Stock Prices in a Perfect Market Let Xn,, be the closing price at the end of day n of a certain publicly traded security such as a share of stock. While daily prices may fluctuate, many scholars believe that, in a perfect market, these price sequences should be martingales. In a perfect market freely open to all, they argue, it should not be possible to predict with any degree of accuracy whether a future price X n+1 will be higher or lower than the current price Xn For example, if a future price could be expected to be higher, then a number of buyers would enter the market, and their demand would raise the current price X Similarly, if a future price could be predicted as lower, a number of sellers would appear and tend to depress the current price. Equilibrium obtains where the future price cannot be predicted, on average, as higher or lower, that is, where price sequences are martingales.
The above is from textbook an introduction to stochastic process. Can anyone please clarify whether or the sequence of prices is martingale. The text seems to be a bit contridictory to me.
In this ideal "Perfect Market", scholars believe that the price sequences should be martingales since you cannot predict the future prices. In other words, if you relate this to part b) of the definition of a martingale provided in the book, we see that $$E[X_{n+1}|X_0,\dotsc,X_n] = X_n.$$