Hi,I have about ten years of historical data for stock prices. I am trying to justify the use of the poisson process for modeling jump event times, and a log-normal distribution for jump sizes.Does anyone know of any simple tests that I can perform to justify the use of the compound poisson process?Many Thanks

- LocalVolatility
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If you want to show why a compound Poisson process with log-normally distributed jumps is a better model for your data than some other process(es) you could estimate them via maximum likelihood and then run selection tests like likelihood ratio, AIC, BIC, ...

Hi,Thank you for your response.I am not sure if I entirely understand your suggestion. To my knowledge, maximum likelihood is to estimate the parameters of a statistical distribution. I want to demonstrate that the compound poisson process is reasonable to use for modeling jumps in equity prices. How can I make use of it?Thanks

Use the same criterion to estimate vicinity of the historical data to jump model and to continuous model. Then we can say which model is better represents historical data on this particular time interval.

To elaborate on the good advice that has already been given by both responders, maximum likelihood yields, no surprise, a likelihood.Everything is relative, so it is a matter of comparing the likelihood to the likelihood of some alternative. Having said that, I suspect the OP is having some doubts about 'jumps' in general for stocks.My advice for that is to simply watch the market at times when sharp moves regularly occur.For single names, this is at the earnings release. For broad-based indices (in the US), thebest examples are at the monthly jobs report and at FOMC announcements.

- LocalVolatility
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My suggestion was to to estimate your model and some alternative models based on the historical returns using maximum likelihood estimation, i.e. fitting the physical return distribution. Then you can make inferences about which model fits your data better and potentially conclude that your jump specification is suitable.

Actually, for the project that I am working on, my objective is to come up with simple test cases that can justify the use of the compound Poisson process for modeling jumps in stock prices. The tests does have not be rigorous, just simple cases that can somewhat justify the usage of the poisson process.Any suggestions?Thanks

The simplest example of the sort of thing you are trying to do would be a stock price that follows a geometric Brownian motion and is subject to jumps at Poisson arrival times. If the jumps are multiplicative, lognormal and have mean equal to 1, then you have three parameters you need to estimate (forget about the drift, at least for now): the regular volatility, the jump volatility, and the intensity of the Poisson process. As a practical matter, I think you need to pick a cut-off, and assume that daily changes smaller than that are diffusive and changes larger than that are the effect of jumps. For a given cut-off, it would seem to be straightforward to estimate the three parameters in question, as well as to do basic specification tests. You should also be able to find the cut-off level that minimizes the probability of misclassifying moves, given the estimated parameters. Regardless of exactly how you go about doing this I would strongly suggest that you explore the properties of your statistical tests on simulated data before starting to draw conclusions about the real world.

Bearish,Thank you very much for the response you provided. Actually the procedure you described is exactly the calibration algorithm I am employing.However, my question actually is, given 10 years of historical stock return data, what statistical tests on the data can I use to justify the usage of the Poisson process to model event times? And what tests can I design to justify that the jump sizes are lognormal ( besides an empirical plot of the actual realized jumps)Many Thanks

Your time between jumps should be exponentially distributed, so you can compare your empirical distribution with the theoretical using the Kolmogorov-Smirnov test. Same thing with the lognormality (or lack thereof of the) jump size. You can also test for things like autocorrelation in jump size and/or frequency, whether the jump size is correlated with the length of time since the last jump, and whether the parameters are different between the first and second half of your sample. Again, I would encourage you to experiment with the tests on simulated data in order to build some intuition for how well they work.

Yes I can do that. But I am concerned that test might be too rigorous, and would invalidate the usage of the Poisson process.Are there any simple tests that I can use to justify the Poisson process is acceptable?Thanks

QuoteOriginally posted by: billyx524Yes I can do that. But I am concerned that test might be too rigorous, and would invalidate the usage of the Poisson process.Are there any simple tests that I can use to justify the Poisson process is acceptable?ThanksThat's not a very scientific spirit.

I know, and I am in complete agreements with you. If I was do it rigorously, I definitely would follow your suggestions. But unfortunately it is not entirely up to me.However, for the purpose of the project that I am working on, it is sufficient to show that the use of Poisson process is acceptable. It has to be practical not to invalidate the current use of Poisson process. Are there any simple statistical tests that I can employ?Thanks

This is becoming a bit trollish IMO. Acceptable for what and to whom?

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