Options on Currencies

Currency options are among the most actively traded derivatives in the world, used by corporations to hedge foreign exchange exposure and by investors to express views on exchange rate movements. Pricing them requires only a small adaptation of the Black-Scholes framework: because holding foreign currency earns interest at the foreign risk-free rate, it behaves like a stock paying a continuous dividend yield. This note develops that idea from the ground up — starting with exchange rate conventions, building the risk-neutral model, and arriving at closed-form prices for European calls and puts.

Exchange Rates

The (nominal) exchange rate is the price of one currency in terms of another. In a currency pair, the base currency is the one being priced and the quote currency is the unit of measurement. The EUR/USD rate, for example, gives the number of US dollars per one euro: a rate of $1.4380/€ means €1 costs $1.4380. Despite the notation “EUR/USD,” it is the euro that is being bought or sold, and the dollar that measures its price.

Example 1 If EUR/USD = $1.47/€, then $1 buys: \frac{1}{1.47} = \text{€}0.68. The exchange rate is a relative price: its reciprocal gives the rate from the other currency’s perspective.

Most major pairs involving the euro or pound sterling (EUR/USD, GBP/USD) quote the dollar as the price currency. For most other pairs the dollar is the base currency instead, e.g., USD/JPY, USD/CNY, USD/CLP. Always check which currency is which before working with a rate.

The Risk-Neutral Process for a Currency

A unit of foreign currency is analogous to a stock that pays a continuous dividend yield. Specifically, investing in a foreign money-market account earns the foreign risk-free rate r^{*} continuously, just as a stock paying a continuous dividend yield q returns q per unit of time. Replacing q with r^{*} in the standard dividend-adjusted GBM gives the risk-neutral process for the exchange rate S: dS = (r - r^{*}) S \, dt + \sigma S \, dW^{*} where r is the domestic risk-free rate. The drift (r - r^{*}) reflects that, under no-arbitrage, the expected appreciation of the foreign currency equals the interest-rate differential.

The forward price with maturity T follows directly: F = S e^{(r - r^{*}) T}

Example 2 The EUR/USD currently trades at $1.18663. The continuously compounded 9-month risk-free rates in USD and EUR are 1.5% and 0.5% per year, respectively. The 9-month EUR/USD forward rate is: F = 1.18663 \, e^{(0.015 - 0.005)(9/12)} = \$1.19556.

Options on Currencies

Currency options have a useful symmetry: a call in one currency’s terms is a put in the other’s. Consider a call on EUR/USD with maturity 1 year, strike $1.25, and notional €1 million. From the perspective of a European investor, that same contract is a put on USD/EUR with the same maturity, strike €0.80, and notional $1.25 million. For this reason it is standard to name both legs explicitly — we call this contract a EUR call / USD put.

Black-Scholes Model for Currencies

The Black-Scholes formulas for European options on currencies (Garman-Kohlhagen model) are: \begin{aligned} C & = S e^{-r^{*} T} \operatorname{\Phi}(d_{1}) - K e^{-r T}\operatorname{\Phi}(d_{2}) \\ P & = K e^{-r T}\operatorname{\Phi}(-d_{2}) - S e^{-r^{*} T}\operatorname{\Phi}(-d_{1}) \end{aligned} where d_{1} = \frac{\ln(S/K) + (r - r^{*} + \frac{1}{2} \sigma^{2}) T}{\sigma \sqrt{T}} \quad \text{and} \quad d_{2} = d_{1} - \sigma \sqrt{T}.

Put-Call Parity

A portfolio that is long a call and short a put with the same strike K and maturity T pays S_T - K at expiry regardless of S_T. That payoff replicates a forward contract, whose present value today is S e^{-r^{*}T} - K e^{-rT}. Hence: C - P = S e^{-r^{*} T} - K e^{-r T}.

An option with strike equal to the forward price, K = F, is called at-the-money-forward (ATMF). Substituting into put-call parity gives C - P = 0: an ATMF call and put have the same value.

Example 3 The EUR/USD spot rate is $1.25. The volatility of the exchange rate is 12% per annum, the USD risk-free rate is 3%, and the EUR risk-free rate is 1%. Price a 6-month EUR call / USD put and a EUR put / USD call, both with strike $1.28.

We identify the inputs: S = 1.25, K = 1.28, r = 0.03, r^{*} = 0.01, \sigma = 0.12, T = 0.5. \begin{aligned} d_{1} & = \frac{\ln(1.25/1.28) + (0.03 - 0.01 + \frac{1}{2}(0.12)^{2})(0.5)}{0.12\sqrt{0.5}} = \frac{-0.0237 + 0.0136}{0.0849} = -0.12 \\ d_{2} & = -0.12 - 0.12\sqrt{0.5} = -0.20 \end{aligned} From the standard normal table: \operatorname{\Phi}(-0.12) = 0.452, \operatorname{\Phi}(-0.20) = 0.421, \operatorname{\Phi}(0.12) = 0.548, \operatorname{\Phi}(0.20) = 0.579.

The call (right to buy EUR, sell USD) is out-of-the-money since K > S: C = 1.25\, e^{-0.01(0.5)} (0.452) - 1.28\, e^{-0.03(0.5)} (0.421) = 1.244(0.452) - 1.261(0.421) = \$0.032. The put (right to sell EUR, buy USD): P = 1.28\, e^{-0.03(0.5)} (0.579) - 1.25\, e^{-0.01(0.5)} (0.548) = 1.261(0.579) - 1.244(0.548) = \$0.049. We can verify using put-call parity: C - P = 0.032 - 0.049 = -\$0.017, which matches S e^{-r^{*}T} - K e^{-rT} = 1.244 - 1.261 = -\$0.017.

Black’s Model

Since F = S e^{(r-r^{*})T}, we can substitute the forward rate directly into the pricing formulas. Because r^{*} is already embedded in F, it drops out and only one discount factor remains:

Forward-Based Form

\begin{aligned} C & = e^{-rT}\left[F\,\operatorname{\Phi}(d_{1}) - K\,\operatorname{\Phi}(d_{2})\right] \\ P & = e^{-rT}\left[K\,\operatorname{\Phi}(-d_{2}) - F\,\operatorname{\Phi}(-d_{1})\right] \end{aligned} where d_{1} = \frac{\ln(F/K) + \frac{1}{2} \sigma^{2} T}{\sigma \sqrt{T}} \quad \text{and} \quad d_{2} = d_{1} - \sigma \sqrt{T}.

This form is preferred in practice for two reasons. First, for most currency pairs, the forward rate is directly observable in the market and often quoted more reliably than the two risk-free rates separately. Second, the formula is identical to Black’s model for options on futures, so one formula covers currency, interest rate, and commodity options alike.

Practice Problems

Problem 1 Suppose that the spot price of the Canadian dollar (CAD) is USD $0.75 and that the CAD/USD exchange rate has a volatility of 4% per annum. The risk-free rates of interest in Canada and the United States are 9% and 7% per annum, respectively. Calculate the value of a European call option to buy one Canadian dollar for USD $0.75 in nine months. Use put-call parity to calculate the price of a European put option to sell one Canadian dollar for U.S. $0.75 in nine months.

Solution

We have that: \begin{aligned} d_{1} & = \frac{\ln(0.75/0.75) + (0.07 - 0.09 + \frac{1}{2} 0.04^{2})(9/12)}{0.04\sqrt{9/12}} = -0.4157 \Rightarrow \operatorname{\Phi}(d_{1}) = 0.339 \\ d_{2} & = -0.4157 - 0.04 \sqrt{9/12} = -0.4503 \Rightarrow \operatorname{\Phi}(d_{2}) = 0.326 \end{aligned} Hence, C = 0.75 e^{-0.09 (9/12)} \operatorname{\Phi}(d_{1}) - 0.75 e^{-0.07 (9/12)} \operatorname{\Phi}(d_{2}) = \$0.01. For the put we have that: P = C - S e^{-r^{*} T} + K e^{-r T} = 0.01 - 0.75 e^{-0.09 (9/12)} + 0.75 e^{-0.07 (9/12)} = \$0.02

Problem 2 Calculate the value of an eight-month European put option on a currency with a strike price of 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, the domestic risk-free interest rate is 4% per annum, and the foreign risk-free interest rate is 8% per annum.

Solution

We have that: \begin{aligned} d_{1} & = \frac{\ln(0.52/0.50) + (0.04 - 0.08 + \frac{1}{2} 0.12^{2})(8/12)}{0.12 \sqrt{8/12}} = 0.1771 \Rightarrow \operatorname{\Phi}(-d_{1}) = 0.430 \\ d_{2} & = 0.1771 - 0.12 \sqrt{8/12} = 0.0791 \Rightarrow \operatorname{\Phi}(-d_{2}) = 0.468 \end{aligned} Hence, P = 0.50 e^{-0.04 (8/12)} \operatorname{\Phi}(-d_{2}) - 0.52 e^{-0.08 (8/12)} \operatorname{\Phi}(-d_{1}) = \$0.02.