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FX Option Forwards, Interest Rate Differentials, and OIS

Article Quant Q&A · Author: Shyam
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Summary

The document explains how forward premium enters currency option pricing and clarifies the role of overnight indexed swaps. In the Garman–Kohlhagen formulation, the domestic-minus-foreign interest rate differential determines the forward exchange rate relative to spot. Equivalently, pricing a Black-style option on the FX forward incorporates that premium, while discounting accounts for rates over the option’s term.

The answers also correct two assumptions in the question: implied volatility is forward-looking rather than historical, and an OIS rate reflects an overnight benchmark but is not identical to the Federal Funds Rate. The discussion provides conceptual relationships rather than market calculations or a full treatment of curve construction. It distinguishes forward formation from discounting, but leaves detailed conventions, collateral assumptions, and benchmark-specific implementation to other sources.

Key ideas

  • The Garman–Kohlhagen rate differential links spot FX to the forward rate.
  • An FX option can be represented as an option on the forward exchange rate.
  • Interest rate differentials capture the forward premium in the pricing framework.
  • OIS rates reflect overnight benchmarks but are not themselves the Federal Funds Rate.
  • Option volatility is implied and forward-looking, rather than simply historical.

Tags

ForexOptionsDerivatives pricingFixed incomeVolatility
Full text
# Black Scholes- Options and OIS


# Black Scholes- Options and OIS












I have 2 questions.

In the Black Scholes formula for currency options, where does forward premium come in? Volatility will be a historic parameter, so which component considers fwd premia.

Typically, is OIS used to fix overnight rate? and the overnight rate would be the Fed Funds rate, right? USDIRS is used to hedge LIBOR.

## Answer by AdB (score 2)

https://quant.stackexchange.com/a/44919

When you say the Black Scholes formula for currency options, I assume you are referring to the Garman-Kohlhagen formula described here. Note that this formula is based on the interest rate differential $r_d - r_f$, which essentially captures the forward premium.

An even more explicit way to see this is to use the Black Model described here. Using this formula with $F_{t,T} = S_t e^{(r_d - r_f)(T-t)}$ being the forward FX rate will yield exactly the same formula as Garman-Kohlhagen (try to verify this algebraically), so it is clear that the option is actually written on the forward exchange rate - hence, it natually captures the forward premium.

I am a little unsure what you are asking in your second paragraph. It is true that OIS (Overnight Indexed Swap) captures the overnight rate. However, the OIS is not the same as the Fed Funds Rate (see this question).

## Answer by AKdemy (score 0)

https://quant.stackexchange.com/a/63969

You can find actual computed examples here and there.

Since you talk about vol, I think you do not mean how the actual forward is related to spot, but forward premium of the option vs spot premium (which is simply discounting, and probably why you ask about OIS rates as well). The second link shows this.

On a side note, VOL is not historic. It is implied and forward looking. See this answer for some details.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.