Martingales, Expected Returns, and Market Efficiency
Summary
The document explains how expected price growth can coexist with a martingale description of prices. A stock may have a positive expected return because investors require compensation for risk; that drift does not by itself imply predictable deviations from expected returns. For martingale reasoning, the relevant process is adjusted for the expected return or discount rate, and the adjustment depends on the measure and pricing framework being used.
The answer distinguishes real-world expected growth, which may include a risk premium, from risk-neutral growth at the risk-free rate. Under either framework, a price discounted at the matching rate can be a martingale. It cites Samuelson's work on properly anticipated prices and gives illustrative discount-rate scenarios, but provides no empirical tests. The discussion is conceptual: conclusions depend on specifying the probability measure, discount rate, and asset or claim being modeled. It also cautions against interpreting the efficient market hypothesis as requiring raw stock prices to have zero drift or be martingales.
Key ideas
- A stock price can have a positive expected return while still having unpredictable deviations from its expected path.
- Martingale properties depend on how prices are adjusted and which probability measure is used.
- Discounting at a rate that matches expected growth can yield a martingale even when that rate includes a risk premium.
- Risk-neutral valuation discounts expected growth at the risk-free rate, while real-world expectations may include compensation for risk.
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Full text
# Efficient market hypothesis and martingales # Efficient market hypothesis and martingales 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 by Mild_Thornberry (score 1) https://quant.stackexchange.com/a/61603 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.
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