(^1) The reasoning stems from the maximization of a linear utility function resulting in
a linear program with constraints. The weights happen to be the values of the dual
variables in the solution of the linear program. This work by Arrow and Debreu was
awarded the Nobel Prize in Economics.
reserve capital requirements for a bank. The requirements for reserve capi-
tal are designed to help markets survive extreme conditions by making sure
(to some extent) that the counterparties in a trade have enough reserve cap-
ital to meet their obligations. Ideally, we would like the reserve capital to be
the proverbial Goldilocks value, not too little and not too much. If the risk
measurement is too conservative, then we will need to post more reserves,
leading to underutilization of capital. If it is too aggressive, then we may not
have enough reserves to meet obligations during extreme moves. We will
propose a practical value at risk measurement approach for risk arbitrage
deals, based on the merger probabilities.
The chapter is organized as follows. First, we discuss briefly the Arrow-
Debreu theory, which forms the basis for our probability measure. We then
describe the single-step model for measuring merger probability success. The
single-step model is then extended to multiple steps. Subsequently, we rec-
oncile between the proposed theory and practice. This is followed by an ex-
ample application to risk management.
Implied Probabilities and Arrow-Debreu Theory
The purpose of this section is not so much to provide a formal description
of the Arrow-Debreu theory as much as to provide a flavor for it. Let us con-
sider the scenario that involves placing bets on a set of outcomes. Examples
of such events could be a boxing match or a horse race. In these cases, the
set of outcomes is finite and well defined. We will use the horse race exam-
ple for purposes of illustration. Important to the discussion is the notion of
betting. If, for example, the bet is placed in favor of a horse and it wins the
race, then the reward is the payoff from the bet. If it happens to lose, then
here, too, the reward is the payoff from the bet, except that the payoff is
probably zero dollars. Thus, a bet is completely defined when we specify the
payoff for every possible outcome. To place a bet, one has to put up the
stake money. This is specified by the bookie.
The Arrow-Debreu theory states that the full and complete specification
of bets with the stake money and the payoff for each outcome automatically
implies a probability for a particular outcome.^1 Additionally, the stake
The Market Implied Merger Probability 173