This is an open question : How would you price an american option on future consistently with the smile ?
For example, when it comes to pricing Eurodollar option, we often see the black formula used in an awkward way :
- There is no margin process for the option, an american option should not be priced as a European.
- The future rate is not a martingale under the "T-forward measure"
- Even if it was a European, payment occurs at the expiring date, not "tenor"-month later.
The american premium is not negligible , hence the need to model the future rate as well as the funding rate.