Cross-Currency Implied Volatility Requires a Correlation Assumption

Summary

The document asks whether one-year volatility for a cross currency pair such as GBP/JPY can be inferred directly from the volatilities of GBP/USD and USD/JPY. The response points to material on converting foreign exchange volatility surfaces between base currencies and on the relationship between cross-rate volatility and correlation.

Its main lesson is that the two component volatilities alone do not determine the cross rate’s volatility: an assumption or estimate for the correlation between the component exchange rates is also needed. The discussion gives no formula, numerical example, market data, or preferred way to estimate that correlation. Consequently, the result depends on the chosen correlation input and on aligning the underlying volatility measures and conventions. The short exchange identifies the modeling dependency but does not provide a complete calculation procedure.

Key ideas

• A cross currency pair’s volatility cannot be inferred from component volatilities alone.
• The correlation between the component exchange rates is a required input.
• Any inferred cross-rate volatility depends on the correlation assumption or estimate.
• The document points to related explanations but gives no calculation or empirical example.

Tags

# Implied Volatility of cross currency pairs


# Implied Volatility of cross currency pairs












Been looking for this...

Is there any way we can infer directly, say GBP-JPY's 1-year volatility from GBP-USD's and USD-JPY's?

Many thanks.

## Answer by Juan Martinez (score 1)

https://quant.stackexchange.com/a/68962

I'd say your question is answered here: How should I convert FX Volatility Surface from one base currency to another?

As they summarise, you need to make correlation assumptions. For an intuitive explanation of why this is the case, there's a nice framework for understanding the connection between volatilities and correlations in cross rates which you can find explained here: https://quantdare.com/volatilities-and-correlations-of-cross-rates-a-geometrical-understanding/

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.