Solvency cone

The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone of portfolios that can be exchanged to portfolios of non-negative components (including paying of any transaction costs).

If given a bid-ask matrix ${\displaystyle \Pi }$ for ${\displaystyle d}$ assets such that ${\displaystyle \Pi =\left(\pi ^{ij}\right)_{1\leq i,j\leq d}}$ and ${\displaystyle m\leq d}$ is the number of assets which with any non-negative quantity of them can be "discarded" (traditionally ${\displaystyle m=d}$), then the solvency cone ${\displaystyle K(\Pi )\subset \mathbb {R} ^{d}}$ is the convex cone spanned by the unit vectors ${\displaystyle e^{i},1\leq i\leq m}$ and the vectors ${\displaystyle \pi ^{ij}e^{i}-e^{j},1\leq i,j\leq d}$.

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