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Why FX Cross-Rate Correlations Cannot Be Recovered from Pair Statistics Alone

Article Quant Q&A · Author: locke84
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Summary

The document raises whether the volatility and correlation statistics for USD/JPY and GBP/USD are enough to calculate the correlation of the derived GBP/JPY cross rate, given that GBP/JPY equals GBP/USD multiplied by USD/JPY. It also asks whether a variance-style cosine rule applies to historical observations, noting that an attempted calculation did not match the time-series result.

No answer or derivation is included, so the document does not establish a formula or explain the source of the discrepancy. Its useful takeaway is the distinction between a cross-rate identity and the statistics needed to characterize its returns: multiplying exchange rates does not by itself provide all return distribution information. A reliable calculation would require careful definitions of returns, volatility, and correlations, but those steps are not supplied here. The question therefore serves as a prompt for statistical investigation rather than a complete method.

Key ideas

  • GBP/JPY is the product of GBP/USD and USD/JPY exchange rates.
  • The document asks whether historical pair statistics suffice to infer the cross-rate correlation.
  • It reports that a cosine-rule-style calculation did not match the historical time-series result.
  • No response is provided, so the document gives no validated formula or derivation.

Tags

ForexStatistics
Full text
# Can you derive fx correlations given the historic correlations of base pairs


# Can you derive fx correlations given the historic correlations of base pairs












Say for USDJPY and GBPUSD, I have the historic data and calc the volatility and correlation between these pairs.

Because GBPJPY = GBPUSD x USDJPY. Can I calculate the correlation for GBPJPY directly from these statistics, without having to calculate from the historic time series data?

I am not sure if its valid to use cosine rule, eg: Vol C ^ 2 = Vol A ^ 2 + Vol B ^ 2 - 2 * rho (A, B) * Vol A * Vol B Or if this is just for implied correlations.

As I tried it for some historic data and it didn't work.

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