# Tagged Questions

A mean reverting process is a process that, over time, tends to drift toward its long-term mean.

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### Variance Ratio on Currency Pair, Validation

Can somebody tell me if my variance ratio (unit root test) test is correct? I don't want to rely on my results until I can find somebody that also has a variance ratio algo that confirms them? Here is ...
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### What is the probability that a OU process hits an upper barrier U before a lower barrier L?

What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?
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### Modeling the Stock Market [closed]

Hi I was wondering what is the model that best describes the price movement of the stock market? A Brownian motion Process with drift? An Ornstein Uhlenbeck_process? (where the long term mean ...
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### cointegration strategy

If it can be proved that two time series $$S_t^1=\alpha + \beta S_t^2 + \xi_t$$ representing stocks are correlated, with $\beta=-2$ and then are proved to be cointegrated, how a portfolio should be ...
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### Mean Crossing for Ornstein-Uhlenbeck

Suppose we have classic Ornstein-Uhlenbeck process. How can we calculate expected number (and variance too) of crossing mean value over the certain period of time? Say, if we have discrete OU process ...
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### How to transform Ornstein-Uhlenbeck parameters from hourly to daily?

I get the parameters (long-term mean, volatility, mean-reversion speed, correlation) of two correlated Ornstein-Uhlenbeck processes via a likelihood estimation from hourly data. If I want to transform ...
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### How to reduce variance of mean reversion estimator with maximum likelihood?

I estimate the parameters of an Ornstein-Uhlenbeck process with Maximum Likelihood. The mean reversion speed estimator is well known to be biased. There are methods to reduce the value of the expected ...
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### How to trade the Ornstein-Uhlenbeck process?

My question comes from this paper, which is a short version of Avellaneda's paper The picture bellow provides a summary of the equations. Do I understand correctly that in order to trade OU process ...
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### How to fit model implied forward curve with market forward curve for Ornstein-Uhlebeck?

I have a spread option model of 2 correlated Ornstein-Uhlenbeck commodity prices that I estimate the parameters of with Maximum Likelihood. What is the formula for introducing the additional ...
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### What is the effect of mean-reversion on an upper barrier knock-out call option?

Consider a mean-reverting normal model for an underlying $dX^{(1)}_t=-\kappa X^{(1)}_tdt+\sigma^{(1)} dW^{(1)}_t$, for fixed time-independent constants, $\kappa$ (mean-reversion) and $\sigma^{(1)}$ (...
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### What is the covariance of two correlated Ornstein-Uhlenbeck processes?

What is the covariance of two correlated Ornstein-Uhlenbeck processes? I was trying correlation(1,2)*Var1^(1/2)*Var2^(1/2), but I am not sure! I took Var1=(sigma1^2/(2*speedofmeanreversion1))*(1-exp(-...
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### How to estimate parameters for 2 correlated Ornstein-Uhlenbeck processes with maximum likelihood?

I would like to use maximum likelihood to estimate the parameters of two correlated Ornstein-Uhlenbeck processes from empirical data. Do you have any good references for this? If you have any hints as ...
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Trying to evaluate model for pairs trading. Consider classic formula: $\frac{dP}{P} = adt+b\frac{dQ}{Q}+dX$, where $P$ and $Q$ are stock prices, and $X$ is a mean reverting process (MRP) and $a$ is ...
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### Calculating half life of mean reverting series with python

I am currently attempting to calculate the halflife of a mean reverting series using python programming language and the theory of the Ornsteinâ€“Uhlenbeck process. I have a series which when plotted ...
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### Consequence of negative mean reversion of hull white one factor model

I tried to calibrate the data for hull-white one-factor model. Sometimes, I get negative estimate of mean reversion factor after the calibration process. When I plug the negative mean reversion factor ...
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### How to optimize an arbitrage portfolio when taking into account different speeds of mean reversion?

In portfolio optimization, it is insufficient to just note the size of price deviation - that only tells the amount of profit if held to maturity. One also needs to take into account reversion speed - ...
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Is there a major article or even better a comprehensive recent review article showing quantitative evidence for the existence of the business cycle and measuring the trending and mean reversion on ...
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### Are there alternatives to the Box-Tiao decomposition in identifying mean reverting portfolios?

As documented in this paper, Box-Tiao decomposition (a way to decompose multiple time series into components with different speeds of mean reversion) can be used to identify mean reverting portfolios. ...
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### Is trading mean reversion of small principal components of prices profitable?

Many have told me that it is a good idea to look at the third principal component (PC) of yield curve movements, as well as third and fourth PC of G10 currencies. They claim these PCs represent "...
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### Mean reversion in seasonal ARIMA model

Hi I'm looking for some help deriving the half life from an ARIMA model. My current best model candidate is (1,1,0)x(1,0,0)[4] and I'm a little confused on this. I've read up on HL for AR(1) models ...
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### Extracting mean reversion and incorporating into formula for general series

I'm looking to derive a general formula for margin reversion like Hussman has done in this article for some projections I'm doing. http://www.hussmanfunds.com/wmc/wmc100802.htm I'm not interested in ...
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### Tests for Mean Reversion in a Portfolio Rebalancing

On a single time series one can run a Dickey-Fuller test to determine if the asset is mean reverting or at least has been mean reverting during your sample. Is there a way to test for mean-reversion ...
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### Does it make sense to interpret autocorrelation and box test on 5 data points?

I am trying to see if after I trade a stock the price movements at 2, 5, 7, 10, 30 and 60 seconds after exhibit any autocorrelation. Below I have the returns from my trade price to the trade 2,5,7,10 ...
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I'm looking for some reference on how to calibrate a non-mean-reverting Ornstein-Uhlenbeck process to historical data using MLE or OLS. The model has the following SDE: $d\lambda(t)=a\lambda(t)dt+\... 1answer 2k views ### Speed of mean reversion of an interest rate model I would like to have a bit more of intuition about the concept of "speed of mean reversion" for an interest rate model, e.g. Vasicek or CIR. In particular, is a negative speed of mean reversion ... 1answer 250 views ### Mean reversion time estimation I am new to mean reversion trading, and I would like to get some good references about how to estimate the time it takes to a mean reverting process to cross its long term mean. 1answer 296 views ### The meaning of Ornstein-Uhlenbeck parameters I am trying to understand theOrnstein-Uhlenbeck process$dX_t = \kappa(\theta-X_t)dt + \sigma dW_t$my question is what is the meaning of the parameters? and assuming that we know those parameters ... 4answers 2k views ### Is a stationary process necessarily mean-reverting? Intuitively, a stationary stochastic process needs to be mean-reverting. This should follow immediately from the definition of stationarity: the mean of the process needs to be constant over time, so ... 2answers 465 views ### How can I estimate the Ornstein-Uhlenbeck paramters of some mean reverting data that I have on R? 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 ... 1answer 242 views ### How to approximate the time to mean reversion for implied volatility Given an option and its implied volatility, and also the mean value of the implied volatility over the last 30 days, if we find that the current IV is significantly (> 1 std dev.) away from the mean, ... 1answer 158 views ### Estimating mean reversion I've read in some places that mean reversion parameters for a rates model, eg Hull White, can be estimated directly from the current yield curve. However I've not been able to find anything more on ... 2answers 125 views ### Why is OU process stationary? The mean and variance of OU process have time dependence (exponentially decay in time). So they are not constant in time. How can it to be stationary? 2answers 212 views ### Getting the next price of a GBM with reversion Here is the "twin" question of Getting the next price of a GBM (Geometric Brownian Motion) but for GBM with reversion As in that case, I'd like to write a formula for the next price, as function of: ... 2answers 518 views ### Why is the mean time-dependent in the Hull-White interest rate model? In the Vasicek interest-rate model, the interest rate reverts to a constant mean. This makes sense to me. In my conception, the mean ought to be time-invariant, since interest rates don't follow an ... 0answers 152 views ### Exact value of mean reversion rate knowing terminal value of the process Let you have the following mean reverting process:$\text{d}x_{t}=a(\theta-x_{t})\text{d}t$, where the diffusion term is absent, that is this process is not stochastic. Let you know the value of$\...
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I have constructed a mean reverting spread using two indexes. I know they have to be mean reverting, but when plotted side by side they are mean reverting for a little bit and then deviate and head ...
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### Statistical significance of a pair trading strategy

How can I test the significance of a pair trading strategy, i.e. that the H0 is "The strategy has no predicting power". I was considering to use the technique in Evidence Based Technical Analysis ...
1k views

### Threshold calculation for buying a mean-reverting asset

I am trying to figure-out an optimal policy for buying a unit when its price follows a mean-reverting price process (Ornsteinâ€“Uhlenbeck), when I have a finite time deadline for buying the unit. I ...
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### Does mean reverting imply mean stationary?

If I have a time series that exhibits mean reverting properties, does it necessarily mean that the time series is mean stationary?
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### How to build a mean reverting basket?

I have been playing with mean reverting pairs, but seems that most of the low hanging fruit (ie pairs) have been squeezed already. I would like to start with mean reverting baskets (>2 securities) in ...
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### Entry and exit points for very short mean-reverting timeseries

I have a model specifying a cointegration relationship on a number of transaction-level timeseries. I would like to specify entry and exit points for trades where these points ideally would be just ...
2k views

### How are momentum and reversion long/short strategies dynamically combined in trading?

I'm trying to understand how to combine two strategies dynamically in trading: one mean-reversion and the other momentum. One way (also the simplest one) of doing this is by scaling/normalizing ...
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### Two prices pass the cointegration test but there is a trend. How to check stationarity?

Below is a spread built with two ETFs that pass the cointegration test i.e. Adjusted Dickey Fuller, adfTest(type="nc") in R's fUnitRoots with a p-value < 0.01. The red line is the trendline. What ...
1k views

### How is mean reversion implied by different valuations of Bermudan swaptions?

Someone told me that mean reversion can be implied by the different valuations of bermudan swaptions when using different methods for volatility calibration. Does anyone know what this means?
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### R code for Ornstein-Uhlenbeck process

Can any one help me with some R code to run Ornstein-Uhlenbeck process?