I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:

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.

  • $\begingroup$ 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. $\endgroup$ – Gordon Mar 11 '19 at 12:53

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