# 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
• No. It has more to do with how Futures work. They do not require an upfront investment, so they do not earn the risk free rate. You should learn about how futures and forwards are traded, they are not "assets" in the traditional sense (things that you buy in return for cash) they are an agreement between two parties where no money changes hands when you enter into it. As such it does not appreciate in value at r. Sep 18 '18 at 23:05
• @AlexC and Ivan: Both of you are right. Thanks.
– Hans
Sep 19 '18 at 23:31
• @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