Sep
27

## Mean Reversion Speed Estimation - What Am I Missing?

I am stuck by the method to

where mu is the mean value of the prices over time, and dW is simply some random Gaussian noise. Given a time series of the daily spread values, we can easily find theta(and mu) by performing a linear regression fit of the daily change in the spread dz against the spread itself, then we are able to calculate the half-life as log(2)/theta, which is the expected time it takes for the spread to revert to half its initial deviation from the mean. This half-life can be used to determine the optimal holding period for a mean-reverting position and as a measure for exit-trading strategy.

with Ito lemma, where the error term is normally distributed, therefore basically regression of dz on returns a value for instead of for theta itself.

In order to check the issue, I simulate a mean reversion process with dt=0.05, theta=0.75, mu=0.02 and run a regression

ols(dz, (mu - z(t-1))) = 0.03767243

It is far away from the true value 0.75, and we get a closer number 0.7680074 if we invert it back based on the equation .

Is it an error of the book or am I missing something? help me.

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**estimate the mean reversion speed**(and hence half life) described in the book Quantitative Trading: How to Build Your Own Algorithmic Trading Business, on page 140 the author said suppose the mean reversion of a time series can be modeled by an equation called the**Ornstein-Uhlenbeck**formula, and denote the mean-reverting process of a stock z be writen aswhere mu is the mean value of the prices over time, and dW is simply some random Gaussian noise. Given a time series of the daily spread values, we can easily find theta(and mu) by performing a linear regression fit of the daily change in the spread dz against the spread itself, then we are able to calculate the half-life as log(2)/theta, which is the expected time it takes for the spread to revert to half its initial deviation from the mean. This half-life can be used to determine the optimal holding period for a mean-reverting position and as a measure for exit-trading strategy.

**So far so good.**But then I am confused by the example 7.5 for theta estimation, where the author simply regresses dz on , is it correct? as we know if a process is mean reverted like above, it can be rewriten aswith Ito lemma, where the error term is normally distributed, therefore basically regression of dz on returns a value for instead of for theta itself.

In order to check the issue, I simulate a mean reversion process with dt=0.05, theta=0.75, mu=0.02 and run a regression

ols(dz, (mu - z(t-1))) = 0.03767243

It is far away from the true value 0.75, and we get a closer number 0.7680074 if we invert it back based on the equation .

Is it an error of the book or am I missing something? help me.

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Hemingway

2010/09/27 11:23 [Add/Edit reply] [Clear reply] [Del comment] [Block]

To first order (1-exp(-b dt)) = b dt. So Chan's method works well for small time steps (we need dt^2<<dt). Your method is correct and works for any time step.

abiao

2010/09/27 11:36 [Add/Edit reply] [Clear reply] [Del comment] [Block]

yes, then he still needs to divide dt, right?

sjev

2010/09/27 15:16 [Add/Edit reply] [Clear reply] [Del comment] [Block]

Personally I did not have any luck with OU process modelling either. The estimates seem to be very unreliable and are sometimes even negative. I wonder if OU-process is a valid model to estimate mr-process.

Hemingway

2010/09/27 15:50 [Add/Edit reply] [Clear reply] [Del comment] [Block]

@abiaoTrue, but that is simply a change of units.By the way, I think that MLE is worth trying too. The likelihood is known in closed form.

abiao

2010/09/27 16:01 [Add/Edit reply] [Clear reply] [Del comment] [Block]

Many thanks, Hemingway, I tried MLE just now, more flexible.

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