A new approach for the solution of the Black-Scholes equation with barrier option constraints

Abstract

This paper introduces an efficient approach to solve the Black-Scholes Partial Differential Equation (BSPDE) with Barrier Option Constraints (BCOs). The approach of the Laplace–Adomian Decomposition Method (LADM), which is the combination of the Laplace Transform Method (LTM) and the Adomian Decomposition Method (ADM) is employed. The LTM is applied to the BSPDE, and the ADM is used for decomposing the solution of BSPDE into an infinite series. Moreover, the approximate solution obtained via LADM is expressed in the form of a convergent series with computed components. An illustrative example is presented, and the results are compared with the Analytical Value (AV). Hence, LADM is found to be effective and a powerful technique for obtaining an approximate solution of BSPDE with BCOs.

Homotopy Analysis Method for Fractional Barrier Option PDE

“…Note that the valuation formula for the classical barrier option model is retrievable from (5) for 1   . which is the same as the result obtained by [19] and [20]…”

“…Consider the following parameters [19]: S= $80, σ = 25% and B u = $85 and by varying the exercise price K, maturity date T, Risk-neutral interest rate r and fractional / non-integer order  . The results generated were displayed in Figures 1-5.…”

“…Figure 1: HAM, LADM [19], CFRDTM [17], AV [22] with T = 2, r = 0.1, σ = 25%, β = 1, Bu = 85, S =80 Figure 2: HAM, LADM [19] , CFRDTM [17], AV [22] with T = 6, r = 0.1, σ = 25%, β = 1, Bu = 85, S =80…”

“…Some researchers have studied the solution of non-integer differential equations, see [16][17][18]. Fadugba and Edeki [19] and Edeki and Fadugba [20] proposed Laplace-Adomian decoposition method and projected differential transformation method, respectively to obtain the solution of barrier option PDE model It is also asummed in this paper that the underlying asset price is driven by marked point process [21]. Other solution methods that can be adopted for the model include [22][23][24].…”