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Questions tagged [random-walk]

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24
votes
5answers
7k views

Proof that you cannot beat a random walk

There is much speculation to what degree financial series are random (and what kind of randomness prevails). I want to turn the question on its head and ask: Is there a mathematical proof that ...
21
votes
6answers
3k views

How random are financial data series?

Pseudorandom number generators are often tested using e.g. a test suite like Diehard tests or Dieharder. If one would run these tests e.g. on stock market time series or other financial data, would ...
14
votes
1answer
1k views

Is creating constrained random portfolios a hard problem?

Creating random portfolios with weights $x_i$ can be thought of as sampling from the surface of a simplex given by $$Ex = f$$ and $$Ax \le b$$ Where $E$ and $A$ are constraint matrices for equality ...
10
votes
2answers
850 views

Proving Random Walk Hypothesis in Stock Market

Given the time series for a particular stock market, what are the statistical weapons one can bring on to prove, or disprove that random walk hypothesis?
8
votes
2answers
3k views

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

According to investopedia efficent market hypothesis is The efficient market hypothesis (EMH) is an investment theory that states it is impossible to "beat the market" because stock market ...
8
votes
4answers
1k views

How is stock data objectively different to this random walk?

I have a random walk that is generated as so using python, numpy, and matplotlib ...
8
votes
1answer
863 views

Connections between random walk and heat equation (Material for ~)

I am preparing an undergraduate lecture in quantitative finance and I am looking for material that combines the topics: random walk and heat equation The material should be accessible (intuitive!), ...
7
votes
1answer
2k views

How to simulate correlated assets for illustrating portfolio diversification?

I have seen multiple instances where people try to explain the diversification effects of having assets with a certain level of correlation, especially in the "most diversified portfolio" literature. ...
5
votes
2answers
364 views

Constructing a Brownian motion from a Simple Random Walk

I'm trying to get my head around how a Brownian motion is formed from a simple random walk. I've seen two similar methods used: Why has one approach used $\frac{1}{\sqrt{k}}$ and the other hasn't? ...
5
votes
1answer
591 views

Coin Toss System

Coin Toss Runs Calculator The expected number of runs for two consecutive heads or tails is 3. Is there an edge if we place a progressive constant size bet(limited to 3 times)for consecutive ...
5
votes
0answers
129 views

Optimal mortgage rate strategy

When buying a mortgage, you can choose to "lock in" a rate at any point within 60 days of your closing date. Once locked in, you can't revert. This makes it a secretary problem - in the traditional ...
4
votes
2answers
4k views

Relationships between white noise and random walk

I would like to ask 5 questions about relations between these processes. 1) Could white noise be also a random walk? 2) Could random walk be also a white noise? 3) Could white noise be stationary? ...
4
votes
1answer
3k views

Mersenne twister random number generator in Java for Monte Carlo Sim.

I am using the Mersenne twister random number generator in Java for a Monte Carlo Simulation. I need a uniform distribution of values between -1 and 1. My code is below (I am importing org.apache....
4
votes
2answers
319 views

What sort of order submission strategy would result in a random walk of trade prices?

I have written a simulation that matches buy and sell orders, keeps track of an order book and simulates trades. My first pass at order submission was to generate random orders around the bid/ask ...
4
votes
2answers
252 views

If the distribution of returns in symmetric, why not use a coin toss, small risk & high reward?

If the distribution of returns is symmetric then why not use a coin toss to decide whether to buy or sell Calculate the average velocity of the market (ATR - in technical analysis) Place a stop loss ...
4
votes
0answers
194 views

Is it random walk?

I would like to ask a question about random walk. Campbell, Lo & Mackinlay defined the random walk, in the following way (RW3): $$ cov[f(r_{t}),g(r_{t+k})]=0,\qquad k\neq0 $$ for all $f(\cdot)$ ...
3
votes
4answers
12k views

Difference between ito process, brownian motion and random walk

Can someone explain to a non-math person (myself) what is the difference between these three? If they are so different that a comparison does not even make sense, please point it out. 1.Ito process 2....
3
votes
2answers
109 views

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

Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
3
votes
1answer
184 views

Determining if a time series is random

I originally posted this in the Data Science Stack Exchange. Another poster suggested I post it here. The idea would be to identify "orderly" segments within a market time series and use them to ...
3
votes
2answers
145 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
3
votes
2answers
141 views

What is the difference between these two equations for GBMs?

The two equations commonly found online for GBM are: $\begin{matrix} S_{ t }=S_{ 0 }\exp\left( \left( \mu -\frac { \sigma ^{ 2 } }{ 2 } \right) t+\sigma W_{ t } \right) \\ S_{ t }=S_{ 0 }\exp\left(\...
3
votes
1answer
365 views

How come the existence of ARCH effect is not a violation of Random Walk Hypothesis 3?

An ARCH (autoregressive conditional heteroscedastic) (1) model is: $r_t=\mu +a_t$, where $a_t=$return residual, and $\mu$ is the drift of the stock return $a_t=\sigma_t\epsilon_t$, where $\sigma_t=$...
2
votes
2answers
704 views

Proof showing that dollar cost averaging always worse than lump sum alternative

I am referring to the article here. In a nutshell the article says that using data based on S&P 500 index going back as far as to 1950, dollar cost averaging is performing worse than a lump sump ...
2
votes
2answers
58 views

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

Suppose I have a strategy, I run a backtest on it in only one symbol (suppose the historic data to backtest is 25000 candles). The results of that backtest is: Total Trades = 50 TakeProfit/...
2
votes
1answer
106 views

Random Walk of N Correlated Assets

I am trying to value an option on N assets, say $S^1, S^2,..., S^N$ that expires in $\Delta T$ years using Monte Carlo simulation. I have read many sources that state I should use the following ...
2
votes
1answer
92 views

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

I've got the following question: Suppose the price of a stock either rises or falls by the same percentage for each day. Suppose there is no dividend and the interest rate is 0. Should I buy ...
2
votes
1answer
60 views

Expected number of days inside a corridor

Is there a simple (ish) approximation for the expected number of steps a random walk is within a set of bounds over a given time period? - in particular if i presume log normal and constant vol. If i ...
2
votes
1answer
34 views

Identity given in Shreve volume 1

in a solution to a question about random walks (5.3 i), Part of the answer includes the identity: $$\ln \frac{1+\sqrt{1-4 pq}}{2p}=\ln\frac{1-p}{p}$$ note that $p+q=1$ and $0<p<1/2<q<1$. ...
2
votes
1answer
58 views

Why are changes in stock market wealth considered permanent?

Assume stock prices follow a random walk. If my investments go up by 1,000 dollars on the stock market today and I keep that money invested, in expectation, how much are my investments worth 1 year ...
2
votes
1answer
110 views

random walk with drift and absorption barrier

Hi: I will explain my question through the use of an ant that only walks in one direction and it's horizontal and to the right. So, assume that I have an ant named slowmo who is sitting at $x = 0$. ...
2
votes
1answer
259 views

Previsibility in Binomial Representation Theorem

I'm working through Baxter and Rennie's "Financial Calculus: An Introduction to Derivative Pricing". It was going very well and I've actually found it an easy read up until the point where they ...
2
votes
1answer
65 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
2
votes
2answers
98 views

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

Is it OK to assume a trading strategy should have a binomial distribution if the price is just a random walk? using p of the event as: $$\frac{AverageStopLossPercent}{AverageStopLossPercent + ...
1
vote
1answer
52 views

Cannot Understand The Ticket Line Question From Interview Book

I'm reading an interview book called A Practical Guide to Quantitative Finance Interviews (nickname: Greenbook) and cannot understand the following question(the question itself instead of its answers):...
1
vote
2answers
114 views

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

This might be against the law of gravity, but I'll give a try 🙂 Is there a way to combine two financial products $p_1$ and $p_2$, into a single product $p_c$ that is more volatile than its ...
1
vote
2answers
106 views

Counting random paths

Assume the path of a certain stock can be modeled using a binomial tree. The initial price of the stock at time $t=0$ is 1024. The upstage factor of the stock price is $x=1.25$ and downstage factor of ...
1
vote
1answer
3k views

How to apply Ljung Box Test?

I am checking the closing prices(about 9000+ prices) of the stocks data to test for randomness. The test I am using is Ljung Box test, in MFE toolbox for MATLAB, I used 300 data of closing prices, ...
1
vote
1answer
553 views

Probability of trade's exit orders being triggered in random-walk market

When placing a trade with Stop Loss and Take Profit orders in a hypothetical random market (i.e. 0.5 probability of up tick and 0.5 probability of down tick), assuming: x is the distance in ticks of ...
1
vote
1answer
149 views

Ergodicity of Random Walk

The Random walk is a special case of AR(1) with $x_t = \phi x_{t-1} + \epsilon_t$ with $\phi = 1$ A process is ergodic if two samples of a stochasitc process sampled far (say j < $\infty$ ) ...
1
vote
1answer
219 views

Monthly returns annualized vs annual returns [closed]

Lets say that I have a stock with annual returns, $a_i $ for year $i\in \left\{1,...n\right\}$ and monthly returns $m_{i,j}$ for month $j\in \left\{1,...12\right\}$. Lets define monthly returns to be ...
1
vote
1answer
172 views

Correlated Random Number Generation using Sobol?

There is a clear theory about generating correlated random numbers using Cholesky decomposition or PCA. I suppose if we apply above methods to random numbers generated using Uniform random numbers ...
1
vote
0answers
64 views

Rare Events in Normal Multivariate distributions

I don't work in finance, but I've stumbled upon a problem that you guys may have to deal with in your jobs. My problem is a random walk in high dim spaces ( > 100), in which I'm looking for vectors ...
1
vote
0answers
71 views

Modelling turnovers with a random walk. Is it right?

I need to analyse a bunch of weekly time series that reflect the turnovers of various companies. I already read that return rates or share prices show stochastic patterns that can be modelled by a ...
1
vote
0answers
40 views

Does the time between prices created from a GBM affect the estimation of parameters of the GBM?

Recently I created a simulation of a GBM. The time between the prices were sampled from an exponential distribution. The log rate of return was sampled from $\sigma \sqrt { { t }_{ i }-{ t }_{ i-1 } } ...
0
votes
1answer
267 views

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

A topic I am struggling with is the implementation of a (for the simplest higher dimensional case) bivariate normal distribution simulation for geometric brownian motion. The clearest explanation by ...
0
votes
2answers
74 views

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

The price of IBM changes from second to second, but there's no way that actual news about IBM is coming out that fast. The information available about IBM changes a lot more slowly than its share ...
0
votes
1answer
524 views

Should I adjust historical data for dividends when estimating drift?

I'm building a Geometric Brownian Motion model which incorporates future dividends which vary over time. Since these should reduce stock price when paid, I can incorporate that into the model, however,...
0
votes
1answer
43 views

Simulating Co-Integrated Assets

I know how to simulate correlated returns, but I do not know how to simulate Co-Integrated assets. I would like to simulate a co-integrated time series where the Beta Co-Efficient is not constant, but ...
0
votes
1answer
577 views

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, ...
0
votes
1answer
274 views

Random walks and using the reflection principle

Consider exercise 5.5 from Shreve volume 1: For part (I), I understand how you can use reflection to show that $P(M_n^*\geq m, M_n=b)=P(M_n=2m-b)$. However, it seems to me that this latter ...