Replication and the no-arbitrage forward price

How can you build a forward from the asset and a loan?

Replication means recreating a position’s cash flows from other assets. For a forward, the building blocks are the underlying asset and a loan at the risk-free rate . Replication is usually used to price or offset a derivative when the law of one price holds and there is no arbitrage.

Take a long forward. You agree today to pay F₀(T) at time T for one unit of the asset. You can get the same exposure without a forward: borrow S₀, buy the asset now, hold it until T, sell it at S_T and repay the loan.

Replication with the asset and a loaneach position has the same payoff as the forward or bond it copies

• Long forward Memorize borrow S₀ at r and buy the asset
• Short forward Memorize sell the asset short and lend S₀ at r
• Risk-free bond Memorize buy the asset and sell it forward

Compare the two positions at expiration. Use an asset at USD 80 and a risk-free rate of 3%, with a forward for delivery in nine months at USD 81.79:

S_T at expiration	Long forward: S_T − 81.79	Borrow and buy: S_T − 80 × 1.03^0.75
70.00	−11.79	−11.79
81.79	0.00	0.00
95.00	+13.21	+13.21

The two columns match whatever S_T turns out to be. Neither needs any cash at the start: the forward is free, and the borrowed money pays for the asset. Two positions with the same payoff and the same cost must have the same price, so the forward price is pinned down.

What is the no-arbitrage forward price?

Set the forward price so that the two positions are equal. The loan costs S₀(1 + r)^T at expiration, so that is what the forward must charge. The result is the no-arbitrage forward price .

$$F_0(T) = S_0\,(1 + r)^T$$

The rate is the risk-free rate, and T is in years. The forward price does not depend on where anyone thinks the asset is going. It depends on the spot price and the cost of money.

For indexes and currencies the curriculum uses continuous compounding instead. The idea is the same, with $e^{rT}$ in place of $(1+r)^T$.

$$F_0(T) = S_0\,e^{rT}$$

Exam tip Single asset and discrete rate: use (1 + r)^T. Index or currency and a continuous rate: use e^(rT). Check which one the question gives you.

What if the quoted forward is off?

If the market quotes a forward that is not S₀(1 + r)^T, the replication argument turns into a trade. The arbitrageur buys the cheaper of the forward and its copy and sells the dearer.

Compare the quote with S₀(1 + r)^Tthe quote and the fair value should be equal

• Quote above fair value Memorize forward is dear
  • Cash-and-carry Memorize borrow, buy the asset, sell the forward
• Quote below fair value Memorize forward is cheap
  • Reverse cash-and-carry Memorize short the asset, lend, buy the forward

What does a hedged position earn?

Buy the asset and sell it forward. This hedged portfolio has a certain outcome: you receive F₀(T) at T whatever S_T is. If F₀(T) is the no-arbitrage price, the return on this position is exactly the risk-free rate.

$$r = \left(\frac{F_0(T)}{S_0}\right)^{1/T} - 1$$

If the same purchase is funded by borrowing at r, the investor puts in nothing and earns nothing: the forward’s gain pays the interest. That is the “zero” some candidates mix up with the hedged return.

What happens when interest rates are negative?

Use it

With negative interest rates the formula does not change. When r < 0, the factor (1 + r)^T is below 1, so the forward price is below spot. The replication shows why: a loan of S₀ is repaid with less than S₀, so borrowing and buying the asset costs less than S₀ at expiration, and the forward price must match it.

Go deeper Why the forward price ignores anyone's view of the future

The replication argument involves no forecast. Whatever S_T turns out to be, the borrow-and-buy position and the forward give the same payoff. A trader who thinks the asset will rise cannot change the price, because the arbitrage that closes any gap works for every possible S_T. This is why derivative pricing differs from valuing a stock: no expected return enters.

Go deeper Discrete or continuous compounding?

The two conventions describe the same growth. A discrete rate of 3% a year compounds as 1.03^T. A continuous rate that gives the same one-year growth is ln(1.03) = 2.96%, so e^(0.0296 × T) is the same factor. The curriculum uses discrete compounding for individual assets and continuous compounding for indexes, commodity or credit indexes, and currencies.