European Call Value as a Function of the Matching Futures Price

Summary

This note asks how the value of a European call on a non-dividend-paying stock changes with the futures price observed before expiry, when the option and futures contract mature together. It relates the futures price to the spot price through the interest rate and applies the chain rule to express the call’s sensitivity to futures in terms of the option’s spot delta.

The answer gives a qualitative graphical intuition: because futures and spot prices are proportional at a given time, the call-value curve has the same general shape as its curve against spot, with a change in the horizontal scale. The note offers no plotted curve, derivation of the full pricing function, or numerical example. The stated intuition is limited to the setup described and depends on the relationship between spot and futures prices used there.

Key ideas

• The futures price is proportional to the spot price under the stated non-dividend-paying stock setup.
• The chain rule converts the call’s spot delta into sensitivity to the futures price.
• The call-value curve retains the same general shape when plotted against futures instead of spot.
• The response offers a qualitative graphing intuition rather than a full derivation or numerical illustration.

Tags

# Graph of European call option value versus future price


# 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.

## Answer by siou0107 (score 4)

https://quant.stackexchange.com/a/50138

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.