FRM Exam Part I · Foreign Exchange Markets

Currency Futures, Options and Cross-Currency Swaps Explained

Q: What is the difference between currency futures and forwards?

Futures are standardised, exchange-traded and cleared by a clearinghouse, with daily marking to market and margin. Forwards are customised OTC contracts settled at maturity and carry counterparty risk. Both fix an exchange rate for a future date.

Q: How is the Garman-Kohlhagen model different from Black-Scholes-Merton?

It prices options on currencies. The foreign risk-free rate takes the role of the dividend yield, so S is discounted at r_f and K at r_d. The structure of d1 and d2 is otherwise the same.

Q: How does a cross-currency swap differ from an interest rate swap?

A cross-currency swap involves two currencies, so the principal is usually exchanged at the start and returned at the end. The interest legs are in different currencies. In a plain interest rate swap the notional is not exchanged.

Q: How do you hedge currency risk with forwards and options?

A forward locks in the exchange rate, so you remove both the loss and the gain from currency moves. An option sets a worst-case rate but lets you benefit if the rate moves in your favour. You pay a premium for that flexibility.

Q: Is the value of a cross-currency swap zero throughout its life?

No. It is zero at inception only if the swap is set at fair market rates. Afterwards it changes with the spot rate and the interest rates in both currencies. Its value is the difference between the two legs, converted at the current spot rate.

Updated 11 October 2026 · Fact-checked

Currency derivatives let you fix or trade an exchange rate. Futures are exchange-traded and marked to market daily. Forwards, options and cross-currency swaps are mostly OTC. Price forwards with interest rate parity, options with Garman-Kohlhagen, and swaps by exchanging principal and interest in two currencies, valued as the difference between two bonds.

Understand Currency Futures, Options and Cross-Currency Swaps

A currency derivative gets its value from an exchange rate. Firms use it to hedge a foreign-currency cash flow. Traders use it to take a view on a currency. Always fix the quote first: S is the price of one unit of foreign currency in domestic currency, for example USD per EUR.

A currency forward is an OTC agreement to exchange currencies at a set rate on a set date. Its price follows covered interest parity: the forward rate equals the spot rate adjusted for the interest rate gap. The currency with the higher interest rate trades at a forward discount. A currency future is the same idea, but standardised, traded on an exchange and cleared by a clearinghouse. It is marked to market every day, so gains and losses are settled daily through margin. Forwards settle once, at maturity, and carry counterparty risk.

A currency option gives the right, not the obligation, to exchange at a strike. A foreign currency behaves like a stock paying a continuous dividend yield equal to the foreign interest rate. That is why the Garman-Kohlhagen model is Black-Scholes-Merton with the foreign rate as the yield. A call on EUR (strike in USD per EUR) is economically a put on USD. Options cost a premium, but they keep the upside while capping the loss.

A cross-currency swap exchanges interest payments, and usually principal, in two currencies. Principal is usually exchanged at the start at spot and returned at maturity at the same rate. Interest can be fixed against fixed, fixed against floating or floating against floating. Unlike a plain interest rate swap, the principal exchange, when it takes place, is real, because the currencies differ. At inception the swap has zero value only if the fixed rates are the market (fair) rates. Later, its value is the present value of the currency you receive minus the present value of the currency you pay, both converted at the current spot rate.

Key formulas to remember

Covered interest parity forward (continuous compounding): F = S × e^((r_d − r_f) × T); S and F are domestic per unit of foreign currency. r_d is the domestic rate and r_f the foreign rate. With annual compounding use F = S × (1 + r_d)^T ÷ (1 + r_f)^T.
Garman-Kohlhagen call: c = S × e^(−r_f × T) × N(d1) − K × e^(−r_d × T) × N(d2); Option on one unit of foreign currency, priced in domestic currency.
Garman-Kohlhagen put: p = K × e^(−r_d × T) × N(−d2) − S × e^(−r_f × T) × N(−d1); Same d1 and d2 as the call.
d1 and d2: d1 = [ln(S ÷ K) + (r_d − r_f + σ² ÷ 2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; Swapping r_d and r_f is the most common error.
Put-call parity for currency options: c − p = S × e^(−r_f × T) − K × e^(−r_d × T); European options with the same strike and maturity.
Option on a currency forward (Black model): c = e^(−r_d × T) × [F × N(d1) − K × N(d2)], with d1 = [ln(F ÷ K) + σ² × T ÷ 2] ÷ (σ × √T); Use when the option is written on a forward or futures rate F.
Value of a cross-currency swap: V = B_D − S × B_F (receive domestic, pay foreign); B_D and B_F are the present values of the domestic and foreign legs, including principal. Reverse the sign if you receive foreign and pay domestic. Value at inception is zero only if the swap is set at market (fair) rates. The worked example does not assume this.
Swap principals: Foreign principal = Domestic principal ÷ S0; Exchanged at the start and returned at maturity at the same rate S0.

How to solve Currency Futures, Options and Cross-Currency Swaps questions

Use this order for any question on currency futures, options or swaps. It stops the usual quote-direction and rate-swap errors.

1Write down the quote convention: which currency is the base (foreign) and which is the domestic (price) currency. Label r_d and r_f accordingly.
2Identify the instrument: forward, future, option or swap. Check whether the question is about price, value, payoff, hedge or cash flows.
3For a forward or future, apply F = S × e^((r_d − r_f) × T) or the discrete version. Check that the higher-rate currency is at a forward discount.
4For an option, compute d1 and d2 with r_d in the discount term of K and r_f in the discount term of S. Then use N(d) values or parity.
5For a swap, list the cash flows in each currency: initial principal, periodic interest, final principal and interest. Value each leg by discounting at that currency's rates.
6Convert the foreign leg at the current spot rate only when you need a single-currency value.
7For hedging questions, compare the outcomes: forward locks the rate, option keeps upside for a premium, futures add daily margin and basis risk.
8Check the answer for sign, units and reasonableness, for example put-call parity or a swap set at market rates being worth zero at inception.

Quickest way: Direction-and-parity shortcut

When to use it: Use it for multiple-choice questions where you can eliminate options before doing heavy calculation.

Decide the forward direction first. If r_d > r_f, then F > S. If r_d < r_f, then F < S. Remove options that go the wrong way.
For puts and calls, use parity instead of computing both N(d) pairs: p = c − S × e^(−r_f × T) + K × e^(−r_d × T).
For swaps, remember that cash flows are fixed amounts in each currency. Interest equals rate × principal in that currency. The final period includes the principal.
For hedging questions, use the rule: forward or future fixes the rate, option sets a floor or cap on the rate.
If a question gives annual rates and one year, use discrete compounding if the text says so. Do not switch to continuous.

Common mistakes in Currency Futures, Options and Cross-Currency Swaps

Using the foreign and domestic rates the wrong way round in the forward or d1 formula.
The quote can be USD per EUR or EUR per USD, and the formula is easy to memorise without labels.
Fix: Write S as domestic per foreign first. Then r_d is the rate of the currency in the numerator of S. Check the direction against the rule that the higher-rate currency trades at a forward discount.
Saying futures and forwards are the same.
Their prices are close when rates are stable, so the differences get forgotten.
Fix: Remember: futures are standardised, exchange-traded, cleared and marked to market daily. Forwards are customised OTC contracts settled at maturity, with counterparty risk.
Treating a cross-currency swap as only an exchange of interest.
It is confused with a plain vanilla interest rate swap, where notional is not exchanged.
Fix: List the initial principal exchange, interest flows and final principal exchange. Principal is usually exchanged at S0 at both ends.
Converting the whole swap at the original spot rate when valuing it later.
The initial rate is used for principals, so it feels right for valuation as well.
Fix: The principals are fixed at S0, but the value of the swap uses today's spot rate to convert the foreign leg: V = B_D − S × B_F.
Forgetting that a call on EUR is also a put on USD, or using the strike the wrong way up.
The option is described from only one currency's viewpoint.
Fix: A call on EUR with strike K USD per EUR equals a put on USD with strike 1/K EUR per USD, with the notional scaled to match (USD notional = K × EUR notional). Rewrite the position in terms of the currency you receive and the currency you pay. Then decide which side is the call and which the put.
Forgetting the foreign-rate discount on the spot rate in Garman-Kohlhagen and in put-call parity.
Students use the stock-option version of the formula with no dividend yield.
Fix: Always multiply S by e^(−r_f × T), as for an asset paying a yield equal to the foreign rate.

Worked examples

Example 1

A European call on EUR with USD as the domestic currency has strike 1.10 USD per EUR, S = 1.10, 1 year to expiry, USD rate 4% and EUR rate 2% (both continuous). Volatility is not given, so take the call price as an assumption: the call is worth USD 0.0700 per EUR. What is the put with the same strike and expiry?

Show the solution

Use put-call parity: c − p = S × e^(−r_f × T) − K × e^(−r_d × T).
S × e^(−r_f × T) = 1.10 × e^(−0.02) = 1.10 × 0.980199 = 1.078219.
K × e^(−r_d × T) = 1.10 × e^(−0.04) = 1.10 × 0.960789 = 1.056868.
c − p = 1.078219 − 1.056868 = 0.021351.
p = c − 0.021351 = 0.0700 − 0.021351 = 0.048649.

Answer: The put is worth about USD 0.0486 per EUR.

Example 2

A US firm enters a 3-year annual cross-currency swap. Spot is 1.10 USD per EUR. The firm pays USD 50 million at the start and receives the EUR equivalent. It receives 5% fixed on USD 50 million and pays 3% fixed on the EUR principal. Find the initial EUR principal and the firm's cash flows at maturity (year 3).

Show the solution

EUR principal = USD 50 million ÷ 1.10 = EUR 45.4545 million. At the start the firm pays USD 50 million and receives EUR 45.4545 million.
Annual USD interest received = 5% × 50 = USD 2.5 million.
Annual EUR interest paid = 3% × 45.4545 = EUR 1.3636 million.
At year 3 the principals are exchanged back at the same rate: the firm receives USD 50 million and pays EUR 45.4545 million.
Year 3 USD flow: receive 50 + 2.5 = USD 52.5 million.
Year 3 EUR flow: pay 45.4545 + 1.3636 = EUR 46.8182 million.

Answer: EUR principal is EUR 45.4545 million. At year 3 the firm receives USD 52.5 million and pays EUR 46.8182 million.

Exam tips

Write the quote convention before touching any formula. Most lost marks come from direction errors.
Know the differences between futures and forwards as a short list: standardisation, clearing, daily settlement, counterparty risk, liquidity.
Use put-call parity to get a put from a call, instead of computing N(−d1) and N(−d2).
In swap questions, always draw the cash flows in two columns, one per currency. Include the initial and final principal.
Hedging questions often test the choice of instrument: a forward gives certainty, an option protects against downside and keeps upside, at a premium.

Practice questions from Foreign Exchange Markets

Currency Futures, Options and Cross-Currency Swaps in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Currency Futures, Options and Cross-Currency Swaps: frequently asked questions

What is the difference between currency futures and forwards?

Futures are standardised, exchange-traded and cleared by a clearinghouse, with daily marking to market and margin. Forwards are customised OTC contracts settled at maturity and carry counterparty risk. Both fix an exchange rate for a future date.

How is the Garman-Kohlhagen model different from Black-Scholes-Merton?

It prices options on currencies. The foreign risk-free rate takes the role of the dividend yield, so S is discounted at r_f and K at r_d. The structure of d1 and d2 is otherwise the same.

How does a cross-currency swap differ from an interest rate swap?

A cross-currency swap involves two currencies, so the principal is usually exchanged at the start and returned at the end. The interest legs are in different currencies. In a plain interest rate swap the notional is not exchanged.

How do you hedge currency risk with forwards and options?

A forward locks in the exchange rate, so you remove both the loss and the gain from currency moves. An option sets a worst-case rate but lets you benefit if the rate moves in your favour. You pay a premium for that flexibility.

Is the value of a cross-currency swap zero throughout its life?

No. It is zero at inception only if the swap is set at fair market rates. Afterwards it changes with the spot rate and the interest rates in both currencies. Its value is the difference between the two legs, converted at the current spot rate.