Efficient market hypothesis and martingales

Question

One of the tasks in the book we´re using in introduction to finance is

Stocks are expected to earn (much) more than the risk-free interest rate. This means that stock prices are expected to increase over time which, in turn, means that stock prices will be positively autocorrelated and that they are not a fair game or a martingale as the EMH claims. Is this reasoning correct?

The answer is

No, over short time intervals (e.g. days) the expected return is so small that it can be ignored in autocorrelation calculations. 20 per cent return per year over 250 trading days means less than 0.1 per cent per day, very small compared to daily price changes. The fair game model does not require returns (price changes) to have zero expectation, but the excess returns, or deviations from the expected returns. Similarly, the EMH does not require stock prices to be martingales but the properly discounted stock prices. The stock prices themselves are expected to increase with required rate of return on the stock.

From the answer on this thread: Is the stock price process a martingale or a random walk in efficient markets? I read that

On martingales: The stock itself is never a martingale in an efficient market. That is a popular misconception. If that were true, the risk premium for the stock would be negative and you would invest in riskless assets instead. Even the discounted stock price shouldn't be a martingale, because, again, that would imply that the risk premium is 0 and again the riskless asset would be a better choice. However, the discounted stock price under risk-neutral dynamics is a martingale if the market is arbitrage-free.

The question in my book says EMH claims that stock prices are martingales. But the answer from that thread seems to contradict that. The answer for the question in the book also says that EMH does not require stock prices to be martingales, but the properly discounted stock prices which is also not in line with the answer given in the linked thread.

What's the difference here? The answer given in the other thread says "the stock itself is never a martingale(…)". Is it that the book refers to the market as a whole when it mentions "stocks" and not individual stocks?

Also can someone clarify: Why is it that if an individual stock was a martingale, the risk premium for the stock would be negative and you would invest in riskless assets instead?

Answer 1

Your book is right. Samuleson--you'll find his name written all over EMH's history--proved it in 1973.

https://www.jstor.org/stable/3003046?seq=1#metadata_info_tab_contents

The pertinent section, "properly anticipated future prices fluctuate randomly--i.e. contain a martingale sequence, or a generalized martingale with with specifiable mean drift."

Derivatives are not priced to generate 0% return; they are priced to generate a return equal to the risk-free rate. The comment referenced in your original post understands this when it says, "the discounted stock price under a risk-neutral measure is a martingale." Well, the same can be said about the discounted stock price under a real world measure, exactly like your book says, "Similarly, the EMH does not require stock prices to be martingales but the properly discounted stock prices".

In a risk-neutral setting, stock prices are expected to grow and be discounted at the risk-free rate. In a real-world setting, stock prices are expected to grow and be discounted at CAPM. As long as your expected growth and discount is the same, you have a martingale.

Just as a simple exercise, imagine if the risk free rate was 8%, like in the 80s. Everyone acknowledges that discounted derivative prices are martingales under risk-neutral pricing, even when your stochastic drift is 8%. Now imagine your discount rate is 8%, calculated using a 2% risk free rate and a 6% risk premium. If your expectation matches your discount, why is your discounted stock price not a martingale? Martingality is not contingent on the size of your drift; you can add a risk premium to your expected return and still be martingale if your discount rate adds the same risk premium.