I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:
$dS_{t}=r_{t}S_{t}d_{t}+\sigma_{t}S_{t}dW_{t}$
When $r_{t}=r$, and $\sigma_{t}=\sigma$, it's trivial that futures price $F_{t,T}$ follows GBM:
$dF_{t}=\sigma F_{t}dW_{t}$
as futures price is a martingale.
I wonder if we can derive an explicit form of SDE for futures price when interest rate and volatility are random processes. I tried myself but failed to do so.
