Research on α-Arbitrage for Uncertain Stock Market Model

This paper introduces the concept of α-arbitrage opportunity based on the uncertain stock model, where the stock price follows an uncertain differential equation driven by the canonical Liu process rather than the traditional Brownian motion. Mainly, a sufficient and necessary condition for the uncertain stock model being strict no-arbitrage, i.e. absent of α-arbitrage opportunity is derived. The uncertain stock model is strict no-arbitrage if and only if the system of linear equations has a solution and the absolute value of the solution does not exceed the given threshold, which depends on the investors' belief degree. Some examples are given to illustrate the strict no-arbitrage theorem, especially to show that an α-arbitrage opportunity is possible in a no-arbitrage market. The numerical analysis shows that the market can be strict no-arbitrage when the belief degree is 99% except for five months and 98% except for seven months. Moreover, the optimal belief degree and the α-arbitrage boundary curve are obtained in the numerical analysis.