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

S.O Edeki,E.A Owoloko,O.O Ugbebor,Kok Lay Teo

doi:10.1080/23311835.2017.1352118

S.O Edeki, E.A Owoloko + Show 2 more

Open Access

https://doi.org/10.1080/23311835.2017.1352118

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Journal: Cogent Mathematics	Publication Date: Jan 1, 2017
Citations: 3	License type: CC BY 4.0

Affiliation: Covenant University, University of Ibadan

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 timefractional order. Our results strongly agree with the associated exact solutions in

Highlights

The term “liquidity” is used in describing the degree to which an underlying asset can be exercised in the market setting in a way that the asset’s price is not affected (Acharya & Pedersen, 2005; Amihud & Mendelson, 1986)
The remaining part of the paper is structured as follows: in Section 2, we give a brief note on the nonlinear option pricing model; in Section 3, we present an overview, the basic theorems of the semi-analytical method and the analysis of its fractional form; in Section 4, the modified differential transform method (MDTM) is applied to the time-fractional order-type nonlinear option pricing model followed by numerical examples for some special cases with graphical interpretations; in Section 5, we give concluding remarks and summary of our results
Concluding remarks In this paper, we considered 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 model assumption through a relatively new semi-analytical method called the modified differential transform method (MDTM)

Summary

Introduction

The term “liquidity” is used in describing the degree to which an underlying asset can be exercised (sold or bought) in the market setting in a way that the asset’s price is not affected (Acharya & Pedersen, 2005; Amihud & Mendelson, 1986). Addressing this leads to a nonlinear option pricing model extended to time-fractional form and solved for approximate-analytical solutions via a proposed semi-analytical method.

Results

Conclusion