25

I can help you beat random walk 'in the way you want', i.e. the expected value $E[\$]$ will always be positive even assuming no drift. However, I have to warn people that $E[\$] > 0$ is NOT really an adequate condition for 'beating' in reality (at least to myself). Let's define some mathematical notations for derivation, and rephrase (simplify) vonjd's ...


20

Variance ratio tests have been used numerous times to show that financial asset prices do not follow a random walk. You can for example look at -Lo and MacKinlay : Stock market prices do not follow a random walk : http://press.princeton.edu/books/lo/chapt2.pdf (US Stocks) -Hoque, Kim, Pyun: A comparison of variance ratio tests of random walk: A ...


9

Your formula looks like cointegration (between the price time series) rather than correlation (between the returns). To simulate "correlated random walks", i.e., random walks built from correlated innovations, you can just build the desired covariance matrix (for instance, put ones on the diagonal and $\rho$ everywhere else), take multivariate gaussian ...


9

Why don't you try it and report back? Recall, though, that while a random walk is often a rather competitive forecast, realized data is understood to have weak dependence especially in higher moments. Having worked a bit with DieHarder, I'd suspect it to reject a number of series. But the proof is in the pudding...


8

I will assume a white noise is a process $(\varepsilon_t)$ with zero mean, no autocorrelation and constant variance $\sigma^2 > 0$ while a random walk is a process $(x_t)$ defined by $$ x_{t+1} = x_t + \varepsilon_{t+1} $$ where $\varepsilon$ is a white noise. 1) No since $Var(x_{t+1}) = Var(x_t) + Var(\varepsilon_{t+1})$ is stricly increasing while ...


7

If the price of every asset follows an independent random walk without drift then every position has an expected return of zero. So, in expectation, there is no combination of positions that has an expectation different from zero.


6

First, for Ito processes and Brownian motion. Ito process is a continuous-time trajectory with random evolution, so non-smooth and very kinky - also has a fractal look: no matter how much you'd zoom in, it will look similar. Ito process consists in fact of two parts: the drift part (deterministic evolution) and the diffusion part (where all the kinkiness and ...


6

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), then future price movements will follow a purely random walk as new and unpredictable information emerges"


5

I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is perfectly symmetric whereas the real one takes longer for going up and then crashes in a relatively shorter time frame. This is a known phenomenon in real financial ...


5

All the ideas above are great ideas. Another kind of test would be an idea borrowed from Random Matrix Theory. Assemble your time-series into a matrix. Evaluate the distribution of the eigenvalues of the matrix vs. the distribution of a random matrix. Turns out that the distribution of eigenvalues in a random matrix conforms to distributions such as the ...


5

I have tested lots of forex data for randomness. Some currency pairs are very close to random walk. And the problem is open question, because there is no uniform explanation what the random walk is. According to Mandelbrot, Taleb and some other authors randomness can be different. Even if the data is not random it doesn't mean it can be effectively traded. ...


5

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 Samuelson in 1965 ("Proof that properly discounted prices...") and E.Fama (1969) as a general empirical/theoretical hypothesis that guided the field for many ...


4

Here is a more or less formal proof of the fact that "the system can't be beaten". The argument works whenever the underlying process is a martingale. In particular, it is valid for a random walk without drift. Let $S=\{S_n\}$ be a discrete-time martingale which represents a series of games played at times $n=1,2,...$. Assume that $S_0=0$ (no game at time $...


4

I would start with explaining random walk (this should be fairly simple) and then making a connection to heat equation in discrete time. This paper is doing exactly this and by leaving out technicalities you should make this pretty intuitive for students. Basically the intuition is as follows: At each integer time unit, the heat at each point is spread ...


4

You have intense academic research on orderbook dynamics simulations, just cite: Econophysics: Empirical facts and agent-based models, by Anirban Chakraborti, Ioane Muni Toke, Marco Patriarca, Frederic Abergel (Arxiv 2010) High Frequency Simulations of an Order Book: a Two-Scales Approach by: Charles-Albert Lehalle, Olivier Guéant, Julien Razafinimanana, In ...


4

That post has been up since March. Either he hasn't figured it out, or he's trying to get people to click through to the book. In the following statement, isn't he implying that "rw" is a return (as in....random walk)? rw <- rnorm(100) In the following statements, isn't he calling a "trade" the DIFFERENCE IN RETURNS? Isn't that meaningless? if(...


4

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 $\sqrt{\frac{t}{n}} \times \epsilon$ where $\epsilon \sim N(0,1)$. I think they leave the $\epsilon$ out because the variance is 1 but showing the consistency is ...


3

Assuming no math at all: Using an Ito process we can describe the return of a stock with two components: an average level (the "drift") plus some uncertainty (the "volatility"). This uncertainty is represented by a Brownian Motion. As written in Wikipedia, A random walk is a mathematical formalization of a path that consists of a succession of ...


3

Using $q = 1-p$ we can work out the root as: $$\sqrt{1-4pq} = \sqrt{1-4p(1-p)} = \sqrt{1-4p+4p^2} = \sqrt{(1-2p)^2}$$ Taking the positive root reduces this to $(1-2p)$. This gives for the fraction: $$\frac{1 + \sqrt{1-4pq}}{2p} = \frac{1 + (1-2p)}{2p} = \frac{1-p}{p}$$ This also holds inside the logarithm.


3

In the Ljung-Box test, the null hypothesis is: $H_0$: The data are independently distributed So, your p-values of 0 indeed indicate that you should reject the null hypothesis, but it means that your data is not independently distributed, and in particular that there is some significant autocorrelation in the process. This is obviously the case, because ...


3

1) The probability of a H or T of any next coin toss (fair coin) is always 0.5 because coin tosses are independent of each other. 2) Stock markets, or for that matter any asset, are an entirely different game. First of all the expectancy is not 0.5 of, for example, experiencing an up or down day tomorrow in a stock simply because the distribution is ...


3

There are also the NIST tests used to examine the random number generators used in cryptographic systems. http://csrc.nist.gov/groups/ST/toolkit/rng/index.html I played around with it a little bit, here is the first test, the monobit test. library(stats) library(quantmod) code_input <- function(sym, fn=Cl) { return(na.omit(as.vector(ifelse(ROC(fn(...


3

would you expect your financial data to qualify as being a good random number generator Financial time-series, specifically price-change series, would make terrible random number generators because they generally contain significant dependencies. or would it fail in many of these tests? If you test for randomness, meaning, initial conditions do not ...


3

You did not carefully read the article you yourself linked to. Dollar cost averaging is a generalized concept. What the author compares is a full-sized investment or time-specific partial investments. So, dca is a concept and you draw conclusions from one single approach to dca. There is no mathematical proof that dca works or not because it is one single ...


3

Not sure about the correctness of the first approach, but second approach uses $1 /\sqrt k$ to scale the variance of the total sum by $k$. So the difference of two processes (say $W_t$ and $W_{t+\Delta t}$) generated by the random walk would have a variation of $\Delta t$, which satisfies one of conditions needed to get a Wiener's process.


3

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}_{b_1 < S_{t_i} < b_2} \right) &=\sum_{i=1}^n E\left(\pmb{1}_{b_1 < S_{t_i} < b_2} \right)\\ &=\sum_{i=1}^n \left[E\big(\pmb{1}_{S_{t_i} >...


3

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 after one day is $$\mathbb E(X_1)=\frac12((1+r)+(1-r))=1.$$ After $n$ days, your capital is $X_1X_2\cdots X_n$, and $$\mathbb E(X_1\cdots X_n)=\mathbb E(X_1)\...


3

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 & \sigma_2^2 \end{array} \right) \end{equation} is given by \begin{equation} B = \left( \begin{array}{c c} \sigma_1 & 0\\ \rho \sigma_2 & \sigma_2 ...


3

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., $$ \frac{{\rm d}S_t^i}{S_t^i}=\mu_i{\rm d}t+\sigma_i{\rm d}W_t^i, $$ where the correlation between $W_t^i$ and $W_t^j$ is supposed to be $$ \text{Corr}\left(W_t^i,W_t^j\right)=\rho_{ij}. $$ ...


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