Markov-Modulated Geometric Brownian Motion and Bollinger Bands

In the stock market, Bollinger bands as a popular technical analysis tool are widely used by traders. There are a lot of models built to forecast the stock price, so it is a significant issue to investigate whether these models have Bollinger band property. Liu, Huang and Zheng (2006) and Liu and Zheng (2006) discussed the Bollinger bands for Black-Scholes model and stochastic volatility model as real stock markets, respectively. The stationarity and the law of large number of the corresponding statistics were proved. In this paper, we extend the above results to the general model of Markov-modulated geometric Brownian motion.