A no-arbitrage theorem for uncertain stock model

Abstract

Stock model is used to describe the evolution of stock price in the form of differential equations. In early years, the stock price was assumed to follow a stochastic differential equation driven by a Brownian motion, and some famous models such as Black---Scholes stock model and Black---Karasinski stock model were widely used. This paper assumes that the stock price follows an uncertain differential equation driven by Liu process rather than Brownian motion, and accepts Liu's stock model to simulate the uncertain market. Then this paper proves a no-arbitrage determinant theorem for Liu's stock model and presents a sufficient and necessary condition for no-arbitrage. Finally, some examples are given to illustrate the usefulness of the no-arbitrage determinant theorem.