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Disclaimer: I understand this is a basic question that gets addressed in most 101 textbooks. Yet I have reviewed many of them not finding a satisfactory answer. Please bear with my ignorance.

Suppose a forward contract enforces the parties to exchange an asset with price $K$ at the delivery date $t=T$, and suppose the spot price of that asset at $t=T$ is $L$ (so both $K$ and $L$ are constant). Then the possessor of the forward contract at $t=T$ is forced to buy the asset at price $K$, and the possessor can also sell the asset immediately to obtain $L$. That means, the possessor at time $t=T$ can immediately earn $L-K$. Therefore anyone who are to buy that future contract at $t=T$ must obviously pay $L-K$. This means that the forward contract at $t=T$ should be of price $L-K$.

However, all textbooks and resources I've seen claim that the forward contract at $t=T$ should be of price $L$ instead of $L-K$.

Why so? What is wrong in my argument?

Another attempt I made to understand this point is by reading [1]. In section 2.3, it provides a more detailed argument:

As the delivery period for a futures contract is approached, the futures price converges to the spot price of the underlying asset. When the delivery period is reached, the futures price equals—or is very close to—the spot price. To see why this is so, we first suppose that the futures price is above the spot price during the delivery period. Traders then have a clear arbitrage opportunity:

  1. Sell (i.e., short) a futures contract
  2. Buy the asset
  3. Make delivery.

But when I carry the cash flow out, I can't make that balance.

First, selling a future contract yields a flow (-1 * future) + (+1 * future-price-at-time-T). Second, buying the asset yields a flow (+1 * asset) + (-1 * spot-price-at-time-T). Third, making delivery yields a flow (-1 * asset) + K. The net balance of three flows is

+ 1 * future-price-at-time-T
- 1 * spot-price-at-time-T
- 1 * future
+ K

I can't tell why if + 1 * future-price-at-time-T - 1 * spot-price-at-time-T > 0 then there is an arbitrage opportunity.

Thanks for your patience and sharing.

Reference

  • [1] Options, Futures, and Other Derivatives by John C. Hull
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    $\begingroup$ $L - K$ is a function of time. As the difference between $t$ and $T$ shrinks, $L-K$ shrinks $\endgroup$
    – amdopt
    May 25 at 13:52
  • $\begingroup$ @amdopt I defined $L$ as the spot price at time $t=T$, and $K$ as the delivery price. Both are constant. $\endgroup$
    – Student
    May 25 at 14:41
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    $\begingroup$ @Student, Yes, they are constant because time has run out on the contract when $t=T$. If you modeled out the relationship over a period of time, you would see that $L$ and $K$ converge to the same price at time $T$. So, if you made them constant at $T$, they are the same and $L-K=0$ $\endgroup$
    – amdopt
    May 25 at 14:54
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    $\begingroup$ My claim and the claim in the textbook are the same. The price of the futures contract when $t=T$ is, and should be the spot price. $\endgroup$
    – amdopt
    May 25 at 16:18
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    $\begingroup$ A forward contract that delivers $K (\neq L)$ must be priced at $L-K$ at $t=T$, as you yourself pointed out. But this is no different to a forward contract that delivers $K=L$ and is priced at $L-K=0$ at $t=T$. It's like saying you get 1000 barrels of oil and \$2000, and I want to see spot oil + \$2 (in $/bbl). $\endgroup$
    – hroptatyr
    May 26 at 6:47

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