Hi all,I want to simulate asset prices using a student distribution instead of a log normal (because I found out that the fat tails of those assets returns were better modeled using the t-distrib).My problem is that whereas in the case of the lognormal random walk: dS/S = mu dt + sigma dWtI can use Ito to integrate it St = So exp ( (mu - 1/2 sigma^2) Dt + sigma N(0,1) Dt)I don't think that I can use Ito in the case of the Student distrib, since I don't have a Brownian anymore?Suppose I am looking at the dynamic: dS/S = mu dt + sigma d(Student)(t)How can I integrate it then?I know some people are using MC with student distrib (esp in credit), so there should be a way to do it, but somehow I feel a bit confused.Thank you for your help.Regis

If your asset is log-T distributed then it is not integrable. This means that call options etc. cannot be priced in the usual way.

But I don't want to price derivatives, just to simulate asset prices.Thanks a lot.

I believe what you are proposing won't work... people simulate distribution of returns (maybe) ( then i guess you just simulate uniform and invert to get t distribution) I don't believe they simulate the process ... i suspect no such process exists ( ie that is t dist for each time interval).

Variations on this theme come up from time to time. Take a look at this thread for some references: http://www.wilmott.com/messageview.cfm? ... &forumid=1

QuoteOriginally posted by: spv205I believe what you are proposing won't work... people simulate distribution of returns (maybe) ( then i guess you just simulate uniform and invert to get t distribution) I don't believe they simulate the process ... i suspect no such process exists ( ie that is t dist for each time interval).I believe that the t distribution is infinitely divisible, so there does exist a Levy process such that every increment of the process over a fixed unit of time has a t distribution.However, the characteristic function of this process is either unknown or very messy.

Quote I believe that the t distribution is infinitely divisible, so there does exist a Levy process such that every increment of the process over a fixed unit of time has a t distribution.However, the characteristic function of this process is either unknown or very messy.That's also what I was thinking of, but as I was reading some papers (mostly commercials, see www.msci.com/resources/new-simulation-m ... chnote.pdf for instance), I wondered if there was not something I was missing.Maybe I can just simulate the distribution of the returns and avoid simulating the process.Thank you for helping me to clarify my thoughts.

If the volatility is distributed Inverse Gamma (IGa) then the returns process will be Student-t. You can then simulate the price process in the usual manner depending on the stochastic volatility (SV) model you fitted. Options should still be possible to price under this framework provided the assumptions of the SV model are met. For example the Barndoff-Neilsen SV model allows the volatility to be distributed IGa.Appologies if I am missing something here... you say that you are interested in simulating the price process (not option pricing) and that you can see the returns are t-distributed. However you managed to 'see' this, fit a model on this data and simulate using this. For example if you looked at log(S[t]/S[t-1]) and can 'see' this is student-t, just estimate this student-t distribution from the data and simulate forward using log(S[t]/S[t-1])~StT? You might want to make sure you have finite variance when you fit this.I know this might not be ok for pricing options, but if he's only interested in simulating the price process, and he knows some model is better than Geometric Brownian motion, surely he just fits his better model and simulates?