As a necessary preprocessing step, we evaluate the equivalent target
share quantities for the fill quantities on the bidder side trades. This is ac-
complished by dividing the bidder fill quantities by the exchange ratio. These
are the share quantities that will be used in the modeling exercise. The net-
work flow problem can now be represented in a pictorial fashion.
Every execution is represented as a node. We pick a node from each of
the two sets (bidder trades and target trades) and determine if the fill prices
on the two executions satisfy the spread constraint. If satisfied, we draw an
edge or line between the two nodes or executions and assign a weight or ca-
pacity to it based on the number of shares that can be matched along that
edge. Thus, each edge represents a potential way to pair the executions. We
wish to match as many shares of the bidder as we possibly can with the tar-
get shares. The edges and the nodes form a network called a bipartite graph
in graph-theory parlance, and the pairing problem is equivalent to maxi-
mizing the flow on this bipartite network. The idea becomes clearer with the
following example:
Example
Consider an order specification with the following data:
Exchange ratio = 1.0
Cash amount = 0.0
Spread value specified = $1.00
Action = put onNow, since we are putting on a spread, we sell the bidder and buy the tar-
get. The spread constraint is therefore met when the calculated spread is
greater than the specified value of $1. The executions list for the bidder and
target stocks when putting on the spread position is listed in Table 10.1.
The pairing problem can now be posed as a max-flow problem on the
network shown in Figure 10.1.
Trade Execution 157
TABLE 10.1 Execution List.
Bidder List Target List
Ratio
Adjusted
Label Quantity Fill Price Quantity Label Quantity Fill Price
B1 100 19.0 100 T1 50 19.0
B2 100 19.5 100 T2 150 18.5
B3 100 20.0 100 T3 150 18.0