Analytical Study And Generalisation Of Selected Stock Option Valuation Models

ABSTRACT

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.