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Log-Return Variance of a Synthetic Currency Cross

Article Quant Q&A · Author: Hiperfly
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Summary

The document explains how covariance between two currency pairs contributes to the variance of their synthetic cross rate. For EUR/USD divided by GBP/USD, the resulting cross is EUR/GBP. Taking logarithms turns the ratio into a difference, so the variance of the cross’s log return equals the sum of the two pairs’ log-return variances minus twice their log-return covariance.

This identity shows that positive covariance reduces the cross’s variance, while negative covariance increases it, all else equal. The answer gives the algebraic relationship but does not provide market data, a numerical example, or a method for estimating the inputs. It applies directly to log returns; it should not be read as an exact formula for the variance of the untransformed price ratio.

Key ideas

  • The logarithm of a price ratio is the difference between the logarithms of its component prices.
  • The variance of the synthetic cross’s log return includes both component variances and their covariance.
  • Positive covariance lowers the cross’s log-return variance relative to the sum of component variances.
  • The stated identity concerns log returns rather than the raw price ratio.

Tags

ForexStatisticsVolatility
Full text
# Is there any relationship between the Covariance(A, B) and the variance of the synthetic asset A/B?


# Is there any relationship between the Covariance(A, B) and the variance of the synthetic asset A/B?












Let's say we have 2 pairs of currencies: EUR/USD and GBP/USD. The cross-asset (or synthetic asset) would be (EUR/USD) / (GBP/USD) = EUR/GBP.

Is there any relationship between the covariance(EUR/USD, GBP/USD) and the variance of EUR/GBP?

## Answer by dm63 (score 3)

https://quant.stackexchange.com/a/76416

We have $$\ln(A/B)= \ln A - \ln B.$$ Hence $$\textrm{Var}(\ln(A/B))= \textrm{Var}(\ln A) + \textrm{Var}(\ln B) - 2\textrm{Cov}(\ln A,\ln B).$$

Hope that helps.

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