Options on Currencies

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2026

Exchange Rates

• The (nominal) exchange rate is the price of one currency in terms of another
  • The base currency is the one being priced; the quote currency is the unit of measurement
  • EUR/USD = $1.4380/€ means €1 costs $1.4380
• Most euro/sterling pairs quote the dollar as the price currency (EUR/USD, GBP/USD)
• Most other pairs use the dollar as the base currency (USD/JPY, USD/CNY, USD/CLP)
• The exchange rate is a relative price: if EUR/USD = $1.47/€, then \$1 = \frac{1}{1.47} = \text{€}0.68

The Risk-Neutral Process for a Currency

• A unit of foreign currency is analogous to a stock paying a continuous dividend yield
  • Investing in a foreign money-market account earns r^{*} continuously
  • Replace dividend yield q with r^{*} in the standard GBM
• The risk-neutral process for the exchange rate S is then: dS = (r - r^{*}) S \, dt + \sigma S \, dW^{*}
• The drift (r - r^{*}) reflects that expected currency appreciation equals the interest-rate differential

Forward Contracts on Currencies

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

Example 1 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, notional €1 million:
  • From a European investor’s perspective: a put on USD/EUR, strike €0.80, notional $1.25 million
  • Same contract — different perspectives
• It is standard to name both legs explicitly: EUR call / USD put

Garman-Kohlhagen Model

• The Black-Scholes formulas for European currency options 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}

Black’s Model

• Since F = S e^{(r-r^{*})T}, substituting gives an equivalent form with only one discount factor: \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}}
• Preferred in practice:
  • Forward rates are directly quoted in the market
  • Identical to Black’s model for futures — one formula covers many asset classes

Put-Call Parity and ATMF

• A long call / short put pays S_T - K at expiry — the same payoff as a forward contract
• Hence put-call parity for currencies: 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: C - P = 0
  • An ATMF call and put have the same value