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### Estimate parameters in SDEs

Posted: **September 1st, 2011, 3:27 am**

by **journeyhome**

Hello everyone,The question bothers me a lot is to estimate the parameters in SDEs:For example, consider Trolle and Schwartz (2009) in RFS about unspanned volatility. The part 3 and part 4 they estimated three versions. SVgen, SV2, and SV1 given the data of NYMEX contracts. Or whatever paper like this one.I am wondering how to do this work?Would anyone please recommend some material on this topic? I really appreciate that!

### Estimate parameters in SDEs

Posted: **September 1st, 2011, 2:14 pm**

by **Alan**

Well, that paper is handling a big model with lots of difficult techniques. It is as if you want tobe a pilot and you want to begin with a Boeing 757. My suggestion is start with learning howto estimate the parameters of the Ornstein-Uhlenbeck process SDE by maximum likelihood. Googling should turn up plenty of discussions of that as it is a standard example.

### Estimate parameters in SDEs

Posted: **September 1st, 2011, 2:41 pm**

by **journeyhome**

Very nice! This is indeed a good starting point. I really appreciate your advice. Honestly, I have not found many materials, books, on this topic. The best one I have is Hamilton 1994 by far. And I am not able to discover the connection between the discrete series and continuous time processes properly right now.

### Estimate parameters in SDEs

Posted: **September 1st, 2011, 3:25 pm**

by **Alan**

You're welcome. The connection you ask about is that, for many time-homogenous processes such as the OU, we know from the continuum model the transition density p(DeltaT,x(t2) | x(t1)) where t2 > t1.This density p also depends upon the parameters of the sde. Then, given a series of discrete obs {x(t1),x(t2),...x(tN)}, you form the Log-likelihoodLL(parms) = Sum_{i=1,N} log p(DeltaT,x(t(i+1)) | x(t(i))) Then you get the MLE estimates by just giving it to an optimizer:Max(parms) LL(parms)That's it!In finance, unfortunately you often have (unobserved) latent variables, which complicate things terribly. But, popular models still have a known transition density. So ... if you can find some good proxy for the latent variables, you are back in business with MLE. Personally, that is the approach I like when stochastic volatilityis the latent variable.

### Estimate parameters in SDEs

Posted: **September 1st, 2011, 3:41 pm**

by **journeyhome**

QuoteOriginally posted by: AlanYou're welcome. The connection you ask about is that, for many time-homogenous processes such as the OU, we know from the continuum model the transition density p(DeltaT,x(t2) | x(t1)) where t2 > t1.This density p also depends upon the parameters of the sde. Then, given a series of discrete obs {x(t1),x(t2),...x(tN)}, you form the Log-likelihoodLL(parms) = Sum_{i=1,N} log p(DeltaT,x(t(i+1)) | x(t(i))) Then you get the MLE estimates by just giving it to an optimizer:Max(parms) LL(parms)That's it!In finance, unfortunately you often have (unobserved) latent variables, which complicate things terribly. But, popular models still have a known transition density. So ... if you can find some good proxy for the latent variables, you are back in business with MLE. Personally, that is the approach I like when stochastic volatilityis the latent variable.Thanks Alan! That's really helpful!

### Estimate parameters in SDEs

Posted: **September 2nd, 2011, 9:03 am**

by **ACD**

Also worth bearing in mind that when the mean reversion is very weak (process is close to having a unit root or actually has one) then the MLE estimate becomes very biased, you'll need to be aware of this when getting/interpreting your results (it can make thing look like they mean revert a lot more than they actually do).

### Estimate parameters in SDEs

Posted: **September 2nd, 2011, 9:21 am**

by **ronm**

Quotehaving a unit root or actually has one) then the MLE estimate becomes very biasedCan you put some example on that? In many time series processes which are integrated, MLE is one of the best choices. Ofcourse, ideally speaking, MLE is performed on the transformed data (like diff. series). Is it that, you meant to say, MLE will be biased if it is operated directly upon the raw data (came from some integrated process?)

### Estimate parameters in SDEs

Posted: **September 2nd, 2011, 9:59 am**

by **ACD**

Ok, to see it simulate 100, 1000, 10000, 100000 points on the following process 10000 times: And set beta to something close to 1 (0.99, 0.999, 0.9999 or even 1). The estimate the value of beta by MLE and plot a histogram of these estimates for each combination of points and pre-chosen value of beta. You'll find that the closer beta gets to one the more negatively skewed the histogram becomes and the mean of the distribution is below the value you chose for the process. Even with a lot of points to estimate from the bias is hard to get rid of. It's the estimation of beta that causes the problem.

### Estimate parameters in SDEs

Posted: **September 2nd, 2011, 1:48 pm**

by **Alan**

Yes, agree that estimated mean reversion speeds, under MLE, often come with large errors. One should alwayscreate sampling distributions and confidence bands. Beyond the statistical errors, a separate problem is thatthe models themselves may be particularly mis-specified with respect to the drift behavior. Basically, it comes downto the fact that variance-covariance structure is relatively easy to estimate, while drift is not. In simpleexamples, one can show that having more data simply doesn't help (with a fixed time horizon).

### Estimate parameters in SDEs

Posted: **September 2nd, 2011, 5:41 pm**

by **Polter**

If you'd like an introductory textbook approach on a level somewhat similar to Hamilton's, then "The Econometrics of Financial Markets" by John Y. Campbell, Andrew W. Lo, & A. Craig MacKinlay might be a choice, or, more precisely, the (rather short) section "Implementing Parametric Option Pricing Models" thereof:

http://press.princeton.edu/TOCs/c5904.htmlFrom the software point of view, R with the yuima package might be of use (go straight to the "Estimation of Financial Models" section of the slides):

http://www.rinfinance.com/agenda/2011/StefanoIacus.pdf Note, that when we say "estimation" (as in the above) we often mean statistical estimation (e.g. MLE) using the physically observed data, i.e., the parameter estimates you obtain are the ones corresponding to the physical measure P. Those might be useful e.g. for risk management.For (risk-neutral) pricing you need the risk-neutral parameters, i.e. the ones under the risk-neutral measure Q.You can make a transition if you know (or assume) something about the market price of risk or, equivalently, the stochastic discount factor (this is discussed in the aforementioned book).You can also calibrate directly under Q, minimizing some mispricing metric, but that's usually not "estimation" in the statistical sense.For instance, Trolle and Schwartz (2009) specify their model under the risk-neutral measure Q (in section "The model under the risk-neutral measure"), while the estimation is carried under the physical measure P ("Estimation procedure"). The link is the market price of risk, denoted with the upper-case Greek letter Lambda ("Market price of risk specification").

### Estimate parameters in SDEs

Posted: **September 2nd, 2011, 5:44 pm**

by **journeyhome**

Thank you everyone for your wonderful help! I am really grateful!