Olabisi O. Ugbebor

The modified Black-Scholes model via constant elasticity of variance for stock options valuation

In this paper, the classical Black-Scholes option pricing model is visited. We present a modified version of the Black-Scholes model via the application of the constant elasticity of variance model (CEVM); in this case, the volatility of the stock price is shown to be a non-constant function unlike the assumption of the classical Black-Scholes model.

Solution of a two-dimensional stochastic investment problem

This paper gives an extension of the work of Kobila [Stochastics Stochastic Rep. 43 (1993) 29-63]. The one-dimensional stochastic differential equation considered in that paper is now extended to two dimensions. The resultant partial differential equation is solved by a finite difference method. Therefore, there exists a two-dimensional stochastic investment problem whose solution could be solved, at least numerically.

A note on Black-Scholes pricing model for theoretical values of stock options

In this paper, we consider some conditions that transform the classical Black-Scholes Model for stock options valuation from its partial differential equation (PDE) form to an equivalent ordinary differential equation (ODE) form. In addition, we propose a relatively new semi-analytical method for the solution of the transformed Black-Scholes model. The obtained solutions via this method can be used to find the theoretical values of the stock options in relation to their fair prices. In considering the reliability and efficiency of the models, we test some cases and the results are in good agreement with the exact solution.

Analytical solutions of a time-fractional nonlinear transaction-cost model for stock option valuation in an illiquid market setting driven by a relaxed Black–Scholes assumption

Abstract In financial mathematics, trading in an illiquid market has become a topic of great concern since assets in such market cannot be sold easily for cash without at least a minimal loss of value. This may be due to uncertainty traceable to factors like lack of interested buyers, transaction cost, and so on. Here, we obtain analytical solutions of a time-fractional nonlinear transaction-cost model for stock option valuation in an illiquid market through a relatively new semi-analytical method: modified differential transform method. Firstly, we considered a nonlinear option pricing model obtained when the constant volatility assumption of the classical linear Black–Scholes option pricing model is relaxed by including transaction cost. Thereafter, we extend, for the first time in literature, this nonlinear option pricing model to a time-fractional ordered form, and obtain approximate-analytical solutions to this new nonlinear model via the proposed technique. For efficiency and reliability of the method, two cases with five examples are considered: case 1 with two examples for time-integer order, and case 2 with three examples for time-fractional order. Our results strongly agree with the associated exact solutions in literature and those obtained via the application of Adomian Decomposition Method (ADM) even though our approximate solutions include only terms up to time power two, accuracy is improved for more terms. This therefore, shows that the result obtained via the ADM is a particular case of this present work when α = 1. Maple 18 software is used for the computations done in this work.

Ergodic properties of Brownian Motion

Since Brownian motion is point recurrent in R 1 , recurrent in R 2 and transient in R n , n ≧ 3 (see (7)), it follows that the total time spent in a bounded open set in R 1 or R 2 is unbounded. With the following ergodic theorems for Brownian motion in R 1 and R 2 as motivation, we examine the rate of convergence in these theorems. Note that there is no ergodic property in R n for n ≧ 3 since Brownian motion is not dense there.