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Replicating a USDCHF Position with USDEUR and EURCHF

Article Quant Q&A · Author: nily
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Summary

The document asks how to reproduce the profit and loss of a USDCHF position with a stated size of 1,000 by trading USDEUR and EURCHF instead. It notes the triangular exchange-rate relationship: USDCHF is the product of USDEUR and EURCHF. The practical question is how to choose the two leg sizes so their combined exposure matches the direct position, and whether those sizes must be adjusted over time when the account currency is USD.

The document does not provide a sizing derivation, worked example, or resolution. It therefore introduces a useful cross-rate and portfolio replication problem but leaves the answer open. Any replication would depend on how the instruments define position units and value gains and losses, as well as on exchange rates changing; the prompt alone is insufficient to establish exact leg sizes or a rebalancing schedule.

Key ideas

  • The USDCHF exchange rate can be expressed as the product of USDEUR and EURCHF rates.
  • Replicating a direct FX position requires matching the profit and loss exposures of both component legs.
  • Position sizes depend on contract conventions and the account currency.
  • Changing exchange rates may affect whether fixed component sizes continue to match the direct position.
  • The document poses these questions but does not provide a solution.

Tags

ForexPosition sizingPortfolio constructionArbitrage
Full text
# Replicating PNLS of FX Portfolio


# Replicating PNLS of FX Portfolio












I need to enter a USDCHF position and the size is 1000.

However I want to replicate the same portfolio pnl by using USDEUR and EURCHF.

I know that due to triangular arbitrage USDCHF=USDEUR*EURCHF as a rate. However I couldn't figure out the sizes that can replicate the pnl that I got from USDCHF and whether it needs regular rebalancing given my account currency is USD.

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