Martingale Stock Prices, Risk Premia, and Market Efficiency

# Martingale Stock Prices


# Martingale Stock Prices












In

http://www.principlesofforecasting.com/files/pdf/Granger-stockmarket.pdf

Granger makes survey of some arguments. In section I there are two hypothesis H01, and H02.

H01: Stock prices are a martingale

H02: A market is efficient with respect to information set I, if it is impossible to make economic profits by trading on the basis of this information set

Maybe I am missing something but stock prices better not be martingales as profitable companies should increase in value as time passes. The price will stay the same if period profits are returned to the shareholders. Otherwise nobody would invest. Hence dividend adjusted prices should be submartingales, at least for profitable companies.

Same goes for H02, I am buying the stock knowing that the company is generating earnings, and even if I am buying at fair value I will have positive returns on the average.

What's the catch here?

Thanks

## Answer by berkorbay (score 1)

https://quant.stackexchange.com/a/14268

As in the vonjd's answer martingale property makes some sense only if considered with the risk premium and risk-free rate ("stochastic discount factor" they say). Discounted stock price process is assumed to be a martingale in many studies.

The root of H02's "evil" is Fama's Efficient Market Hypothesis Survey. It is the most clear and comprehensive survey about the efficient markets hypothesis in 1970s. Both in the paper that you referred to and Fama's paper H01 is not highly regarded.

Concisely he divides the hypothesis into three: Weak form, semi strong form and strong form. Weak form deals only with the past price information, semi strong form adds market reaction to public information (i.e. earnings, dividends, etc.) and the strong form adds the information homogeneity (i.e. no part of the market has private information that can be used to obtain extra gains). To the impatient reader, he defends weak form and semi strong form but he admits deficiency to some extent in the strong form (though he still defends it is enough for him to deem the markets efficient).

Most of what he did is to survey previous studies (including his and his colleagues) and draw conclusions by compare and contrast. Weak form is the most interesting since most of the modern mathematical models has at least a notion of it, including option pricing models. If the memory serves, first a term "fair game" is coined. Then skepticism towards random walk models are discussed and random walk theories are deemed not so solid and not even necessary for a market to be efficient. There are also discussions about the Gaussian assumption (in a negative way).

Semi strong form says that the market adjusts for publicly available information and the market is unbiased towards these information. For stock splits, dividends and earnings he cites studies that even if there is extra movement caused by these kind of information it is confined to a very small subset of assets.

Fama says there is limited evidence against strong form, indicating specialists (aka market makers) have access to privileged information (i.e. order limits). But it was not much of an interest to me so I urge you to read from the source.

Final word. You should only take these as an idea and search for more. Well, it is a model and "some" representation of the market. Academics are usually lazy enough to take those for granted and have 'reasons' to do so ("the guy and followers of the theory earned many Nobel prizes").

ps. I love Taleb's remarks (although with some reservation due to my far less than full grasp about the field) about Nobel prizes related to asset price theoreticians. See some here.

## Answer by Yannis (score 1)

https://quant.stackexchange.com/a/14312

The fundamental theorem of mathematical finance states that under the assumption of no arbitrage (which is by the way an idealization and not possible in the real world), a probability measure exists under which all "relative asset prices" are martingales.

What is meant by "relative asset prices" is the asset prices divided by the price of one particular selected asset being referred as "numeraire".

So H01 is incorrect on two grounds:

1) The "stock price" is universally defined as the number of currency units required to buy one unit of the referred stock. This is in no way a ratio of any two asset prices and therefore does not qualify as a "relative asset price". Note here a "currency" itself is not considered an "asset" in mathematical finance". A bank account where such a currency is deposited is considered an "asset".

2) Any statement of the form: "A stochastic process X is a martingale" is meaningless if the underlying probability measure is left unspecified. In the stated hypothesis H01, X stands for the stock price. The correct statement would instead be "The stock price (divided by the numeraire) is a martingale with respect to a particular probability measure, called the "risk neutral measure". In the quite different "real world measure" the relative stock price exhibits a positive drift, which compensates the risk averse investors for assuming increased risk.

Similar comments apply to H02.

By the way, I have just looked at the mentioned paper http://www.principlesofforecasting.com/files/pdf/Granger-stockmarket.pdf and I could not spot any reference to the measure being used. This indicates lack of mathematical maturity since it is a well known fact (see Girsanov theorem) that by changing the measure one may achieve any drift. In other words, the "martingale property" is measure-dependent.

To make matters even worse, the paper's context seems to imply that all equations refer to the "real world measure". If this is indeed the case, then it is ludicrous to state that stock prices are martingales, since almost nothing - save for dice and fair coin tosses - behaves as martingale in the real world!