X dominates the world

For more than a decade, collateral discounting as been the de facto standard for derivative valuation. The central technical concept of this approach is the expectation under a currency — but not collateral mechanism — dependent measure.

That measure is denoted

in part of the literature. This is the notation I used in my Multi-Curve Framework with Collateral paper in 2023. This is the paper that developed the details of the “collateral square” valuation, when the collateral is is an asset that can itself be used into a repo to obtain cash. The generic formula in that latter case is (originally described in formula 13 in the above paper).

The measure X appears in the expectation and is central to the approach, based on replication.

It mathematics, it is traditional to add some marker (like a tilde or a bar) on some symbols to indicate another object similar or generated from the first one. One symbol that I like is the "check", which I called when I was lecturing in french "oiseau" for it resemblance to a flying bird. I will try in one of my forthcoming papers to use the following symbol for a measure:

This is a measure that will be called X-blue-bird. Obviously this is not related in any way to the recent social media related news!

Reference:

Henrard, Marc. Multi-Curve Framework with Collateral (July 21, 2013). OpenGamma Quantitative Research, n.13, July 2013, Available at SSRN: https://ssrn.com/abstract=2302278.

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