scholarly journals Comparing algebraic and numerical solutions of classical diffusion process equations in computational financial mathematics

2001 ◽  
Vol 6 (3) ◽  
pp. 157-169
Author(s):  
Andreas Ruffing ◽  
Patrick Windpassinger ◽  
Stefan Panig
Keyword(s):  
Diffusion Equation ◽  
Diffusion Process ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Financial Mathematics ◽  
Classical Diffusion ◽  
Black Scholes ◽  
Algebraic Properties

We revise the interrelations between the classical Black Scholes equation, the diffusion equation and Burgers equation. Some of the algebraic properties the diffusion equation shows are elaborated and qualitatively presented. The related numerical elementary recipes are briefly elucidated in context of the diffusion equation. The quality of the approximations to the exact solutions is compared throughout the visualizations. The article mainly is based on the pedagogical style of the presentations to the Novacella Easter School 2000 on Financial Mathematics.

Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Yufeng Xu ◽  
Om Agrawal
Keyword(s):  
Finite Difference ◽  
Diffusion Process ◽  
Weight Function ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Weight Functions ◽  
Linear Recurrence ◽  
Scale Function ◽  
Special Cases ◽  
Wide Range

AbstractIn this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.

scholarly journals On a time fractional diffusion with nonlocal in time conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nguyen Hoang Tuan ◽  
Nguyen Anh Triet ◽  
Nguyen Hoang Luc ◽  
Nguyen Duc Phuong
Keyword(s):  
Fourier Series ◽  
Diffusion Equation ◽  
Convergence Rate ◽  
Exact Solutions ◽  
Mild Solution ◽  
Integral Condition ◽  
Fractional Diffusion ◽  
Well Posedness ◽  
Nonlocal Integral Condition ◽  
Regularized Solution

AbstractIn this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

SOLVING THE GENERALIZED BURGERS–KdV EQUATION BY A COMBINATION METHOD

2008 ◽  
Vol 22 (21) ◽  
pp. 2021-2025 ◽  
Author(s):  
YUANXI XIE
Keyword(s):  
Exact Solutions ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Combination Method ◽  
Generalized Kdv Equation ◽  
Generalized Burgers Equation

In view of the analysis on the characteristics of the generalized Burgers equation, generalized KdV equation and generalized Burgers–KdV equation, a combination method is presented to seek the explicit and exact solutions to the generalized Burgers–KdV equation by combining with those of the generalized Burgers equation and generalized KdV equation. As a result, many explicit and exact solutions for the generalized Burgers–KdV equation are successfully obtained by this technique.

Value-at-Risk and Expected Shortfall approaches for option premiums and the probability of default estimation based on ARMA models

2021 ◽  
Vol 57 (3) ◽  
pp. 126
Author(s):  
Nikolai Berzon
Keyword(s):  
Time Series ◽  
Option Pricing ◽  
Discrete Time ◽  
Value At Risk ◽  
Probability Of Default ◽  
Pricing Models ◽  
Arma Process ◽  
Underlying Asset ◽  
Black Scholes

The need to address the issue of risk management has given rise to a number of models for estimation the probability of default, as well as a special tool that allows to sell credit risk – a credit default swap (CDS). From the moment it appeared in 1994 until the crisis of 2008, that the CDS market was actively growing, and then sharply contracted. Currently, there is practically no CDS market in emerging economies (including Russia). This article is to improve the existing CDS valuation models by using discrete-time models that allow for more accurate assessment and forecasting of the selected asset dynamics, as well as new option pricing models that take into account the degree of risk acceptance by the option seller. This article is devoted to parametric discrete-time option pricing models that provide more accurate results than the traditional Black-Scholes continuous-time model. Improvement in the quality of assessment is achieved due to three factors: a more detailed consideration of the properties of the time series of the underlying asset (in particular, autocorrelation and heavy tails), the choice of the optimal number of parameters and the use of Value-at-Risk approach. As a result of the study, expressions were obtained for the premiums of European put and call options for a given level of risk under the assumption that the return on the underlying asset follows a stationary ARMA process with normal or Student's errors, as well as an expression for the credit spread under similar assumptions. The simplicity of the ARMA process underlying the model is a compromise between the complexity of model calibration and the quality of describing the dynamics of assets in the stock market. This approach allows to take into account both discreteness in asset pricing and take into account the current structure and the presence of interconnections for the time series of the asset under consideration (as opposed to the Black–Scholes model), which potentially allows better portfolio management in the stock market.

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

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