Options on Currencies

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

Exchange Rates

• The (nominal) exchange rate between two currencies is the number of domestic currency units per unit of foreign currency.
  • You could always define it the other way around (indirect-quotes)
• Consider the EUR/USD exchange rate:
  • The quote currency is the US dollar (USD)
  • The base currency is the Euro (EUR)
• If the EUR/USD exchange rate is $1.47/€
  • For a US investor, 1 Euro is worth $1.47
  • In Europe, how many Euros is worth $1? \$1 = \frac{1}{1.47} = \text{€} 0.68/\$.
• The exchange rate is a relative price.

Direct Quotes for Exchange Rates

• Remember the street market convention:
  • A direct quote is the price of 1 unit of base currency expressed in the quote currency
  • For example, the direct quote of the EUR/USD could be S = \$ 1.4380 / \text{€} and represents the price in USD of 1 EUR.
• The market convention of calling this exchange rate EUR/USD might be misleading
  • It really represents the number of USD per EUR, i.e. \$1.4380 \Leftrightarrow \text{€}1.
• Some currency pairs such as EUR/USD or GBP/USD use the USD as the quote currency.
• However, most currency pairs are expressed using the dollar as the base currency, i.e., USD/JPY, USD/CNY, USD/CLP, etc.

The Risk-Neutral Process for a Currency

• When working with currencies, it is usually convenient to denote by r the risk-free rate of the quote currency and by r^{*} the risk-free rate of the base currency.
• The risk-neutral process for an exchange-rate S expressed with direct-quotes is then: dS = (r - r^{*}) S dt + \sigma S dW^{*}

Forward Contracts on Currencies

• Using the previous notation, the forward price with maturity T for the currency is: 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 then: F = 1.18663 e^{(0.015 - 0.005) (9/12)} = \$1.19556 or +89.3 forward-points.

Options on Currencies

• Options on currencies reveal an interesting relationship between the underlying asset and the numeraire used to express the price of the asset.
• Consider an American investor analyzing a option on the EUR/USD with maturity 1-year, strike price $1.25 over a notional of €1 million.
• From the point of the view of a European investor, that option is really a on the USD/EUR with same maturity, strike price €0.80 over a notional of $1.25 million.
• Hence, it is convenient to be explicit about the currency being bought and the one being sold when specifying the contract, i.e., we will talk about a when describing the previous contract.

Option Pricing Formulas for Currencies

• It is common to express the Black-Scholes formulas for options on currencies as a function of the corresponding forward price: \begin{align*} C & = F e^{-r T} \mathop{\Phi}(d_{1}) - K e^{-r T}\mathop{\Phi}(d_{2}) \\ P & = K e^{-r T}\mathop{\Phi}(-d_{2}) - F e^{-r T}\mathop{\Phi}(-d_{1}) \end{align*} where \begin{align*} F & = S e^{(r-r^{*}) T} \\ d_{1} & = \frac{\ln(F/K) + \frac{1}{2} \sigma^{2} T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma \sqrt{T}. \\ \end{align*}

At-The-Money-Forward Options

• An option with a strike price equal to its corresponding forward price is called at-the-money-forward (ATMF).
• Remember put-call parity for currencies: C - P = S e^{-r^{*} T} - K e^{-r T}
• When K = S e^{(r - r^{*}) T} we have that C - P = 0, i.e., when the strike price is equal to the forward price a call and a put with the same maturity are worth the same.