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Choosing the Correct Delta for a Foreign-Currency Option

Article Quant Q&A · Author: HeadTfewn
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Summary

The document considers how to define delta for an FX option when its premium is denominated in a foreign currency but the holder evaluates value in a domestic currency. It contrasts differentiating the option’s foreign-currency present value directly with differentiating a value converted into domestic currency by dividing by spot.

The answer’s guiding principle is to take the derivative of the value the trader ultimately monitors. If the relevant book value is represented by f(s), its spot sensitivity is f'(s); hedging a different currency-converted quantity can leave sensitivity in the value that actually matters. The response recommends a numerical example as a way to make the distinction concrete, but does not provide one. It offers no pricing model, derivation, or specific hedge calculation, so the answer is a conceptual rule rather than a full treatment of FX translation risk.

Key ideas

  • Define delta with respect to the value the trader intends to manage.
  • A book’s spot sensitivity may differ from the sensitivity of its value after currency conversion.
  • Hedging the derivative of an irrelevant valuation measure can leave risk in the relevant one.
  • A numerical example could clarify the distinction, but none is included.

Tags

ForexOptionsDerivatives pricingRisk management
Full text
# How does a delta for an fx option take into account translation risk?


# How does a delta for an fx option take into account translation risk?












Say I buy a FX option on DOM/FOR and the premium is received in, say, FOR, and my base currency is DOM.

What is the delta of this option?

Is it $f'(s)$ where $f$is the pricing formula and $s$ is the spot rate, or is it the derivative of $f(s)/s$, i.e. the value of the fx option in domestic currency?

## Answer by Andrea (score 1)

https://quant.stackexchange.com/a/81058

A concrete example will help to understand, with real numbers.

But, you need to take the derivative of what you ultimately care about. So if $f(s)$ is the PV of the book (the thing you look at), then $f'(s)$ is more correct.

Otherwise, when you hedge, you zero the sensitivity of the "wrong" thing.

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.