Editing Efficient-market hypothesis (section)

== Theoretical background ==
Suppose that a piece of information about the value of a stock (say, about a future merger) is widely available to investors.  If the price of the stock does not already reflect that information, then investors can trade on it, thereby moving the price until the information is no longer useful for trading.

Note that this thought experiment does not necessarily imply that stock prices are unpredictable. For example, suppose that the piece of information in question says that a financial crisis is likely to come soon. Investors typically do not like to hold stocks during a financial crisis, and thus investors may sell stocks until the price drops enough so that the expected return compensates for this risk.

How efficient markets are (and are not) linked to the [[random walk hypothesis|random walk]] theory can be described through the [[fundamental theorem of asset pricing]].  This theorem provides mathematical predictions regarding the price of a stock, assuming that there is no [[arbitrage]], that is, assuming that there is no risk-free way to trade profitably.  Formally, if arbitrage is impossible, then the theorem predicts that the price of a stock is the discounted value of its future price and dividend:

:<math>P_t = E_t[M_{t+1} (P_{t+1}+D_{t+1})]</math>

where <math>E_{t}</math> is the expected value given information at time <math>t</math>,  <math>M_{t+1}</math> is the [[stochastic discount factor]], and <math>D_{t+1}</math> is the dividend the stock pays next period.

Note that this equation does not generally imply a [[random walk hypothesis|random walk]].  However, if we assume the stochastic discount factor is constant and the time interval is short enough so that no dividend is being paid, we have

:<math>P_t = M E_t[P_{t+1}]</math>.

Taking logs and assuming that the Jensen's inequality term is negligible, we have

:<math>\log P_t = \log M  + E_t [\log P_{t+1}]</math>

which implies that the log of stock prices follows a [[random walk hypothesis|random walk]] (with a drift).

Although the concept of an efficient market is similar to the assumption that stock prices follow:

<math>E[S_{t+1}|S_{t}]=S_{t}</math>

which follows a [[Martingale (probability theory)|martingale]], the EMH does not always assume that stocks follow a martingale.