On the Dual of the Solvency Cone

TL;DR

A characterization of its dual cone in terms of extreme directions and some consequences are discussed, among them: an algorithm to construct extreme directions of the dual cone when a corresponding "contribution scheme" is given.

Abstract

A solvency cone is a polyhedral convex cone which is used in Mathematical Finance to model proportional transaction costs. It consists of those portfolios which can be traded into nonnegative positions. In this note, we provide a characterization of its dual cone in terms of extreme directions and discuss some consequences, among them: (i) an algorithm to construct extreme directions of the dual cone when a corresponding "contribution scheme" is given; (ii) estimates for the number of extreme directions; (iii) an explicit representation of the dual cone for special cases. The validation of the algorithm is based on the following easy-to-state but difficult-to-solve result on bipartite graphs: Running over all spanning trees of a bipartite graph, the number of left degree sequences equals the number of right degree sequences.

Figures (1)

• Figure 1: Left: The digraph $G(P,N)$ for $P= \left\{ 1,4 \right\}$ and $N= \left\{ 2,3,5 \right\}$, where the arcs $ij \in P \times N$ are associated with market prices $\pi_{ij}$. Right: A spanning tree $T$ of $G(P,N)$ and a vector $y=(y_1,y_2,y_3,y_4,y_5)^\top$ generated by $T$.

Theorems & Definitions (39)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark 3
• Proposition 4
• proof
• Example 5
• Theorem 6
• proof