ANALYTICAL STUDY AND GENERALISATION OF SELECTED STOCK OPTION VALUATION MODELS
No Thumbnail Available
Date
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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.
Keywords
Q Science (General), QA Mathematics