# FX Option pricing on Forward vs. Spot

In a GBM world with riskless domestic and foreign interest rates, what would be the correct model for a FX plain vanilla option given the statement that this option is priced on the forward? I guess it would be the Garman Kohlhagen model or the Black (76) model but I'm a bit confused between the two in the context of pricing on spot vs. pricing on forward. I would appreciate an answer that scetches the main differences.

• Black76 formula uses "F" and GK uses "S", but they are the same when you make the substitution $F \leftrightarrow S e^{(r-q)T}$ – Alex C Jan 26 '17 at 23:34
• @Alex C, that's what I was thinking as well. I just didn't believe it;) For the sake of clarity, pricing on forward or spot is a matter of convinience depending on the inputs used, right? – Tim Jan 27 '17 at 8:55
• Yes, I find that one or the other is more convenient in a given situation. For example if it is difficult to know r and q I prefer to work in terms of $F$ rather than $S$. – Alex C Jan 28 '17 at 4:39
• @Alex C, thx, if you could write up a short answer, also for the reference to others, I can accept this. For me it was really a matter of confusion about the mentioned statements. – Tim Jan 28 '17 at 10:19
• @Alex C, I posted a follow up question, you may have a quick look. – Tim Jan 28 '17 at 11:26

## 2 Answers

The Black76 formula uses "F" (the forward price) and Garman-Kohlhagen uses "S" (the spot price), but they are the same formula when you make the substitution $F ↔ Se^{(r−q)T}$

Which formula to use then becomes a matter of convenience. In some cases F is publicly quoted but r and q would have to be estimated, in that case (being lazy) I prefer to use F directly in the relevant formula.

Pricing on spot takes the mid price of all the bids and offers available in the market and then skews that mid according to any proprietary position (ie you're getting too long EUR so to maintain a less risky position you want to sell EUR so lower your price in the market and participants will be attracted to buy your EUR off you), and then adds a spread to cover hedging costs and client counterparty risks depending on the market venue. Pricing on the forwards takes that spot price and adds in money market interest rate differentials. You can see more at FX / MM training

• I don't think this answers the question. You describe how you can compute a spot or forward price. The question was about pricing options on them. – LocalVolatility Jan 27 '17 at 13:42
• Thank you for the additional information. I aggree with the comment of @LocalVolatility, however, I also find your infos about "microstructure" interesting. In principle, you seem to agree with the comment from Alex C written above? – Tim Jan 28 '17 at 10:27
• Yes, a forward is derived from spot & interest rate differential, so the answer to your question about options pricing models that use either spot or forward is they are "substitutable". The basic point of options pricing is intrinsic value + time value, and the latter is fraught with risk due to volatility. See this answer: quant.stackexchange.com/questions/30987/… – rupweb Jan 30 '17 at 10:15