Deriving Futures Price Dynamics with Stochastic Rates and Volatility

Summary

The document poses a derivatives-pricing question: how to derive the stochastic differential equation for a futures price when the underlying stock follows geometric Brownian motion but its interest rate and volatility vary randomly over time. It notes the familiar constant-parameter case, where under the risk-neutral measure the futures price is a martingale and its diffusion has the stock’s volatility.

No derivation or answer is included, so the document does not specify the dynamics for stochastic rates or explain how correlations among the stock, rate, and volatility processes affect the result. Those details matter: a risk-neutral measure must be defined consistently with the chosen numeraire, and a futures price need not have the same dynamics as a forward price when rates are stochastic. The material is therefore useful as a focused problem statement, but it provides no evidence, solution method, or assumptions sufficient to produce a unique SDE.

Key ideas

• Under constant interest rates and volatility, the question identifies the futures price as a martingale under the risk-neutral measure.
• The document asks how random interest rates and volatility alter futures price dynamics.
• It provides no derivation or answer for the stochastic-parameter case.
• A complete model would need assumptions about the random processes and their dependence, as well as a specified pricing measure.

Tags

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


# 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.

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