An alternative approach to solving the Black–Scholes equation with time-varying parameters

Rodrigo, Marianito R.; Mamon, Rogemar

doi:10.1016/j.aml.2005.06.012

Cited by 40 publications

(19 citation statements)

References 4 publications

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“…However, when transformations are introduced, one needs to keep track of changes in the variables and terminal condition and then solve the resulting initial-value problem for the heat equation. A more direct approach was developed recently in [9]. In this approach, the Black-Scholes PDE with time-varying parameters is transformed into a BlackScholes PDE with time-independent but arbitrary parameters.…”

Section: Introductionmentioning

confidence: 99%

An Application of Mellin Transform Techniques to a Black–scholes Equation Problem

Rodrigo

Mamon

2007

Anal. Appl.

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In this article, we use a Mellin transform approach to prove the existence and uniqueness of the price of a European option under the framework of a Black-Scholes model with time-dependent coefficients. The formal solution is rigorously shown to be a classical solution under quite general European contingent claims. Specifically, these include claims that are bounded and continuous, and claims whose difference with some given but arbitrary polynomial is bounded and continuous. We derive a maximum principle and use it to prove uniqueness of the option price. An extension of the put-call parity which relates the aforementioned two classes of claims is also given.

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Section: Introductionmentioning

confidence: 99%

An Application of Mellin Transform Techniques to a Black–scholes Equation Problem

Rodrigo

Mamon

2007

Anal. Appl.

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“…We have established three new solutions given in (26), (28), (33) for the BST model (6) by utilizing the potential symmetries. These solutions are not reported in Naz and Johnpillai [23].…”

Section: Exact Solutions Via Potential Symmetries Of Potential Systemmentioning

confidence: 99%

Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries

Naz¹,

Naeem²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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We analyze the local conservation laws, auxiliary (potential) systems, potential symmetries and a class of new exact solutions for the Black-Scholes model time-dependent parameters (BST model). First, we utilize the computer package GeM to construct local conservation laws of the BST model for three different forms of multipliers. We obtain two conserved vectors for the second-order multipliers of form Λ(x, u, ux, uxx). We define two potential variables v and w corresponding to the conserved vectors. We construct two singlet potential systems involving a single potential variable v or w and one couplet potential system involving both potential variables v and w. Moreover, a spectral potential system is constructed by introducing a new potential variable pα which is a linear combination of potential variables v and w. The potential symmetries of BST model are derived by computing the point symmetries of its potential systems. Both singlet potential systems provide three potential symmetries. The couplet potential system yields three potential symmetries and no potential symmetries exist for the spectral potential system. We utilize the potential symmetries of singlet potential systems to construct three new solutions of BST model.

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“…In literature, a lot of solution methods have been considered by a good number of researchers. These include: Adomian decomposition method (ADM), variational iteration method (VIM), Modified ADM (MADM), homotopy perturbation method (HPM), Differential transformation method (DTM), projected DTM (PDTM) [18][19][20][21][22][23][24][25]. He's polynomials method was initiated in [26,27] by Ghorbani et al, where the nonlinear terms were expressed as series of polynomials calculated with the aid of HPM.…”

Section: Introductionmentioning

confidence: 99%

Closed-form Solutions of the Time-fractional Standard Black-Scholes Model for Option Pricing using He-separation of Variable Approach

Edeki

Akinlabi

Egara

et al. 2020

Wseas Transactions on Environment and Development

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The Black-Scholes option pricing model in classical form remains a benchmark model in Financial Engineering and Mathematics concerning option valuation. Though, it has received a series of modifications as regards its initial constancy assumptions. Most of the resulting modifications are nonlinear or time-fractional, whose exact or analytical solutions are difficult to obtain. This paper, therefore, presents exact (closed-form) solutions to the time-fractional classical Black-Scholes option pricing model by means of the He-Separation of Variable Transformation Method (HSVTM). The HSVTM combines the features of the He's polynomials, the Homo-separation variable, the modified DTM, which increases the efficiency and effectiveness of the proposed method. The proposed method is direct and straight forward. Hence, it is recommended for obtaining solutions to financial models resulting from either Ito or Stratonovich Stochastic Differential Equations (SDEs).

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An alternative approach to solving the Black–Scholes equation with time-varying parameters

Cited by 40 publications

References 4 publications

An Application of Mellin Transform Techniques to a Black–scholes Equation Problem

An Application of Mellin Transform Techniques to a Black–scholes Equation Problem

Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries

Closed-form Solutions of the Time-fractional Standard Black-Scholes Model for Option Pricing using He-separation of Variable Approach

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