How FX Currency Triangles Relate Exchange Rate Volatility

Summary

The document asks whether the volatility of a cross rate such as EUR/GBP can be derived from the volatilities of GBP/USD and EUR/USD, together with their correlation. It starts from the spot-rate identity that expresses one leg of a currency triangle as the ratio of the other two, then asks whether a comparable relationship or useful bounds apply to volatility. The question raises the importance of the dependence between the two dollar pairs when assessing the cross-rate risk.

No derivation, formula, numerical example, or empirical evidence is included in the document. The accepted response points readers toward a discussion of currency-trio volatility in a book and a research paper, but does not summarize their results. As provided, this is a research prompt rather than a complete method: readers need an external derivation to learn the exact relationship and its assumptions. It does identify a useful modeling problem in FX risk analysis, but offers no stated volatility bounds or trading strategy.

Key ideas

• A cross exchange rate can be expressed as a ratio of two rates quoted against a shared currency.
• The document asks how the volatility of that cross rate depends on the component volatilities and their correlation.
• The response refers readers to external material but gives no volatility formula, bounds, or worked example.

Tags

# Triangular Arbitrage In FX Volatility


# Triangular Arbitrage In FX Volatility












If I know the price of $GBPUSD$ and $EURUSD$, I can retrive the $EURGBP$ price simple by $EURGBP = \frac{GBPUSD}{EURUSD}$.

Is there something equivalent to FX Volatility? Knowing the $\sigma_{GBPUSD}$, $\sigma_{EURUSD}$ and myabe $\rho[GBPUSD,EURUSD]$, is there a direct relationship or bounds to the value of $\sigma_{EURGBP}$?

I tried to think in terms of generic numeraie, like $USDX$ and $GBPX$, and take expectaions of the ratios, but I found no solution.

Any idea or clue would be helpful.

## Answer by user42108 (score 2, accepted)

https://quant.stackexchange.com/a/68826

This is covered in the book "FX Derivatives Trader School" (and in other places, too, e.g. "The shape of things in a currency trio", a paper by Walter and Lopez, CS and FRB SF link).

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.