APPROXIMATE SOLUTION OF FRACTIONAL BLACK-SCHOLE’S EUROPEAN OPTION PRICING EQUATION BY USING ETHPM

Pradip R. Bhadane,KIRTIWANT P. GHADLE,AHMED A. HAMOUD 경남대학교 수학교육과 2020 Nonlinear Functional Analysis and Applications Vol.25 No.2

We proposed a new reliable combination of new Homotopy Perturbation Method(HPM) and Elzaki transform called as Elzaki Transform Homotopy Perturbation Method(ETHPM) is designed to obtain a exact solution to the fractional Black-Scholes equationwith boundary condition for a European option pricing problem. The fractional derivativeis in Caputo sense and the nonlinear terms in Fractional Black-Scholes Equation can behandled by using HPM. The Black-Scholes formula is used as a model for valuing Europeanor American call and put options on a non-dividend paying stock. The methods give ananalytic solution of the fractional Black-Scholes equation in the form of a convergent series. Finally, some examples are included to demonstrate the validity and applicability of theproposed technique.

ON THE PARAMETIC INTEREST OF THE BLACK-SCHOLES EQUATION

Amnuay Kananthai 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.3

We have discovered some parametics λ in the Black-Scholes equation which depend on the interest rate r and the Volatility σ and later is named the parametic interest. On studying the parametic interest λ, we found that such λ gives the sufficient condition for the existence of solutions of the Black-Scholes equation which is either weak or strong solutions.

ON THE PARAMETIC INTEREST OF THE BLACK-SCHOLES EQUATION

Kananthai, Amnuay The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.3

We have discovered some parametics $\lambda$ in the Black-Scholes equation which depend on the interest rate $\gamma$ and the Volatility $\sigma$ and later is named the parametic interest. On studying the parametic interest $\lambda$, we found that such $\lambda$ gives the sufficient condition for the existence of solutions of the Black-Scholes equation which is either weak or strong solutions.

미분가능 신경망을 이용한 옵션 가격결정

지상문 한국정보통신학회 2021 한국정보통신학회논문지 Vol.25 No.4

Neural networks with differentiable activation functions are differentiable with respect to input variables. We improve the approximation capability of neural networks by using the gradient and Hessian of neural networks to satisfy the differential equations of the problems of interest. We apply differential neural networks to the pricing of financial options, where stochastic differential equations and the Black-Scholes partial differential equation represent the differential relation of price of option and underlying assets, and the first and second derivatives of option price play an important role in financial engineering. The proposed neural network learns - (a) the sample paths of option prices generated by stochastic differential equations and (b) the Black-Scholes equation at each time and asset price. Experimental results show that the proposed method gives accurate option values and the first and second derivatives. 신경망은 미분가능한 활성화 함수를 사용하는 경우에는 입력변수에 대하여 미분가능하다. 본 연구에서는 신경망의 근사 능력을 향상시키기 위하여 신경망의 그래디언트와 헤시안이 블랙-숄즈 미분방정식을 만족하도록 한다. 본 논문은 확률 미분방정식과 블랙-숄즈 편미분 방정식이 옵션 가격과 기초자산의 미분관계를 표현하는 옵션 가격결정에 제안한 방법을 사용한다. 이는 옵션 가격의 일차와 이차미분은 금융공학에서 중요한 역할을 하므로 미분 값을 쉽게 얻을 수 있는 제안한 방법을 적용할 수 있기 때문이다. 제안한 신경망은 (1) 확률 미분방정식이 생성하는 옵션가격의 샘플 경로와 (2) 각 시간과 기초자산 가격에서 블랙-숄즈 방정식을 만족하도록 학습한다. 실험을 통하여 제안한 방법이 옵션가격과 일차와 이차 미분 값을 정확히 예측함을 보인다.

Comparison of numerical schemes on multi-dimensional Black-Scholes equations

조중리,김용식 대한수학회 2013 대한수학회보 Vol.50 No.6

In this paper, we study numerical schemes for solving multi-dimensional option pricing problem. We compare the direct solving meth\-od and the \emph{Operator Splitting Method}(OSM) by using finite difference approximations. By varying parameters of the \emph{Black-Scholes equations} for the maximum on the call option problem, we observed that there is no significant difference between the two methods on the convergence criterion except a huge difference in computation cost. Therefore, the two methods are compatible in practice and one can improve the time efficiency by combining the OSM with parallel computation technique. We show numerical examples including the \emph{Equity-Linked Security}(ELS) pricing based on either two assets or three assets by using the OSM with the \emph{Monte-Carlo Simulation} as the benchmark.

An adaptive finite difference method using far-field boundary conditions for the Black--Scholes equation

Darae Jeong,하태영,Myoungnyoun Kim,신재민,In-Han Yoon,김준석 대한수학회 2014 대한수학회보 Vol.51 No.4

We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.

민감도(Greeks)를 해(Solution)로 갖는 블랙숄즈 편미분방정식(PDE)

김영성,서기수,강희건 융복합지식학회 2024 융복합지식학회논문지 Vol.12 No.2

파생상품과 같은 조건부청구권(Contingent Claim)의 민감도(Greeks)를 산출하는 것은 위험관리에서 매우 중요하고, 헤지 운용을 위한 수익창출에도 매우 중요하다. 본 연구에서는 가격이 아닌, 민감도가 해(solution)인 편미분방정식(partial differential equation)을 이용하여 민감도를 계산하는 방법을 유한차분법(finite difference method)을 적용하여 제안한다. 또한, 실제 폐쇄형해(closed form solution)와 비교하였다. 분석결과, 민감도를 해로 설정하여 편미분방정식으로 계산한 값(numerical solution)과 실제 폐쇄형해에서 산출된 값이 수치적으로 동일하게 나타났다. 또한, 1일을 격자 1개로 설정하여 계산한 경우보다 3개의 격자로 세분하여 계산한 값이 더 정확하게 나타났다. 그리고 변동성이 높을수록 격자의 범위를 높여야 하며 500%까지 설정한 부분에서 오차가 현저하게 떨어졌다. 따라서 하루의 격자를 적절하게 설정하고 변동성에 따라 기초자산 가격의 범위만 잘 정의할 수 있다면, 본 연구에서 제시한 민감도 자체의 편미분방정식(PDE)의 해는 실제 폐쇄형해와 일치한다는 것을 모델링하여 검증하였다. 이 방법을 이용하면 향후 시간대비 효율적인 민감도를 산출 할 수 있을 것이라 기대한다. Calculating the sensitivities(Greeks) of contingent claims such as derivatives is very important in risk management and also in generating profits for hedging operations. In this study, we propose a method of calculating sensitivity using the finite difference method using a partial differential equation in which sensitivity itself, not price, is the solution. Additionally, it was compared with an actual closed form solution. As a result of the analysis, by setting the sensitivities to the solution, the value calculated from the partial differential equation(numerical solution) and the value calculated from the actual closed form solution were numerically identical. In addition, the value calculated by dividing one day into three grids appears more accurately than when calculating by setting one day to one grid. Also, the higher the volatility, the higher the grid range should be, and the error dropped significantly when set to 500%. Therefore, if the daily grid can be properly set and the range of the underlying asset price can be well defined according to volatility, the solution of the partial differential equation(PDE) of the sensitivities presented in this study is actually a closed form solution. It was verified by modeling that it appeared consistent with the solution. Using this method, it is expected that efficient sensitivities compared to time will be calculated in the future.

COMPARISON OF NUMERICAL SCHEMES ON MULTI-DIMENSIONAL BLACK-SCHOLES EQUATIONS

Jo, Joonglee,Kim, Yongsik Korean Mathematical Society 2013 대한수학회보 Vol.50 No.6

In this paper, we study numerical schemes for solving multi-dimensional option pricing problem. We compare the direct solving method and the Operator Splitting Method(OSM) by using finite difference approximations. By varying parameters of the Black-Scholes equations for the maximum on the call option problem, we observed that there is no significant difference between the two methods on the convergence criterion except a huge difference in computation cost. Therefore, the two methods are compatible in practice and one can improve the time efficiency by combining the OSM with parallel computation technique. We show numerical examples including the Equity-Linked Security(ELS) pricing based on either two assets or three assets by using the OSM with the Monte-Carlo Simulation as the benchmark.

BLACK-SCHOLES EQUATION

Lee, Young S. Chungcheong Mathematical Society 2019 충청수학회지 Vol.32 No.1

The purpose of this paper is to present approximation of $C_0$-sequentially equicontinuous semigroups on a sequentially complete locally convex space X.

COMPARISON OF NUMERICAL METHODS (BI-CGSTAB, OS, MG) FOR THE 2D BLACK-SCHOLES EQUATION

정다래,김성기,최용호,황형석,김준석 한국수학교육학회 2014 純粹 및 應用數學 Vol.21 No.2

In this paper, we present a detailed comparison of the performance ofthe numerical solvers such as the biconjugate gradient stabilized, operator splitting,and multigrid methods for solving the two-dimensional Black-Scholes equation. Theequation is discretized by the finite difference method. The computational resultsdemonstrate that the operator splitting method is fastest among these solvers withthe same level of accuracy.