QuoteOriginally posted by: justintimeNetwalker,Do you have the code of the stochastic experiment you described?Yes, sure. Matlab or C++?

- justintime
**Posts:**7**Joined:**

Netwalker,Matlab would be fine. Thanks.

QuoteOriginally posted by: justintimeNetwalker,Matlab would be fine. Thanks.Done! You’ll need the GARCH toolbox http://econ.ucsd.edu/~ksheppar/ucsd_garch.htm

- justintime
**Posts:**7**Joined:**

Netwalker,Thanks a lot.

- justintime
**Posts:**7**Joined:**

The expected kurtosis is indeed slightly grater than 3. It looks like “overfitting”. The expected variance of z(t) series is hardly distinguishable from 1. That’s not the case in the real world.

QuoteOriginally posted by: justintimeThe expected kurtosis is indeed slightly grater than 3. It looks like “overfitting”. The expected variance of z(t) series is hardly distinguishable from 1. That’s not the case in the real world.Any model is just an approximation of a real world phenomenon. Don’t forget that human’s perceptions are limited. How can one express something that he doesn’t know? The answer is: We use a pure intuition and a language such as mathematics for moving out of the limits of our perceptions. Without "a language" the intuition is nothing.

- justintime
**Posts:**7**Joined:**

QuoteOriginally posted by: NetwalkerQuoteFor empirical evidence, take a look at Andersen et al. (2000) Exchange rates returns standardised by realised volatility are (nearly) Gaussian, Multivariate Finance Journal 4, 159-179 (or National Bureau of Economic Research, Working Paper No. 7488). By the way, I’ve read the Andersen at al (2000) paper and the kurtosis of conditional on realized volatility returns there is actually less than 3. Is that a result of overlooking something that is out of our perceptions? This is highly unusual to see platykurtosis in the area of finance.

I hope I'm not missing the point here - let me know of any corrections that I need to make here. I too would appreciate anyone's views on these pdfs, their use, etc. as per James' original question.For stocks/ IR:Obviously log-normal ... see Wilmott!!Variance-Gamma - See P Carr web site for refs and papersGARCH - see Duan paper from 1999 (not available electronically as far as I can make out)CIR - interest ratesCEV - ?? still pops up(These two are both cases of a mean-reverting process with power vol. The pdf involves the modified bessel function of the first kind. For some cases, e.g. SRCEV, you can reduce the option pricing formula quite nicely. See Cox 1976, Schorder 1989, google)Quadratic Volatility - not entirely sure what the pdf looks likeBessel processes - there are some papers by Lipton in RISK. I'm still stuggling with these, and in particular as to what form the pdf takes!There was another distribution recently published in Quant Finance. Can't remember the author name - last year. Tsallis distribution. I believe it is okay, but not (yet) brilliant for stocks.Inverse Gamma was used for Asians (??) Correct me if I'm wrong!!I suppose that one gist here is that any process you can dream up has an associated probablity distribution, even if it can't be written down.

Of course, the normal is just the continuous limit of the binomial. The binomial makes sense locally as we think of stock prices as moving up or down at each tick, but can also be generalized to non-constant volatility implied distributions, for example, in Rubinstein (1994).

What about the Poisson, it is a limitting case of the binomial?

QuoteOriginally posted by: pburnsThe Efficient Market Hypothesis implies that returns are unpredictable -- it does not imply anything about their distribution. You would need to impose conditions on the information in order to deduce a distribution of the returns in an efficient market. If you look at 1-second returns, they will clearly be far from Gaussian as they will be noticeably discrete.While I find it a plausible hypothesis that standardized returns would be close to normal if the true volatility were known, what I've seen personally doesn't lend much credence to that. I'm wondering what "firm empirical evidence" you are referring to.Here is one more objective evidence on normality of financial returns. I would add that we can use not the normality itself but rather deviations from a normal.

Last edited by Netwalker on February 10th, 2004, 11:00 pm, edited 1 time in total.