Interpreting Currency Call Option Payoffs and Their Units
Summary
The document explains why a currency call option payoff can be calculated by subtracting a strike exchange rate from the market exchange rate. In the example, both values are quoted in euros per dollar, so their difference is also an exchange-rate amount, not yet a payoff in euros. Multiplying that difference by the number of dollars covered by the option converts it into the payoff currency. For a notional covering one dollar, the numerical rate difference corresponds to the euro amount paid toward that dollar’s cost.
The response uses an example of a buyer seeking protection against the euro cost of purchasing a dollar, with another party covering the amount above the chosen rate. It notes that a real option requires an upfront premium and that contract notional determines the total payoff. The explanation is conceptual and uses a simplified one-dollar example; it does not discuss settlement conventions, option style, contract specifications, or the effect of premium on net profit.
Key ideas
- A currency option strike and spot rate must use the same quotation convention to be subtracted.
- The difference between rates is denominated as a rate, not by itself as a cash payoff.
- Multiplying the rate difference by the dollar notional gives a payoff in euros.
- The option premium is paid separately and affects the buyer’s net result.
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Full text
# Understanding the payoff of currency options
# Understanding the payoff of currency options
I am self-studying for an actuarial exam and I am having a hard time understanding what happens when a currency option pays off.
Consider the below problem. The payoff at $C_u$ would be $\max(x_u - K, 0) = \max(1.045\text{ €/\$} - €1, 0)$. The author claims that this payoff is €0.045.
I don't understand how we can subtract €1 from a rate of 1.045€/$. One is a currency and one is an exchange rate on currencies, so how is one able to subtract if the units are different and then arrive at a payoff in the unit of the strike?
## Answer by nbbo2 (score 2, accepted)
https://quant.stackexchange.com/a/29905
You are a European citizen of Italian nationality who has decided to purchase an item in 3 months time. This item is a United States 1 Dollar bill, a piece of paper with a picture of george washington on one side and the US Treasury building on the other. Due to a decision made by President Nixon in 1971 the price of this item is not fixed but fluctuates every day. You are quite certain you will purchase the item but you are concerned that if the price of the dollar rises in the meantime your cost in euros will increase. In particular you do not wish to pay an exchange rate greater than 1.0000 Euros to the dollar; a lower rate would be acceptable.
You have a wealthy uncle who has made the following generous and valuable offer: he will cover the difference if the exchange rate goes above the 1.0000 eur per dollar that you have in mind. So for example if the exchange rate ends up at 1.045€/$, you will put up 1 € and your uncle will give you 0.045€ out of his pocket, the combined 1.045€ will be used to purchase the bill in the free market and you will have paid out of pocket no more than the 1€ you had in mind.
This uncle does not really exist, but it describes the payoff of a call option on one dollar. But the option is not given to you for free, it is a valuable piece of paper and you have to pay a premium up front to obtain it from a greedy investment bank. The payoff is the difference between actual and predetermined (strike) exchange rates multiplied by the notional amount of dollars (in this toy example one dollar bill, but in real markets a suitcase containing one million bills is the standard size).
## Answer by Rosen Petroff (score -1)
https://quant.stackexchange.com/a/29902
Having a strike price of 1€ for a currency option means that you will be allowed to buy 1\$ for 1€ e.g. it is an exchange rate. Therefore the payoff will be the difference between the two rates.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.