American FX Options

[Adam]

In Section 17.6 of Hull (9th global), it is said that "In some circumstances, it is optimal to exercise American currency and index options prior to maturity. Thus, American currency and index options are worth more than their European counterparts. In general, call options on high-interest currencies and put options on low-interest currencies are the most likely to be exercised prior to maturity. The reason is that a high-interest currency is expected to depreciate and a low-interest currency is expected to appreciate. Similarly, call options on indices with high-dividend yields and put options on indices with low-dividend yields are most likely to be exercised early."

My understanding of the above is as below. Is it correct, please?

• "a high-interest currency is expected to depreciate" is due to covered interest rate parity;
• "call options on high-interest currencies ... the most likely to be exercised prior to maturity" means that once spot exchange rate is higher than the strike, then the call holder might want to exercise instead of waiting any longer since he/she expect the currency to depreciate on average. This leads to the early exercise.

 

[mugono]

In Section 17.6 of Hull (9th global), it is said that "In some circumstances, it is optimal to exercise American currency and index options prior to maturity. Thus, American currency and index options are worth more than their European counterparts. In general, call options on high-interest currencies and put options on low-interest currencies are the most likely to be exercised prior to maturity. The reason is that a high-interest currency is expected to depreciate and a low-interest currency is expected to appreciate. Similarly, call options on indices with high-dividend yields and put options on indices with low-dividend yields are most likely to be exercised early."

My understanding of the above is as below. Is it correct, please?

• "a high-interest currency is expected to depreciate" is due to covered interest rate parity;
• "call options on high-interest currencies ... the most likely to be exercised prior to maturity" means that once spot exchange rate is higher than the strike, then the call holder might want to exercise instead of waiting any longer since he/she expect the currency to depreciate on average. This leads to the early exercise.

I see that nobody has provided an answer to this difficult question .

In answer to your questions:

1. I think it's less to do with any particular equilibrium condition: for e.g. put-call parity does not hold exactly for American options. And covered interest rate parity is a relationship to explain the relationship between the spot, forward, foreign and base currency interest rate.

It could be optimal to e.g. exercise the calls on a high-interest currency if the loss of time value in the calls was less than the interest earned on the base (spot) currency. This is analogous to exercising the calls on a dividend paying stock prior to ex-dividend in order to receive the dividend. It would be sub-optimal to exercise the calls if the time value left in the calls exceeded the impending dividend. In this case, you would be better off selling the calls in the open market and purchasing the underlying directly.

2. A high-interest currency is expected to depreciate because there will be an increase in demand for the currency from around the world. You have to physically hold the currency in order to earn the interest. As per 1, it would be optimal to exercise the calls if the loss of time value was less than the interest you would earn by holding the physical currency.

I hope that helps.

 

[Adam]

I see that nobody has provided an answer to this difficult question .

In answer to your questions:

1. I think it's less to do with any particular equilibrium condition: for e.g. put-call parity does not hold exactly for American options. And covered interest rate parity is a relationship to explain the relationship between the spot, forward, foreign and base currency interest rate.

It could be optimal to e.g. exercise the calls on a high-interest currency if the loss of time value in the calls was less than the interest earned on the base (spot) currency. This is analogous to exercising the calls on a dividend paying stock prior to ex-dividend in order to receive the dividend. It would be sub-optimal to exercise the calls if the time value left in the calls exceeded the impending dividend. In this case, you would be better off selling the calls in the open market and purchasing the underlying directly.

2. A high-interest currency is expected to depreciate because there will be an increase in demand for the currency from around the world. You have to physically hold the currency in order to earn the interest. As per 1, it would be optimal to exercise the calls if the loss of time value was less than the interest you would earn by holding the physical currency.

I hope that helps.

Thank you, mugono.
What you wrote makes sense to me. Just want tot clarify one more thing. The book said "The reason is that a high-interest currency is expected to depreciate". However, it appears to me that there is no such causal relationship between a currency depreciation and option exercise. Could you elaborate a bit on this, please? Thank you.

 

[mugono]

Thank you, mugono.
What you wrote makes sense to me. Just want tot clarify one more thing. The book said "The reason is that a high-interest currency is expected to depreciate". However, it appears to me that there is no such causal relationship between a currency depreciation and option exercise. Could you elaborate a bit on this, please? Thank you.

Yes, sure.

The preceding sentence to the one quoted above says " In general, call options on high-interest currencies [are] most likely...". It is therefore reasonable to conclude that the relationship between a currency depreciation and option exercise isn't causal. As explained in my previous post, whether exercising is optimal would require a comparison between the time value lost and interest gained from holding the currency directly.