✎ Random Walks [The Physics Travel Guide]

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====== Random Walks ======

<blockquote>Imagine a man coming out of a bar in cancun, an open bottle of sunscreen dribbling from his back pocket. He walks forward for a few steps, and then there’s a good chance that he will stumble in one direction or another. He steadies himself, takes another step, and then stumbles once again. the direction in which the man stumbles is basically random, at least insofar as it has nothing to do with his purported destination. If the man stumbles o en enough, the path traced by the sunscreen dripping on the ground as he weaves his way back to his hotel (or just as likely in another direction entirely) will look like the path of a dust particle  oating in the sunlight. [...]

Imagine that the drunkard from cancun is now back at his hotel. He gets out of the elevator and is faced with a long hallway, stretching to both isle and his right. At one end of the hallway is room 700; at the other end is room 799. He is somewhere in the middle, but he has no idea which way to go to get to his room. He stumbles to and fro, half the time moving one way down the hall, and half the time moving in the opposite direction. Here’s the question that the mathematical theory of random walks allows you to answer: Suppose that with each step the drunkard takes, there is a 50% chance that that step will take him a little further toward room 700, at one end of the long hallway, and a 50% chance that it will take him a little further toward room 799, at the other end. What is the probability that, a er one hundred steps, say, or a thousand steps, he is standing in front of a given room?

<cite>The Physics of Wall Street by Weatherall</cite>
</blockquote>

* For a nice description see chapter 23 and 24 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan
  * and also [[http://pages.physics.cornell.edu/~sethna/StatMech/EntropyOrderParametersComplexity.pdf|Statistical Mechanics: Entropy, Order Parameters, and Complexity]] by Sethna
  * B. Hughes (1995), "Random Walks and Random Environments"
  * Elements of the Random Walk by Rudnick

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

Random walks are essential to understand Brownian motion, to model stock movements in finance and to understand the [[theories:quantum_mechanics:path_integral|path integral]] of quantum theories.

--> Why do stocks follow random walks?#

<blockquote>
Sure, you might say, I believe the mathematics. If stock prices move randomly, then the theory of random walks is well and good. But why would you ever assume that markets move randomly? Prices go up on good news; they go down on bad news. there’s nothing random about it. Bachelier’s basic assumption, that the likelihood of the price ticking up at a given instant is always equal to the likelihood of its ticking down, is pure bunk. this thought was not lost on Bachelier. As someone intimately familiar with the workings of the Paris exchange, Bachelier knew just how strong an e ect information could have on the prices of securities. And looking backward from any instant in time, it is easy to point to good news or bad news and use it to explain how the market moves. But Bachelier was interested in understanding the probabilities of future prices, where you don’t know what the news is going to be. Some future news might be predictable based on things that are already known. A er all, gamblers are very good at setting odds on things like sports events and political elections — these can be thought of as predictions of the likelihoods of various outcomes to these chancy events. But how does this predictability factor into market behavior? Bachelier reasoned that any predictable events would already be re ected in the current price of a stock or bond. In other words, if you had reason to think that something would happen in the future that would ultimately make a share of Microso  worth more — say, that Microsoft  would invent a new kind of computer, or would win a major lawsuit — you should be willing to pay more for that Microso  stock now than someone who didn’t think good things would happen to Microso , since you have reason to expect the stock to go up. Information that makes positive future events seem likely pushes prices up now; infor- mation that makes negative future events seem likely pushes prices down now.

But if this reasoning is right, Bachelier argued, then stock prices must be random. think of what happens when a trade is executed at a given price. this is where the rubber hits the road for a market. A trade means that two people — a buyer and a seller — were able to agree on a price. Both buyer and seller have looked at the available information and have decided how much they think the stock is worth to them, but with an important caveat: the buyer, at least according to Bachelier’s logic, is buying the stock at that price because he or she thinks that in the future the price is likely to go up. the seller, meanwhile, is selling at that price because he or she thinks the price is more likely to go down. taking this argument one step further, if you have a market consisting of many informed investors who are constantly agreeing on the prices at which trades should occur, the current price of a stock can be interpreted as the price that takes into account all possible information. It is the price at which there are just as many informed people willing to bet that the price will go up as are willing to bet that the price will go down. In other words, at any moment, the current price is the price at which all available information suggests that the probability of the stock ticking up and the probability of the stock ticking down are both 50%. If markets work the way Bachelier argued they must, then the random walk hypothesis isn’t crazy at all. It’s a necessary part of what makes markets run. [...] This way of looking at markets is now known as the **efficient market hypothesis**. The basic idea is that market prices always reflect the true value of the thing being traded, because they incorporate all available information.<cite>The Physics of Wall Street by Weatherall</cite></blockquote>

<--
<tabbox History> 
  * An entertaining description can be found in "On the wonderful world of random walks" by  E.W. Montroll and M.F. Shlesinger.
</tabbox>

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