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Efficient Markets, Random Walks, and Martingale Pricing

Article Quant Q&A · Author: Claire
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Summary

The document distinguishes the efficient market hypothesis (EMH) from the random walk hypothesis (RWH). EMH describes prices incorporating available information continuously; RWH says price changes are not predictable and follow a random walk. A random walk can be consistent with efficiency if future information itself cannot be predicted, but the two ideas are not identical definitions.

It also separates real-world expected returns from risk-neutral pricing. Under the answer’s account, a stock price in an efficient market need not be a martingale because investors expect compensation for risk. Nor should its discounted price generally be treated as a martingale under real-world probabilities. In an arbitrage-free model, however, the discounted stock price is a martingale under risk-neutral dynamics. This is a concise conceptual response, not a derivation; the claims depend on the probability measure and assumptions used in the pricing model.

Key ideas

  • EMH concerns how available information is reflected in prices, while RWH concerns price predictability.
  • A random walk may be consistent with market efficiency without being its definition.
  • Expected risk premia mean stock prices need not be martingales under real-world probabilities.
  • In an arbitrage-free model, discounted prices are martingales under risk-neutral dynamics.

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EquitiesStatisticsDerivatives pricing
Full text
# Is the stock price process a martingale or a random walk in efficient markets?


# Is the stock price process a martingale or a random walk in efficient markets?












What is the difference between RWH and EMH?

In efficient market, the price will be fully reflected by available information.

If there is no news, the price would be unchanged. If there is a news, the price would immediately adjust to a new price reflecting the price. This is the same as the idea of RWH.

However, it is not necessary for efficient efficient market to have random walk prices. I do not totally understand the difference between two hypothesis.

Also, when will the stock prices follow martingale property in efficient market? Only in risk-neutrality?

Which one is the better model for EMH?

## Answer by Slug Pue (score 2)

https://quant.stackexchange.com/a/19335

EMH: An asset always trades at its fair value. That is, all information is continuously being priced in.

RWH: The asset price is not predictable and follows a random walk.

So RWH is a hypothesis which is consistent with EMH. If every piece of information is being priced in continuously, and you cannot predict what information will become available, then from your standpoint the price follows a random walk.

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.