Final_1.pdf

(^1) Since these trades are all done to fill the same order, we can expect the trade direc-
tion to be uniform across all executions for a given ticker.
(^2) For an example, see Umesh Vazirani, “Rapid Mixing Markov Chains,” Proceedings
of the Symposia in Applied Mathematics, Vol. 44, 1991, pp. 99–121.
against the average achieved spread calculated using the execution prices of
the two stocks. We could then use the difference as a measure of execution
efficiency. The question therefore is, which measurement approach do we
choose? We recommend going with the second method; that is, the one
based on the average execution prices. We will justify our recommendation
by demonstrating that the first idea of pairing up executions to judge effi-
cacy is a process fraught with inconsistencies. To do just that, let us go
through the exercise of pairing the executions. This exercise in addition to
proving our point also provides insight into how we may characterize ag-
gressive and conservative trading.
Let us start with the pairing process. The input to the pairing exercise is
the list of executions. Each execution is characterized by the stock ticker, the
direction of execution (long or short),^1 the fill price, and the fill quantity.
The executions may therefore be partitioned into two sets based on the stock
ticker. The idea is to pair the executions of one ticker against the executions
of the other ticker such that the spread and ratio constraints are met. We for-
mulate the trade pairing exercise as a network flow problem.
156 RISK ARBITRAGE PAIRS
HISTORY
One of the important subject areas of the latter part of the twentieth
century that has found successful application in a multitude of situa-
tions has been the study of network flows. The subject area was pio-
neered by Ford and Fulkerson. In their seminal paper published in
1956, they framed the network flow problem as an integer linear pro-
gram. Besides detailing an algorithm to solve for the optimal way to
route the flow across a network, they also provided a novel interpre-
tation of the dual of the network flow linear program, now famously
known as the max flow–min cut theorem.
Since then, network flows have found application in a wide vari-
ety of situations. Of relevance to finance, a subclass of the general
max–flow problem, known as matching, has found application in
game theory and to the theory of auctioning. Other applications of net-
work flows range from something as practical as logistic planning to
something as abstract as theorem proving,^2 making this a fascinating
subject area for study in its own right.