Valuing a Forward Contract on Another Forward Contract

Article Quant Q&A · Author: user52360

Summary

The question considers a contract that transfers ownership of an existing forward at a later date, and asks whether pricing it creates arbitrage or reduces it to a direct forward on the asset. In the example, one forward promises delivery of a non-dividend-paying stock at a fixed price. A second agreement transfers that first contract for a fixed amount at the first contract’s delivery date.

The proposed reasoning treats the first forward’s initial value as zero and then applies the standard forward-pricing relation to conclude that the second contract’s fixed amount should also be zero. This is a useful prompt to distinguish a forward’s delivery price from its current value: a newly entered fairly priced forward has zero initial value, but its value at a later date is not generally zero. The text gives no accepted resolution, and its proposed conclusion therefore needs scrutiny of settlement timing and contract terms before it can establish equivalence or rule out arbitrage.

Key ideas

The example asks whether a forward contract can have another forward contract as its underlying.
A fair forward delivery price and the current value of a forward contract are distinct quantities.
A forward that starts at zero value can have a nonzero value later as the underlying price and time to delivery change.
Any valuation comparison must account for the timing and terms of both contracts.

Tags

Derivatives pricing Equities Arbitrage

Full text

# 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?

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