the hedge models only discuss the discrete-time processes. In this paper, we construct the minimum variance model for the estimation of the optimal hedge ratio based on the stochastic differential equation. At the same time, also by considering memory effects, we establish the continuous-time hedge model with memory based on the fractional order stochastic differential equation driven by a fractional Brownian motion to estimate the optimal dynamic hedge ratio. In addition, we carry on the empirical analysis to examine the effectiveness of our proposed hedge models from both in-sample test and out-of-sample test.

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