Random Walks Do Not Require Symmetric Price Changes
Summary
The document clarifies that a random walk does not inherently require prices to rise as often as they fall, or upward and downward moves to have equal sizes. It presents a stock-price model in which log prices evolve through independent, identically distributed normal increments, then explains that symmetry is an additional modeling assumption rather than a defining feature of random walks.
The discussion uses an analogy to repeated coin flips: the probability of each outcome and the associated payoffs need not be equal for a process to be a random walk. It also notes that modeling choices differ across assets. For example, lognormal price behavior can imply asymmetry in price changes, and debt returns can have infrequent, large losses from default. These are explanatory examples rather than a test of any market’s returns; the simple model’s distributional assumptions may not capture real-world dependence or changing return behavior.
Key ideas
- A random walk does not require equal probabilities of upward and downward moves.
- The probability and payoff size of each step can be asymmetric.
- A common stock model uses independent, identically distributed normal increments in log price.
- Asset-specific features such as default losses can produce asymmetric outcomes within a random-walk framework.
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Full text
# Does the random walk theory assume a simple symmetric random walk?
# Does the random walk theory assume a simple symmetric random walk?
Does the random walk theory assume a simple symmetric random walk? In other words: does the random walk theory assume that the price rises as often as it falls? I've been looking for an answer for a while but I'm still not sure. Maybe this question is very simple, but I think that it will then be easier for other people to find an answer to this question.
Note: I‘m an undergraduate economics student.
## Answer by Sane (score 0, accepted)
https://quant.stackexchange.com/a/69061
Random walk theory assumes that stock price can be modeled by:
$$log(S_t)=log(S_{t-1})+\epsilon_t, \epsilon_t~ iid N(0, \sigma^2).$$
In other words, stock price follows a random walk, as described above.
## Answer by demully (score 0)
https://quant.stackexchange.com/a/69077
For the stockmarket, this is often a first-pass default assumption. Add in a bit of financial spice, however, and you distribute stock returns lognormally (as opposed to normally), then strictly speaking, they cease to be symmetrical. But 99% of people could not tell the difference 99% of the time.
Turning to debt markets, this rapidly falls apart. 99% of the time, the company does not default and I get my coupon. 1% of the time, they do default and I lose 70-75% of my capital. This is clearly not symmetrical. But this can still represent a random walk!
Think of the random walk as a repetitive coin-flipping exercise. At each step, you flip a coin. But the probability of heads does not have to be 50%; and the relative payoff of heads:tails does not have to be 1:-1. Nothing in the maths of random walks requires symmetry.
Except this is often lazily assumed because it makes the entire subject much easier to think about :-)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.