# Is there a forward contract the underlying of which is another forward contract?

Is there a forward contract on a forward contract?

Let us take a simple example: Persons $$A$$ and $$B$$ agree that $$A$$ sells $$B$$ some asset tomorrow at the fixed price $$K_1$$. This is a normal forward contract on the asset.

Let us then assume that persons $$B$$ and $$C$$ agree that $$B$$ sells $$C$$ the previously described forward contract tomorrow at the fixed price $$K_2$$.

This would mean that person $$C$$ pays the amount of $$K_2$$ to $$B$$ to receive the first forward contract, and then $$C$$ pays the amount of $$K_1$$ to $$A$$ to acquire the asset. To summarize:

• Person $$C$$ pays $$K_1+K_2$$ to acquire the asset,
• Person $$A$$ sells the asset and receives $$K_1$$,
• Person $$B$$ gains $$K_2$$.

But wouldn't this be arbitrage (meaning that person $$B$$ makes money out of nothing)?

Would this kind of agreement make sense? And more importantly, how this contract would be valued?

If we assume that the asset in this example is a non-dividend-paying stock $$S$$, then the fair value of $$K_1$$ would be the forward price $$F_0=S_0 e^{rT}$$, where $$r$$ is the risk-free interest rate and $$T$$ is the time to maturity.

With similar reasoning, the fair value of $$K_2$$ would be the forward price $$F_0^\ast = x_0 e^{rT}$$, where $$x_0$$ denotes the initial value of the underlying of this forward, i.e. $$x_0$$ denotes the initial value of the first forward, which is $$0$$. Thus $$K_2=0$$.

Is this reasoning valid? Can we conclude that a forward contract on another forward contract is just a regular forward contract on the asset?

• Your reasoning is a bit hard for me to follow, but your conclusion is correct. The "forward on a forward" is just the same as a "forward". That's why you don't see "forwards on a forward" traded in real life. That second level of indirection does not accomplish anything. – noob2 Feb 4 at 13:10
• There's no arbitrage because the forward contracts, as described, won't be worth 0 at inception. Also the formula for the fair strike shouldn't have a minus in the exponent. Also, make the second contract expire before the first, otherwise it's just a regular forward – user357269 Feb 4 at 13:46
• Thank you for your comments, @noob2 and @user357269! I edited the minus signs. Indeed, I was curious about the case when the maturities of the forwards are the same. – user52360 Feb 4 at 13:58
• I think by the towering law of expectations you can easily show that a forward on a forward is ‘just’ a forward. A different result emerges for an option on an option. – Kermittfrog Feb 4 at 16:45