Using Student-t Innovations in Return Simulations

Summary

The document discusses generating Student-t random variates for use as innovations in a Brownian motion or geometric Brownian motion simulation. It notes that statistical software can generate these variates and gives examples of programming environments and a package function. It also observes that Student-t innovations are used in GARCH modeling, while questioning their fit with the standard geometric Brownian motion setup, which assumes normally distributed increments.

The response does not explain how to calibrate the Student-t distribution, choose its degrees of freedom, or convert sampled values into returns with desired variance and time scaling. It also does not resolve whether the proposed process should be treated as Brownian motion, a jump or Lévy process, or another model. The key modeling choice is therefore not just how to sample a distribution, but whether that distribution and process match the return behavior being represented.

Key ideas

• Geometric Brownian motion is defined with normally distributed increments.
• Student-t random variates can be used as innovations in a simulation, but this changes the model assumptions.
• Student-t innovations are mentioned as an option in GARCH modeling.
• The response does not specify calibration or scaling needed for a complete return simulation.

Tags

# How to simulate asset returns using student t?


# How to simulate asset returns using student t?












I am currently trying to simulate an asset return using the student-t distribution, but I can't find how I should do this. I began with the Geometric Brownian motion and just changed in order that epsilon follows the student-t distribution instead of the normal distribution, but I found out that this is not the correct way, I read a lot about levy-processes, but I don't know exactly how do simulate such returns.

Thank you very much in advance

## Answer by Jan Sila (score 1)

https://quant.stackexchange.com/a/28145

You will need a 'pseudo' random number generator - most stats programming languages have them (Matlab, R, Python...). But GBM is defined with Normal increments $N(0,\sigma^{2}(T-t))$ so I dont think using Student's t distribution is a good idea, never seen it in any literature/applications. It is however used for instance in GARCH modelling....

Random variates: For instance in R, you can get random variates from student distribution from the fGarch package - rstd() function. In Matlab it is described here

Then you can use this as innovations in your BM or GBM motion simulation. How to do this was answered here few times see answer here.

If you elaborate further, what language you are using, or even post your code attempt so far, we might be able to help more...

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.