The forward price is an argument, not a forecast

The first time I saw a forward price formula, I assumed it was a prediction: the market’s best guess of where the price will be. It isn’t. It’s the conclusion of an argument, and the argument is the part worth remembering.

The trade that can’t lose

Suppose a stock trades at $S_0$ today, pays no dividends, and you can borrow or lend at a continuously compounded rate $r$. Someone offers to agree today on a price $F_0$ at which you’ll buy the stock at time $T$.

You can build the same position yourself: borrow $S_0$, buy the stock, hold it. At $T$ you own the stock and owe $S_0{e}^{rT}$. A forward contract and this “cash-and-carry” position end in the same place, so they must cost the same:

If the quoted $F_0$ is higher, sell the forward and do the cash-and-carry. You lock in $F_0-S_0{e}^{rT}$ with no risk. If it’s lower, do the reverse: short the stock, lend the proceeds, buy the forward. Either way the mispricing pays for itself, and traders keep doing it until the gap closes.

What isn’t in the formula

The striking thing is what’s missing. There’s no expected return, no opinion about the company, no forecast of any kind. Two people who disagree completely about where the stock is going still have to agree on $F_0$, because both of them can run the same arbitrage.

That changed how I read the rest of the book. Each new pricing result is worth asking about in the same way: what is the replicating trade, and what does it cost? Once you can answer that, the formula usually follows.

Where it bends

The clean version breaks in familiar ways, and each fix is still the same argument:

• Income (dividends, coupons) reduces what it costs to carry the asset, so you subtract its present value $I$: $F_0=(S_0-I)e^{rT}$.
• A yield $q$ works the same way in continuous form: $F_0=S_0{e}^{(r-q)T}$. Currencies are the cleanest example, with $q$ as the foreign interest rate.
• Storage costs add to the cost of carry. Convenience yield is what holders of a physical commodity are willing to give up to keep it on hand, and it’s why commodity forwards can sit below full carry.

I rebuilt this chapter as a small tool in the interactive Hull: change $r$, $q$ or $T$, and the arbitrage table shows which way to trade when the quote is off.