Option Pricing Under an Abnormal Economy: using the Square Root of the Brownian Motion

Abstract

Purpose: The literature on option pricing is typically suitable to usual circumstances (normal
economy). However, in general, under unusual economic states, the traditional models of
options are not suitable. Therefore, there is a need to consider alternative stochastic processes
and models that captures the unusual states of the economy.

Design/methodology/approach: In this connection, we bridge the gap in the literature by
providing a simple, explicit pricing formula for the European option under both normal and
abnormal economies.

Findings: In this paper, we first discuss the background theory for the Black-Scholes model
under a normal economy when there are no unusual changes in the price of the underlying so
that Brownian motion works well. We then provide a simple, explicit pricing formula for the
European option under both normal and abnormal economies. This formula is as simple as
the classical Black-Scholes formula and there is no need for computational methods. In doing
so, we utilize a nontraditional process (the square root of the Brownian motion) and complex
analysis. We also rely on a non-traditional stochastic process. Thereafter, we construct three
examples to illustrate the use of our proposed model.

Originality/Value: Practical implications: The theory developed in this paper is used for investors for their investments
and is useful for policy-makers in setting up some rules for the options markets.