Why are two currency options available when paying via card at airports abroad?

The service you are referring to is called "dynamic currency conversion" and is sold as an extra convenience in locales with a heavy tourist presence, not just in airports. Many people are tempted to opt for their home currency because they assume that it will relieve them of inter-bank fees and disadvantageous exchange rates.

Starting with your second question...

And what are the benefits of choosing one over the other?

The benefit of paying in local currency is obvious: i.e., you know what the transaction involves. Generally it is picked up by the card issuer and passed through with predictable rates and fees that are governed by the regulator in your home country. So while the end-user faces uncertainty in knowing the precise exchange rate that will appear on their bill, they have the certainty of knowing that it will be set within a regulatory framework and hence 'near the market'.

When you opt to pay in your own currency abroad the exchange rate is established right then by whichever bank is servicing the retailer, and you have no control over it. This can lead to an uncomfortable difference in what you have actually paid.

The regulators in Europe require those offering 'dynamic currency conversion' to exhibit the exchange rate prior to the customer's approving the purchase.

Dynamic currency conversion is legal in the UK and across Europe, as long as traders display not just the price but also the exchange rate being used before the payment is made.

Source: Tourists warned over exchange rate costs (BBC, 24 July 2017)

BUT, they do not regulate how the exchange rate is presented to the customer, and therein lies a problem. What may be presented is the inverse rate of the bank's counterparty's bid and this is a rate applicable to the currency wholesale markets and not available to an ordinary mortal. So in order to evaluate it, the customer has to take the inverse of the rate and then compare it to the relative savings they could achieve by using an ATM. People are not generally willing to go through that kind of hassle.

Your other question...

Why are there two options?

Merchants love it because they receive a commission from the DCC provider.

Merchants that operate a global presence, like Costco for example, will avoid offering the service because it interferes with their internal hedging strategies. Amazon, on the other hand, will offer DCC to customers whose credit cards are denominated in multiple currencies; my observation is that Amazon likes to display a bid-side retail rate derived from yesterday's rate (fair enough for most purposes).

This shows a live screen grab from Amazon's DCC. The exchange rate works out to 500 pips over the prior day's mid-market rate. You can't purchase in this quantity (it's too low) from a retail bank for a rate that attractive, so in this case DCC makes sense. Caveat emptor and your mileage may vary.

Extended discussion on the advantages and disadvantages in the DCC Wiki Article.

It is always most beneficial to use the local amount and not the converted-on-spot amount.

I get asked this a lot as I work in electronic banking.

There are a few things that determine how you are charged:

1. The currency your card is issued in.
2. The local currency of the merchant's acquiring bank.
3. The schema (VISA, MasterCard) currency being used for interchange.
4. The interchange fees.
5. The foreign transaction fees.
6. The currency of the account your card is linked with (if it is not a credit card).

The bottom line is that all reconciliation between banks and the card networks (called schemes) happens in major trading currencies; usually it is the USD, but sometimes in Euros. There is an interbank rate which is updated twice in a business day.

So all transactions have to be charged and billed in this major currency. Lets say, its the USD.

So if your card is issued in INR, and its a credit card and you choose the local currency at the point-of-sale device then:

1. The local currency amount is converted to USD at the rate of the merchant's bank's agreement with the schema (card network). Usually, it is a markup over the midmarket or bloomberg rate.

2. This amount is charged to your card.

3. Your bank then takes this amount in USD and converts it back to INR at some other rate.

4. The bank then adds the transaction and network fees.

5. This total amount is then charged against your account or card balance.

If your card is issued in INR, and you chose INR at the point-of-sale:

1. You are getting the INR equivalent of the amount in local currency, converted to USD (at the merchant's bank rate), then converted to INR again (at the merchant's bank rate). Again, this rate is a midmarket rate.

2. This INR amount is then converted to USD at the network (schema) rate and charged to your card.

3. The normal steps from above are followed.

If your card is issued in a major currency, like the USD - then you are spared a few rates.

Some banks want an extra fee if you pay in a foreign currency, and the exchange rate can be worse than the official one.
So, paying in your currency is cheaper.

On the other side, the shop or ATM may charge you, too if you pay in your currency.

For example, getting money from a Swiss ATM in SFR is free of charge with my credit card, but definitely not with my debit card. Charging the card in EUR on the same ATM would be free for both cards, while the ATM itself would take a fee for this. That fee is cheaper than the fee for by debit card, but using the credit card is for free.

Usually (depending on your card) your bank offers a better exchange rate than the merchant, so choosing the merchant's currency is cheaper than choosing your card's currency.

Problems with refunds

However, one thing to consider is what happens if the product you bought is broken or the merchant fails to deliver the product. You could object against your credit card statement, but the easier and faster way is to talk directly to the merchant. Now the merchant will usually refund the amount of money he received in his currency, and the credit card company will convert it back to your currency. This means that the merchant will not refund the fee you paid to your bank when you bought the product AND the bank will charge an additional fee for the money transferred back from the merchant to your credit card (because the bank converts it from the merchant's currency to your currency)