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Formulating Currency-Pair Exposure as a Constrained Linear System

Article Quant Q&A · Author: nily
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Summary

The document asks how to express a target set of net currency positions using the fewest foreign-exchange pairs while also limiting the absolute size of positions. It illustrates that different combinations of currency trades can produce the same net exposures, while one example uses more trades and capital than another. The practical problem is therefore to map pair positions into currency-level exposures and choose among equivalent solutions.

The response recommends forming a transformation matrix from pair positions to currency positions and solving the resulting linear equations. It notes that the matrix is poorly conditioned and has an eigenvalue close to zero, leaving one degree of freedom in the solution. An additional constraint, such as minimizing total trading position, can select a particular solution. This is a starting point rather than a full optimization recipe: the document does not specify an objective function, execution costs, constraints on trade direction, or a method for balancing few pairs against small exposure size.

Key ideas

  • FX pair positions can be mapped to net currency exposures with a transformation matrix.
  • Different combinations of pair trades can generate the same target currency positions.
  • A nearly singular matrix can leave a degree of freedom among valid solutions.
  • An added constraint, such as minimizing trading position, can choose among equivalent solutions.
  • The discussion does not define a complete optimization objective or account for execution costs.

Tags

ForexStatisticsPosition sizing
Full text
# Achieving desired fx exposure with using minimum pairs possible


# Achieving desired fx exposure with using minimum pairs possible












Let say my algorithm tells me to get the following positions through opening fx positions:

CUR NET POSITIONS

GBP 236.96379

USD -310.58000

CHF 0.02000

There are 2 ways to achieve this:

- Long 1000 GBP/USD, Long 1000 USD/CHF and Long 1000 CHF/GBP given the rates are 1.310580(GBPUSD),0.999980(USDCHF) and 0.763036(CHFGBP)

- Long 236.96379 GBP/USD and Short 0.02 USDCHF. same rates.

So I replicated the same pl but the first option uses more capital and positions meanwhile the second one is optimal.

I want to develop an optimization that tries to satisfy my required currency positions by using as little forex pairs as possible and minimize the absolute value of exposure as well. I read that Bellman-Ford equations can be helpful in finding the shortest possible way but most of the examples try to find a triangular arbitrage instead of the optimization I am after. Are there any examples out there that I can use or any resource, an idea will be helpful.

## Answer by XYQ (score 0, accepted)

https://quant.stackexchange.com/a/44946

if you form a transform matrix from the input to the output currency position. This will be a problem to solve linear equations。the matrix condition number is very big. actually one eigen value is almost zero. so you solution has one degree of freedom. you can add additonal constrain to solve it like minimal trading position

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