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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?

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    $\begingroup$ The strike price is 1.0 €/$ $\endgroup$ – Alex C Aug 30 '16 at 7:26
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    $\begingroup$ And the contract is implicitely set up for 1$. $\endgroup$ – Quantuple Aug 30 '16 at 7:38
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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).

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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.

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