Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black–Scholes Equation for European Option Valuation. The results obtained converge faster to their associated exact solution form; these easily computed results represent the analytical values of the associated European call options, and the same algorithm can be followed for European put options. It is shown that PDTM is more efficient, reliable and better than the classical DTM and other semi-analytical methods since less computational work is involved. Hence, it is strongly recommended for both linear and nonlinear stochastic differential equations (SDEs) encountered in financial mathematics

ANALYTICAL STUDY AND GENERALISATION OF SELECTED STOCK OPTION VALUATION MODELS

In this work, the classical Black-Scholes model for stock option valuation on the basis of some stochastic dynamics was considered. As a result, a stock option val- uation model with a non-�xed constant drift coe�cient was derived. The classical Black-Scholes model was generalised via the application of the Constant Elasticity of Variance Model (CEVM) with regard to two cases: case one was without a dividend yield parameter while case two was with a dividend yield parameter. In both cases, the volatility of the stock price was shown to be a non-constant power function of the underlying stock price and the elasticity parameter unlike the constant volatility assumption of the classical Black-Scholes model. The It^o's theorem was applied to the associated Stochastic Di�erential Equations (SDEs) for conversion to Partial Dif- ferential Equations (PDEs), while two approximate-analytical methods: the Modi�ed Di�erential Transformation Method (MDTM) and the He's Polynomials Technique (HPT) were applied to the Black-Scholes model for stock option valuation; in both cases the integer and time-fractional orders were considered, and the results obtained proved the latter as an extension of the former. In addition, a nonlinear option pric- ing model was obtained when the constant volatility assumption of the classical linear Black-Scholes option pricing model was relaxed through the inclusion of transaction cost (Bakstein and Howison model). Thereafter, this nonlinear option pricing model was extended to a time-fractional ordered form, and its approximate-analytical solu- tions were obtained via the proposed solution technique. For e�ciency and reliability of the method, two cases with �ve examples were considered: Case 1 with two ex- amples for time-integer order, and Case 2 with three examples for time-fractional order, and the results obtained show that the time-fractional order form generalises the time-integer order form. Thus, the Black-Scholes and the Bakstein and Howison models for stock option valuation were generalised and extended to time-fractional order, and analytical solutions of these generalised models were provided

Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments

He’s Polynomials for Analytical Solutions of the Black-Scholes Pricing Model for Stock Option Valuation

The Black-Scholes model is one of the most famous and useful models for option valuation as regards option pricing theory. In this paper, we propose a semianalytical method referred to as He’s polynomials for solving the classical Black-Scholes pricing model with stock as the underlying asset. The proposed method gives the exact solution of the solved problem in a very simple and quick manner even with less computational work while still maintaining high level of accuracy. Hence, we recommend an extension and adoption of this method for solving problems arising in other areas of financial engineering, finance, and applied science

Solution of a barrier option Black-Scholes model based on projected differential transformation method

In this article, the solution of the linear variant of a Barrier Option Black-Scholes Model (BOBSM) is considered via a semi-analytical approach referred to as the Projected Differential Transformation Method (PDTM). Similar to the traditional Differential Transformation Method, this new approach demonstrates feasible progress and efficiency of operation. For simplicity of illustrative, the BOBSM is converted to an equivalent heat-like form, and a series-form of the solution (root) is successfully obtained. Hence the PDTM is suggested for both pure and functional sciences for strongly nonlinear differential models with financial applications

Valuation of options for hedging against exchange rate exposure

>Magister Scientiae - MScThe risk associated with currency exposure is one of the main sources of risk in terms of internationally diversi ed portfolios. Controlling the risk is important for improving the performance of international investments. One approach to hedging against exchange rate exposure is by employing financial derivatives, particularly, foreign currency options. Currency options provide insurance against unfavorable exchange rate fluctuation, but also make provision to lock in a pro t when the exchange rate fluctuation are favorable. However, these instruments cannot be traded or managed without the relevant valuation techniques. In this dissertation we discuss one of the approaches to cover the risk associated with currency exposure. In particular, we focus on the partial differential equation (PDE) valuation of currency options by employing various finite difference schemes. We commence by introducing the mathematical tools required for the valuation of financial derivatives. Thereafter we study the valuation of European options. This involves deriving the famous Black-Scholes PDE for pricing options on stocks that do not yield dividends. Using the Black-Scholes PDE we derive the Black-Scholes formula for pricing European options. This derivation involves transforming the Black-Scholes PDE into the heat equation and by solving the heat equation we obtain the Black-Scholes formula. After completing the pricing of European options we now move to the pricing of American options. The early exercise facility associated with American options, leads to a free boundary problem which makes the pricing process of American options a challenging task. As in the case of the European options, we first derive the Black-Scholes inequality for American options and then transform this inequality for application to the heat equation to value American options. In the absence of an explicit formula for pricing American options we use numerical methods. Thus, we discuss the finite difference methods quite extensively with a focus on the implicit and Crank-Nicholson finite difference methods

Método dos elementos finitos baseado em polinómios de Hermite cúbicos, para resolução da equação de Black-Scholes não linear com opções europeias

Foi desenvolvido um algoritmo numérico para resolver uma equação diferencial parcial generalizada de Black-Scholes, que surge na precificação de opções europeias, considerando os custos de transação. O método Crank-Nicolson é usado para discretizar no tempo e o método de interpolação cúbica de Hermite para discretizar no espaço. A eficiência e precisão do método proposto são testadas numericamente e, os resultados confirmam o comportamento teórico das soluções, que também se encontra em boa concordância com a solução exata.A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution

Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems