when using the student's-t distribution in volatility models you need to modify it to include a scale parameter, \(\gamma\):
\(f(x|\nu,\sigma) = \frac{\Gamma\large(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi\gamma^2}\Gamma\large(\frac{\nu}{2}\right)}\large(1+\frac{x^2}{\nu\gamma^2}\right)^{-\frac{\nu+1}{2}}\)
the scale parameter is related to the variance by:
\(\sigma^2=\gamma^2\frac{v}{v-2}\ for\ v>2\)
Then if you sub for the variance, \(\sigma^2\):
\(f(x|\nu,\sigma) = \frac{\Gamma\large(\frac{\nu+1}{2}\right)}{\sigma\sqrt{\pi(\nu-2)}\Gamma\large(\frac{\nu}{2}\right)}\large(1+\frac{x^2}{(\nu-2)\sigma^2}\right)^{-\frac{\nu+1}{2}}\)
the innovations should have unit standard deviation because you specify the volatility in the volatility equation so just eliminate the standard deviation by setting it equal to 1.
(I got this all from wikipedia and a book by Carol Alexander "Practical Financial Econometrics", part II of the "Market Risk Analysis" series)
