13 votes

Do markets really follow a random walk or is this idea outdated?

No, the idea is not outdated. Fama (1970, JF) summarises the early research on stock price behaviour and defines an efficient market as follows: A market in which prices always ''fully reflect'' ...
Kevin's user avatar
  • 15.9k
6 votes

Do efficient market hypothesis and random walk theory convey the same concept?

They are different concepts, and the relation between them can be described as a conditional: "if EMH holds (all available information about future price movements is already priced into the market), ...
Andrew Maliska's user avatar
6 votes

Fama: Efficient Capital Markets: A Review of Theory and Empirical Work - are martingales incorrect?

The way I understand it is: In equation 2 $x_{j, t + 1}$ is defined as the change in of $p_j$ over the period $t$ to $t + 1$. The formula says that the expectation of the change is zero which is the ...
Bob Jansen's user avatar
  • 8,542
6 votes
Accepted

How to model stock price for a monte carlo simulation with fat tails and asymmetric risk

I did a related project years ago and an example from Matlab's website, Using Extreme Value Theory and Copulas to Evaluate Market Risk, proved particularly helpful as the starting point. At a high ...
Helin's user avatar
  • 11.7k
5 votes

Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?

I don't think I was clear in my comment so I'm putting it in an answer to have more space. The variance of a brownian motion, z, is $t$. (i.e: $E(z^{2}) = t$ ). Notice that $R_{i}$ really equals $\...
mark leeds's user avatar
  • 1,102
5 votes

Do efficient market hypothesis and random walk theory convey the same concept?

Historically the RWT (Random Walk Theory) came first, as empirical observations by for example M.F.M. Osborne (1959) and others in the 1960s. The EMH came about as a result of theoretical work by ...
nbbo2's user avatar
  • 11.2k
5 votes
Accepted

Cannot Understand The Ticket Line Question From Interview Book

If you allow for exchanges between the people in the queue, then there is no problem to be solved. So the implicit assumption must be that people in the queue only deal with the counter.
Magic is in the chain's user avatar
4 votes

Determining if a time series is random

So there are several issues with your posting that you will need to resolve. The first one is your concept of randomness and distinguishing between a random event and a non-random event. To ...
Dave Harris's user avatar
  • 4,389
4 votes
Accepted

some doubts about answers to ticket line question from interview book

The way I understand this approach: you start at $A = (0, 0)$. Every time a 5\$ person wants to buy a ticket you move one unit to the right and unit up. Every time a 10\$ person wants to buy a ticket ...
Cettt's user avatar
  • 1,446
4 votes
Accepted

Proving Scaled Random Walk Approaches Normal Distribution

$X_j$ can be either 1 or -1 with 50% probability each. So this step is just applying the expectation to both possible cases. See definition of the Expectation... \begin{align} {\mathbb E}\bigl[ X \...
StackG's user avatar
  • 3,016
3 votes
Accepted

Why are changes in stock market wealth considered permanent?

I guess the concept you're looking for are martingales. These are stochastic processes which remain on their current level (in expectation!). Ignoring some technical conditions, a stochastic process $...
Kevin's user avatar
  • 15.9k
3 votes

How to estimate the probability of Clustering illusion in our backtest result?

False positives and false negatives are concepts from Frequentist statistical inference. They depend on people not data mining. Using Keynesian notation, they test $\Pr(X|\theta)$. That is to say, ...
Dave Harris's user avatar
  • 4,389
3 votes

Expected number of days inside a corridor

This looks to me like a range accrual. Let $t_1, \ldots, t_n$, where $0 < t_1 < \cdots < t_n$ be business days that are being considered. We compute \begin{align*} E\left(\sum_{i=1}^n \pmb{1}...
Gordon's user avatar
  • 21.1k
3 votes
Accepted

Random Walk of N Correlated Assets

The correlation matrix refers to the correlations between the asset returns. In fact, it can be seen as follows. Each asset follows a geometric Brownian motion, i.e., $$ \frac{{\rm d}S_t^i}{S_t^i}=\...
hypernova's user avatar
  • 396
3 votes

Generally how to simulate bivariate (or multidimensional) BM sample paths?

For the two-dimensional case, the Cholesky decomposition of the covariance matrix \begin{equation} \Sigma = \left( \begin{array}{c c} \sigma_1^2 & \rho \sigma_1 \sigma_2\\ \rho \sigma_1 \sigma_2 &...
LocalVolatility's user avatar
3 votes
Accepted

The right choice when the price of a stock follows a random walk

It makes no difference. Starting with a capital of 1, let $X_i$ be the multiplying factor for the $i$th day, so $X_i\in\{1+r,1-r\}$ with each possibility having probability 1/2. The expected capital ...
Bjørn Kjos-Hanssen's user avatar
3 votes

Simulating artificial asset prices: Random walk vs Brownian motion?

Echoing some of the comments to the OP above, the only real difference between random walks and Brownian motions is a question of time frequency. IE a Brownian motion is just an aggregation of a (...
demully's user avatar
  • 5,071
2 votes

Counting random paths

As I mentioned above, I am not sure what the variable $r$ is. If we ignore that, or assume the questioner wanted to say its the risk free interest rate, then it has no effect on the number of paths. ...
Borun Chowdhury's user avatar
2 votes
Accepted

random walk with drift and absorption barrier

Firstly, this is very normal fare for Stochastics which studies properties of such processes. So if you're not finding anything, look up any stochastics textbook. One thing that is not clear from ...
Phil H's user avatar
  • 3,669
2 votes

How to create a volatile market, by combining less volatile markets?

Let $X_1$ and $X_2$ be your two assets and $C$ your financial product. For now we only assume products which are a linear combination of $X_1$ and $X_2$ with no shorting allowed, hence: $$\begin{align}...
Daneel Olivaw's user avatar
2 votes

How to create a volatile market, by combining less volatile markets?

Throw in correlation as the additional variable. Similarly, volofvol could be another candidate to play with. A spread option could have a larger volatility than either of the two underliers.
bhutes's user avatar
  • 996
2 votes

How does the efficient market hypothesis fit with the rapid changes in prices?

It depends on which form of EMH you're considering as to provide a rationale. Strong-form EMH could or would assert there's private information potentially changing hands that agents are acting on ...
Chris's user avatar
  • 1,643
2 votes

Rare Events in Normal Multivariate distributions

If you’re comfortable making the assumption of multivariate normality (I’m not sure that you are), then this seems like a perfect place to use Mahalanobis distance. One of the first facts that ...
Dave's user avatar
  • 353
2 votes

Forget Kelly, forget fractional sizing. Where is the general theory?

I've recently had to do quite a bit of work on position sizing. Leonard C MacLean, Edward O Thorp, and William T Ziemba have written an incredible amount of literature on this. The following text ...
Jacques Joubert's user avatar
2 votes
Accepted

Variance of Random Walk with Drift

Setup Let $$Z_n\equiv \prod\limits_{i=1}^n(1+x_i)$$ where each $x_i$ is iid normally distributed as $x_i\sim \mathrm{N}\left(\tilde{\mu},\sigma\right)$. For simplicity, and with some abuse of ...
Kermittfrog's user avatar
  • 6,554
1 vote
Accepted

Accumulation Rate of Variance in Random Walk

If we denote the random walk with $(X_k)_{k \in \mathbb{N}}$ than for all $k$ the random variable $\Delta X_k := X_{k} - X_{k-1}$ has mean zero and variance one: \begin{align} \mathbb{E}[\Delta X_k] = ...
Cettt's user avatar
  • 1,446
1 vote

How to estimate the probability of Clustering illusion in our backtest result?

I think what you have in mind is similar to the Cowles Test, described by Alfred Cowles in his 1933 paper Can Stock Market Forecasters Forecast? (link) Cowles wanted to evaluate the trading ...
Alex C's user avatar
  • 9,372
1 vote

If price is a random walk, is ok to use the binomial distribution to estimate a trading strategy?

It is a good idea to make an assumption of "no informational content" on prices to have a reference level for this $H_0$ hypothesis. The best is probably to make Monte-Carlo simulations, i.e. to ...
lehalle's user avatar
  • 12.1k
1 vote
Accepted

How does the efficient market hypothesis fit with the rapid changes in prices?

You‘re right. The "true price" should only jump when news arrives but in practice, market participants need time to arrive at a new equilibrium, i.e. the market needs some time until it is clear how ...
Kevin's user avatar
  • 15.9k

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