How do I derive the no-arbitrage price of a currency forward?

Question

AcadiFi · Accepted Answer

The way to never forget which rate goes on top is to think of the forward as the outcome of two equivalent arbitrage portfolios.

Suppose Kenzaburo Trading needs to lock in a USD/JPY rate 6 months forward. Today's spot is 148.00 JPY per USD. The 6-month USD risk-free rate is 5.20%; the 6-month JPY rate is 0.40%. Portfolio A: borrow 1 USD today, convert to 148 JPY, invest JPY at 0.40% for 6 months, yielding 148 × (1 + 0.0040 × 180/360) = 148.296 JPY. Owe back 1 × (1 + 0.0520 × 180/360) = 1.026 USD. Portfolio B: enter a 6-month forward to buy 1.026 USD in 6 months at rate F.

Both portfolios discharge the same USD obligation at maturity. The JPY cost must be identical: 148.296 = 1.026 × F, giving F = 144.54 JPY per USD. The forward is cheaper than spot in JPY terms because the USD interest rate is higher.

The general formula using price-currency notation (JPY per USD here): F = S × (1 + r_price × t) / (1 + r_base × t). The price currency (JPY) rate goes on top; the base currency (USD) rate goes on the bottom. A mnemonic: high-rate currency trades at a forward discount (it weakens in the forward). USD has the higher rate so USD weakens from 148 to 144.54.

This is covered interest rate parity — violations would be instantly arbitraged away in liquid markets.

How do I derive the no-arbitrage price of a currency forward?

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