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Hedging Cross-Currency Option Volatility with a EURJPY Straddle

Article Quant Q&A · Author: bng
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Summary

The discussion considers a foreign-exchange volatility market maker who is long a EURUSD straddle and short a USDJPY straddle, then must quote a EURJPY straddle. It asks whether the existing positions imply an appropriate trading bias for the cross-currency option, given the desire to reduce risk across the currency triangle.

The answers say the hedge depends on how EURJPY volatility is marked and modeled. Deriving cross rates from dollar pairs requires assumptions about the relationship among the three currency pairs and their volatilities, including correlation. One response frames the hedge in terms of the possible EURJPY move: if it is driven more by EURUSD, a short cross straddle may reduce risk; if USDJPY dominates, a long cross straddle may help. Relative implied volatilities may offer a clue, but the discussion provides no quantitative hedge ratio or validated rule, so the positions should not be treated as automatically offsetting.

Key ideas

  • The appropriate EURJPY straddle hedge depends on how cross volatility is quoted or derived.
  • Inferring cross-pair volatility from dollar pairs embeds assumptions about currency correlation.
  • A short EURJPY straddle may offset risk when EURUSD is the dominant source of cross-rate movement.
  • A long EURJPY straddle may help when USDJPY is the dominant source of movement.
  • Relative implied volatilities can inform the assessment but do not establish a definitive hedge.

Tags

ForexOptionsVolatilityRisk management
Full text
# hedging out of cross-ccy vol risk using direct ccy options


# hedging out of cross-ccy vol risk using direct ccy options












Lets suppose a G10 FX vol market-maker starts out with a flat book. During the day, the market-maker bought a EURUSD 1 week ATM straddle from one client while sold USDJPY 1 week ATM straddle from another client, with same USD notional.

So the entire book looks like this:

- long EURUSD 1 week ATM straddle (same USD notional)

- short USDJPY 1 week ATM straddle (same USD notional)

Also, lets suppose the EURUSD 1 week smile is downward-sloping, and USDJPY 1 week smile is downward-sloping (or JPYUSD 1 week smile is upward-sloping).

Then, another client comes in RFQing a two-way price on EURJPY 1 week ATM straddles, coincidently for the same USD-equivalent notional.

If the market-maker does not have any views in the next 1 week and wants to be flat (maybe needs to go on vacation), is the market-maker axed a certain way on EURJPY 1 Week straddles? For simplicity, assume everything is done simultaneously, so no market moves would impact the decision.

My guess would be that if the market-maker is sitting in a US domiciled desk, then the market-maker would be axed to sell EURJPY 1 week ATM straddles. Given no market move, being able to sell EURJPY will not only flatten out the delta/gamma risk but also roughly flatten out the vanna risk

Or would the preference be to trade out of each of the EURUSD option and the USDJPY option separately, since these vols trade differently? And quote the EURJPY option without any axe, so all three 1 week options would be managed separately?

Look forward to your thoughts.

## Answer by Mercadian (score 0)

https://quant.stackexchange.com/a/72051

I'd say it depends on how you consider EURJPY vols, do you actively mark them? or do you derive them based on some model assumptions off USD pairs? based on this you should go through the USD pairs or not for consistency.

You inherently will be making a correlation assumption about the currency triangle and their corresponding vols and that would Lead to determining which side you’d be axed.

There are some other threads that discuss this:

how-should-i-convert-fx-volatility-surface-from-one-base-currency-to-another

triangular-arbitrage-in-fx-volatility

implied-volatility-of-cross-currency-pairs

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/72058

The question is, would having a position in EUR/Yen straddles help reduce risk? Consider the possible movements of EURJPY. If it doesn’t move, then having the EURJPY straddle is not affecting the overall payoff. If EURJPY does move, the move can be broken down into two separate components I.e. the move in EURUSD and the move in usdjpy. If the EURUSD move is larger, then being short the EURJPY straddle will be risk reducing. If the usdjpy move is larger, then being long the EURJPY straddle will be risk reducing. Hence I would say it depends which of these two is the more likely. If there is a large difference in implied volatility of the EURUSD and usdjpy , that will indicate which move is more likely, for example.

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