# SDE of futures price under non-constant interest rate and volatility process

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.

• I do not think it is possible for general random $r$ and $\sigma$. However, it is possible for random $r$ and $\sigma$ with certain specific dynamic forms. – Gordon Mar 11 '19 at 12:53