FX Digital Option Delta and Its Vanilla Option Hedge
Summary
The document addresses a discrepancy between a calculated FX option delta and a market terminal’s value. The key clarification is that the instrument is a digital vanilla, or binary, option, so ordinary vanilla Black–Scholes delta inputs and intuition do not directly resolve the calculation. The response describes valuing the digital through a tight call spread, with separate implied volatilities for the two legs that may need to be inferred from the market display.
It also explains that payoff currency and notional currency matter: a digital quoted with notional in the first currency can behave as an asset-or-nothing payoff rather than cash-or-nothing, making the final exchange rate relevant. The hedge delta is then related to the combined hedge values of the vanilla legs. The source offers practical interpretation rather than a worked numerical derivation, and it does not provide the missing domestic and foreign rates or independently verify the terminal’s result.
Key ideas
- The instrument described is a digital FX option, not a standard vanilla option.
- A tight call spread can represent the digital, with each leg using its own implied volatility.
- The notional currency affects whether the payoff is cash-or-nothing or asset-or-nothing.
- Digital option delta reflects the hedge values of the component vanilla options.
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# Calculating the Delta of FX option # Calculating the Delta of FX option I'm tying to reconcile the delta value for an FX option. I'm comparing the results to Bloomberg to verify our calculation is correct. I've looked at this - Quantlib: Greeks of FX option in Python but it doesn't show where Rd (domestic interest rate) Rf (foreign interest rate) came from. The option I'm trying to calculate the Delta for is as follows: The Black-Scholes formula for delta is as follows: where: Using the information for the ScreenShot I get: S = 108.947 X = 83.200 T = 83 / 365 = 0.2274 years σ = 15.703% = 15.703 / 100 = 0.15703 Where can I find the Rd (domestic interest rate) Rf (foreign interest rate) from the screen shot? Do I need to access this from another screen in Bloomberg? ## Answer by AKdemy (score 2) https://quant.stackexchange.com/a/76365 What you look at here is not a normal option. The type is called DIVA, which stands for digital vanilla, which is a binary option. Bloomberg computes this via a tight spread. However, you have a few difficulties here: - The spread will use two different IVs, which is not displayed on the GUI. You would need to get this by applying the spread (1% of strike in BBG's case) and reading of the IV of each individual option. - Notional is usually in ccy2 in a digital: JPY in your case - Since your notional is in ccy1 you actually no longer have a cash or nothing, but an asset or nothing binary option. The final exchange rate will matter for your actual payoff in this case, which is why you need to scale it by a vanilla call option. - Delta (by the way, d1 is not delta anyways, also not in the vanilla case) will work very differently here. It will be the sum of both vanilla option hedge values. In all honesty, I'd just trust Bloomberg here.
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