By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in Matlab.

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Can someone please help me out? I am new to stochastic calculus, so please help me understand step by step. The Wikipedia article you cite provides everything you need to evaluate the analytical solution of the Ornstein—Uhlenbeck process. However, for a beginner, I agree that it may not be very clear. You should first be familiar with how to simulate this process using the Euler—Maruyama method.

The stochastic differential equation SDE. This can be simulated in Matlab very easily using randn to generate standard normal variates:. Note that the above is not the most efficient code. To obtain a numerical solution in Matlab with this, you'll need need to numerically approximate discretize the integral term using an SDE integration scheme like Euler—Maruyama described above:.

To compute the full analytical solution of an Ornstein—Uhlenbeck process for a given time series and corresponding Wiener increments, you'll need use a "scaled time-transformed" Wiener process :. See Doob for further details and a derivation. To calculate this in Matlab:. This can be implemented without a for loop using cumsum and diff :.

I also recommend reading the following excellent article for further details on SDEs and simulating them in Matlab:. Desmond J. The URL to the Matlab files in the paper won't work — they can be found here now. Note, however, that some of the Matlab syntax particularly related to random number generation and seeding is a bit outdated as this was written nearly 15 years ago.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 4 years, 11 months ago. Active 2 years, 4 months ago. Viewed 20k times. Active Oldest Votes. Simulating the Ornstein—Uhlenbeck process You should first be familiar with how to simulate this process using the Euler—Maruyama method. I also recommend reading the following excellent article for further details on SDEs and simulating them in Matlab: Desmond J.

To do it numerically, you can look at 2 above the for loop case is simpler to adapt. The diffusion function only differs by a sign in the exponent.

Be careful if your exponential terms get very large or small. However plotting seems to show the process has much higher variance than the plot in the wikipedia article, regardless of the realisation of the process. Would you be able to explain why? Are the parameters given in the wikipedia article possibly wrong?

I looked into it and it appears that the plot on Wikipedia is not being calculated properly. This can be seen by examining the "R-Quelltext" source code section for the original image. This approach, however, seems to produce different results to the one outlined in your post and I am not quite sure why this is.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics.

It only takes a minute to sign up. I have mean reverting data Difference of 2 stock prices, that I want to do pairs trading on. I want to simulate my own mean reverting data as similar as possible to the real data that I have. The approach that I want to take is Least Squares Regression.

Maximum Likelihood is too complicated. Similar to Juan Gil's answer but a bit differently I would say the following based on this :. I have found this with R code, there an MLE approach is used. You find various solutions in this Stack Overflow question.

For a Ornstein-Uhlenbeck process, the maximum likelihood parameters are the ones from least squares regression. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 5 years ago. Active 5 years ago. Viewed 7k times. Thank you all. JonSnow93 JonSnow93 51 1 1 silver badge 2 2 bronze badges.

Active Oldest Votes. Ric Ric Sign up or log in Sign up using Google.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics.

It only takes a minute to sign up. I am currently attempting to calculate the halflife of a mean reverting series using python programming language and the theory of the Ornstein—Uhlenbeck process. Which obviously looks rather mean reverting. The code runs fine, however for this series I am getting a halflife of Full reversions are happening within a fraction of that time frame.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Calculating half life of mean reverting series with python Ask Question. Asked 4 years ago. Active 4 years ago. Viewed 8k times.

I have a series which when plotted looks like: Which obviously looks rather mean reverting. Could someone please advise me as to where I am going wrong with this? Any help much appreciated! Thanks for your reply anyway - much appreciated indeed!

Active Oldest Votes. This derivation is typical from the ARMA methods. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.

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Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. I know that we can convert this problem into a regression problem or an AR 1 fitting problem and back out the parameters. Basically, it is posted on this link. However, in some situation, the parameters in the regression problem are negative and can not compute log of it?

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Asked 4 months ago. Active 4 months ago. Viewed times. Demebleeee Demebleeee 1. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook.

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It only takes a minute to sign up. I am currently attempting to calculate the halflife of a mean reverting series using python programming language and the theory of the Ornstein—Uhlenbeck process.

Which obviously looks rather mean reverting. The code runs fine, however for this series I am getting a halflife of Full reversions are happening within a fraction of that time frame. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Calculating half life of mean reverting series with python Ask Question. Asked 4 years ago. Active 4 years ago. Viewed 8k times. I have a series which when plotted looks like: Which obviously looks rather mean reverting.

Could someone please advise me as to where I am going wrong with this? Any help much appreciated! Thanks for your reply anyway - much appreciated indeed! Active Oldest Votes. This derivation is typical from the ARMA methods. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.

Post as a guest Name. Email Required, but never shown. The Overflow Blog. Q2 Community Roadmap. Featured on Meta. Community and Moderator guidelines for escalating issues via new response….Hi all, welcome back.

Once we have the script working for a single input of one pair of symbols, we can very easily adapt it at the end to work with the symbol pairs function created previously. Use the hedge ratio to generate the spread between the two prices, and then plot this to see if it looks in any way mean reverting.

We will also calculate the Hurst exponent of the spread series. It looks like the prices are definitely correlated to a degree, and generally move in the same direction. But just how strong is this correlation? We can see from the information provided in the jointplot that the Pearson Correlation coefficient is 0. This sets the pair up as a potentially good fit for a mean reversion strategy. What we need to do now is create the spread series between the two prices by first running a linear regression analysis between the two price series.

Right so we have now managed to run the regression between the ETF price series; the beta coefficient from this regression was then used as the hedge ratio to create the spread series of the two prices. So it looks relatively mean reverting. But sometimes looks can be deceiving, so really it would be great if we could run some statistical tests on the spread series to get a better idea. The test we will be using is the Augmented Dickey Fuller test.

You can have a quick read up about it here if you need to refresh your memory:. From this we can see that the test statistic of The p-value of 0.

The last thing we will do is run a quick function to calculate the Hurst exponent of the spread series. For info on the Hurst Exponent please refer to: this article. To simplify things, the important info to remember here is that a time series can be characterised in the following manner with regard to the Hurst exponent H :.

### Python Backtesting Mean Reversion – Part 2

The Hurst Exponent is under the 0. This means that the spread series looks like a definite candidate for a mean reversion strategy, what with the spread series being mean reverting and all. Please click here for more info on half-life. We can calculate this by running a linear regression between the spread series and a lagged version of itself.

The Beta coefficient produced by this regression can then be incorporated into the Ornstein-Uhlenbeck process to calculate the half-life. Please see here for some more info. So according to this result, the halflife of mean reversion is 40 days. Ideally the half-life would be as short as possible so as to provide us with more profitable trading opportunities but there you have it, 40 days is what we have.

We will then begin the actual backtest itself using Pandas and see if we can produce something that is any way profitable. If anyone has any comments, please leave them below — always eager to hear the thoughts of others.

Hello, you are doing a great job here quoting excellent examples, but here i wanted to understand what has changed as I am unable to find update on difference function :. It would be a nice touch to show how from the OLS regression of the spread and the lag one could use the OU dynamics to simulate the data.

It seems like one of the values you are passing to numpy. Trading Strategy Backtest. You may also like. October 23, April 19, February 3, Gerardo Flores July 11, - pm It would be a nice touch to show how from the OLS regression of the spread and the lag one could use the OU dynamics to simulate the data Loading Andreas Prosswimmer October 30, - am Great blog!!By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in Matlab. Can someone please help me out?

I am new to stochastic calculus, so please help me understand step by step. The Wikipedia article you cite provides everything you need to evaluate the analytical solution of the Ornstein—Uhlenbeck process. However, for a beginner, I agree that it may not be very clear. You should first be familiar with how to simulate this process using the Euler—Maruyama method.

The stochastic differential equation SDE. This can be simulated in Matlab very easily using randn to generate standard normal variates:. Note that the above is not the most efficient code. To obtain a numerical solution in Matlab with this, you'll need need to numerically approximate discretize the integral term using an SDE integration scheme like Euler—Maruyama described above:. To compute the full analytical solution of an Ornstein—Uhlenbeck process for a given time series and corresponding Wiener increments, you'll need use a "scaled time-transformed" Wiener process :.

See Doob for further details and a derivation. To calculate this in Matlab:. This can be implemented without a for loop using cumsum and diff :.

I also recommend reading the following excellent article for further details on SDEs and simulating them in Matlab:. Desmond J. The URL to the Matlab files in the paper won't work — they can be found here now.

Note, however, that some of the Matlab syntax particularly related to random number generation and seeding is a bit outdated as this was written nearly 15 years ago.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 4 years, 10 months ago. Active 2 years, 4 months ago. Viewed 20k times.

**Ornstein Uhlenbeck (OU) Process: solution, mean, variance, covariance, calibration, and simulation**

Active Oldest Votes. Simulating the Ornstein—Uhlenbeck process You should first be familiar with how to simulate this process using the Euler—Maruyama method. I also recommend reading the following excellent article for further details on SDEs and simulating them in Matlab: Desmond J. To do it numerically, you can look at 2 above the for loop case is simpler to adapt.

The diffusion function only differs by a sign in the exponent.

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