# Graph of European call option value versus future price

Given a standard European call option on a non-dividend-paying stock. Draw the graph of call price at time $$t$$ versus the future price $$F(t,T)$$. The future price $$F(t,T)$$ is observed at time $$t$$, prior to maturity. The futures contract and the option both mature at the same date $$T.$$

Note that $$F(t,T) = S(t)e^{r(T-t)}$$ where $$S(t)$$ is the stock price at time $$t$$ and $$r$$ is interest rate.

Let $$c$$ be the call option value and $$F$$ be the future price. By Chain rule, we have $$\frac{\partial c}{\partial F} = \frac{\partial c}{\partial S} \cdot \frac{\partial S}{\partial F} = \Delta e^{-r(T-t)} = N(d_1) e^{-r(T-t)}.$$

Initially I thought that I can just solve the differential equation above and obtain $$c$$ in terms of $$F.$$ But it seems that it is not so straightforward.

Any hint is appreciated.

• Your call value at time $t$ is wrong : here you just take intrinsic value + difference between value of the strike price at times $t$ and $T$ (why?) – siou0107 Dec 6 '19 at 18:02
• @siou0107 that is a typo. I deleted that part from my question. – Idonknow Dec 6 '19 at 18:05

Since you have that proportionality between the stock price $$S$$ and the futures price $$F = Se^{rT}$$, you just have to slightly shift your graph but the shape is the same.