Can someone elaborate the difference between the two, and what is the typical convention used in markets? If there is a mathematical relationship. Any helpful links/guides would be appreciated as well, all in the context of European FX Options.

I am assuming you do NOT refer to Investopedia: Forward Premium which does not change the way you value an option.

I assume you mean the following:
Wikipedia: Garman-Kohlagen

The call formula (similar for put) can also be expressed in terms of Fwd instead of S (covered interest rate parity) which yields:

$$exp^{-r_d*t}[FN(d_1) - KN(d_2()]$$ where $$r_d$$ is domestic interest rate (in case of EURUSD, the USD rate). I prefer to use ccy1 and ccy2 (CCY1CCY2) to avoid any potential confusion.

If you exclude discounting $$(exp^{-r_{ccy2}*t})$$ you get forward premium: discount this to your premium date(usually standard T+2 for many currencies), you get spot premium.

Note that t in the discount factor is time to delivery, which is days/365 (assuming 365 daycount). Days is computed as actual days between delivery date and premium (or spot date).

• My reading of the OP's question was that it's about the timing of the payment of the premium when trading FX options. Spot premium: paid upfront, i.e. at time of trade. Forward premium: paid at expiry. Perhaps the OP can clarify. Apr 18, 2021 at 16:32
• I think so too, in which case my answer is correct. I only used two dates, delivery and spot settlement - but t in the discounting can be anything. Apr 18, 2021 at 20:33