# Option price of a future

This must be a dumb question. Consider a European option $V$ on a (stock) futures $F$. The hedging condition seems to be the same as that for a stock $$d\Big(V-\frac{\partial V}{\partial F}F\Big)=r\Big(V-\frac{\partial V}{\partial F}F\Big)dt$$ for the riskless short interest rate $r$, since the portfolio $\Pi:=V-\frac{\partial V}{\partial F}F$ is a riskless traded asset and thus should grow at the riskless interest rate. I will get the same PDE as that of a stock. However, one should obtain the risk-neutral growth rate of $F$ as $0$. What is the catch?

The catch is the future is an unfunded position (I’m disregarding the margin here) and as a result the term $r\frac{\partial{V}}{\partial{F}}Fdt$ does not exist. F is actually a contract worth 0.
• Could you please elucidate this point a bit more? Is the reason for excluding $F$ from the right-hand side of the hedging equation that the portfolio $\Pi$ cannot be considered to be self-financing? – Hans Sep 18 '18 at 22:33
• @AlexC and Ivan: I am revisiting this question. While it is correct that the futures return rate is zero, what Ivan says about $F$ being a contract worth $0$ is wrong. The futures is traded on the market at price $F$. Do you agree? – Hans May 17 '19 at 2:37
• No it is a contract to receive the underlying in exchange for the price $F$, at a future date. $F$ is set so that the contract is worth 0. That is the definition of a future. If you buy a future at $F$ you do not pay $F$, you agree to buy the underlying at $F$ in the future. – Ivan May 18 '19 at 10:51