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.
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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).
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$\begingroup$ 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. $\endgroup$ Apr 18 at 16:32
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$\begingroup$ 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. $\endgroup$– AKdemyApr 18 at 20:33