16
May 18

## Efficient market hypothesis is subject to interpretation

The formulation on Investopedia seems to me the best:

The efficient market hypothesis (EMH) is an investment theory that states it is impossible to "beat the market" because stock market efficiency causes existing share prices to always incorporate and reflect all relevant information. According to the EMH, stocks always trade at their fair value on stock exchanges, making it impossible for investors to either purchase undervalued stocks or sell stocks for inflated prices. As such, it should be impossible to outperform the overall market through expert stock selection or market timing, and the only way an investor can possibly obtain higher returns is by purchasing riskier investments.

This is not Math, and the EMH interpretation is subjective. My purpose is not to discuss the advantages and drawbacks of various versions of the EMH but indicate some errors students make on exams.

## Best(?) way to answer questions related to EMH

Since there is a lot of talking, the best is to use the appropriate key words.

Start with "The EMH states that it is impossible to make economic profit".

Then explain why: The stock market is efficient in the sense that stocks trade at their fair value, so that undervalued or overvalued stocks don't exist.

Then specify that "to obtain economic profits, from the revenues we subtract opportunity (hidden) costs, in addition to direct costs, such as transaction fees". What on the surface seems to be a profitable activity may in fact be balancing at break-even.

Next is to address the specification by Malkiel that the EMH depends on the information set $\Omega_t$ available at time $t$.

Weak form of EMH. The information set $\Omega_t^1$ contains only historical values of asset prices, dividends (and possibly volume) up until time $t$. This is basically what an investor sees on a stock price chart. Many students say "historical information" but fail to mention that it is about prices of financial assets. The birthdays of celebrities are also historical information but they are not in this info set.

Semi-strong form of EMH. The info set $\Omega_t^2$ is all publicly available information. Some students don't realize that it includes $\Omega_t^1$. The risk-free rate is in $\Omega_t^2$ but not in $\Omega_t^1$ because 1) it is publicly known and 2) it is not traded (it is fixed by the central bank for extended periods of time).

Strong form of EMH. The info set $\Omega_t^3$ includes all publicly available info plus private company information. Firstly, this info set includes the previous two: $\Omega_t^1\subset\Omega_t^2\subset\Omega_t^3$. Secondly, whether a certain piece of information belongs to $\Omega_t^2$ or $\Omega_t^3$ depends on time. For example, the number of shares Warren Buffett purchased today of the stock is in $\Omega_t^3$ but over time it becomes a part of $\Omega_t^2$ because large holdings must be reported within 45 days of the end of a calendar quarter. If there are nuances like this you have to explain them.

## Implications for time series analysis

Conditional expectation is a relatively complex mathematical construct. The simplest definition is accessible to basic statistics students. The mid-level definition in case of conditioning on a set of positive probability already raises questions about practical calculation. The most general definition is based on Radon-Nikodym  derivatives. Moreover, nobody knows exactly any of those $\Omega_t$. So how do you apply time series models which depend so heavily on conditioning? The answer is simple: since by the EMH the stock price "reflects all relevant information", that price is already conditioned on that information, and you don't need to worry about theoretical complexities of conditioning in applications.

11
Mar 18

## Option chain and efficient market hypothesis

Figure 1. Option chain for AAPL with 26 days to expiration

In charting software, option prices for a given stock and expiration date are given in a table called an option chain.

## Option chain description

1. The light colored column in the center contains strike prices. They take discrete values. The tick (the step between two nearest strike prices) depends on the stock price. It is large for expensive stocks and small for inexpensive ones.
2. As one can see in the upper right corner, the current price of Apple stock is \$176.94. This would be the theoretical at the money strike. However, since strikes can take only discrete values, the one closest to the current stock price is said to be at the money. It is \$177.5. At this value the option chain is divided by color in two parts: with in the money calls (the upper part of the chain) and out of the money calls (the lower part).
3. The leftmost column shows the open interest, which is the number of options currently in circulation. You can see that for the 155 strike the open interest is zero.
4. In column 2 you can see the trade volume for the day.
5. In the next column you can see the bid size (total number of buy orders).
7. The fifth column contains ask sizes.
8. The sixth column shows the implied volatility, the king among option price determinants.
9. The next three columns contain option greeks, which are derivatives of the option price with respect to various variables. We'll talk about them separately. The look of the option table depends on the software and user's choices. I am using the Trader WorkStation from Interactive Brokers, and my option chain doesn't contain one more greek - Gamma.
10. I chose to see strikes within two standard deviations from the current price (see the upper right corner). The option "SMART" means that my broker will automatically choose the best price from several exchanges. The "100" is the multiplier, which shows that one option commands 100 shares of stock, and it implies that the option price listed as, say, 6.90 means that in fact the option costs \$690.

## Efficient market hypothesis

This hypothesis is a standard assumption in Economics. It states that market prices reflect "all relevant" information. The exact definition of the "relevant" information depends on how strong economists want the hypothesis to be. The main practical implication is that it's impossible to make profits in financial markets. I fail to logically connect the statement "market prices reflect all information" with the statement that "it's impossible to make profits". But my main point is not this.

The main point is that economists themselves indicate reasons why the efficient market hypothesis may not hold. One is the cost of obtaining information. The option chain for Apple for each expiration date contains dozens of strikes, and each strike means a separate market, although there is a high correlation between these markets. Multiply that number of markets by the number of expiration dates (weekly options give about 50 expiration dates and monthly options give another 12, that's for one year), and you will see that Apple alone generates hundreds, if not thousands, of markets.

Another reason why the hypothesis may not hold is transaction costs. The bid-ask spread (which is the difference ask-bid)  in the lower part of the above option chain is \$7 and in the upper part is \$50. In the options markets, the transaction costs and the cost of obtaining information will prevent any big player from attempting to capture all profits.

Dare, and you will be rewarded!