In section 4.2 we have seen how financial agents would price such ILCs on the one hand and how the central bank would determine the present monetary value of ILCs according to its subjective measure on the other hand. Financial industry participants would like to buy ILCs as long as they consider the price as fair - i.e. the price is equal to the discounted expected payoff of the derivative under the agent’s risk neutral measure. In the previous section we have seen that πe increases in K and decreases in N. Further the bank’s incentive to deliver a more moderate inflation decreases in K and increases inN. That is true if the economy-wide shock is sufficiently negative whilst there is no incentive to deviate from the bank’s behavior if the shock is highly positive. Hence intuitively the expected payoff ˜E(π(N)(T −s)−K(T −s))+ is also decreasing in K and N. This means when the central bank sets a certain strike K and a certain price p for the ILCs on the financial market there will be demand for ILCs until N is such that the price equals the discounted expected payoff. Then from the financial market side an equilibrium is attained. Such demand curves are given by figure 4.2.
However we have seen that the central bank always expects a monetary loss when it sells ILC’s on the market for an (in financial terms) appropriate price. This price cannot compensate the central bank for her (subjectively) expected monetary loss due to the obligation at maturity to pay the holders of the ILC’s. Hence there seems to be no incentive for the bank to sell ILC’s. The key is the credibility the bank obtains when issuing ILC’s. Previously we have seen that private agents will choose a smaller πe when they know the bank’s obligation at maturity. This then gives the
lower cost coming along from the term π2 inV
2. Hence
Theorem 4.4.1 The central bank faces an incentive to trade the ILC since this in- creases its credibility as a conservative central bank. In particular we see
selling ILCs⇒N ↑⇒πe↓⇒The loss function V2 ↓.
In that sense the parameterdis more than just a trade off parameter between economic cost and monetary payoff aversion. As it also determines the expected monetary loss for the central bank dimplicitly evaluates the value of credibility to the bank. However
d has to be a reasonable (small, but greater zero) number.
This gain in the V2-part of V1 has to be traded off with the (expected) monetary loss described above. In fact, when making the pricing decision and setting the strike K the central bank faces the loss function
˜ V1 := 1 2λE(a(π(N, K)−π e(N, K)) +ε−k)2+ 1 2Eπ(N, K) 2+dN(M(N)−p(N)) =1 2λE(a(π(N, K)−π e(N, K)) +ε−k)2+ 1 2Eπ(N, K) 2 +dN e−ri(T−s) E(π(N, K)(T −s)−K(T −s))+−E˜(π(N, K)(T −s)−K(T −s))+.
The actual parameters given to the central bank are the price p and the strike K. Each combination results in a specific demanded number N. Therefore what is left to the bank are N ILC’s, each with strike K. Hence the optimal inflation rate is a function of N and K. This justifies the notation π(N, K) from above.
Now we are in a position to set up the whole game of our model in detail:
• There are three different players
– Financial market agents
– Private agents
• In the beginning there isnoinformation asymmetry - i.e. each parameter in the bank’s loss function is known by the bank and also by the other two players. Also no one knows the outcome of the shock ε, however everyone knows its distribution.
• The bank sets the strike K and the price for the derivative optimally - it opti- mizes ˜V1.
• The number N of the sold ILC’s affects the central bank’s preference via the linear factor d.
• Since financial agents know that private agents form their expectation ratio- nally they expect πe to be such that (4.3) holds. The solution to this is a
function of N andd. Thisπe(N) then determines the expected terminal payoff
E(π∗(N)(T −s)−K(T −s))+. As already seen this payoff then is a decreas- ing function of N. Therefore market participants will choose N such that p=e−ri(T−s)E˜(π∗(N)(T −s)−K(T −s))+ holds. Here pis the price set by the central bank and ˜E is the expectation formed with respect to the risk neutral measure ˜P.
• Then after trading private agents form their rational expectation πe as antici- pated by financial agents. Further the central bank observes the actual shock ε and delivers the actual optimal inflation π∗ (with Eπ∗ =πe). Financial agents
then get their transfer payment, which under risk neutral expectation equals the price they have payed in the beginning.
In fact from the diagram below we can see: At first the central bank makes a move (i.e. to set price pand strike K), then market participants make the next move (i.e. to buy N ILC’s). After that private agents make a move (i.e. they choose and announce πe) and last but not least the central bank makes her second move (i.e.
to observe the shock and deliver π∗ accordingly). So once the first move has been done any further move by any of the players such as the outcome of the game is predetermined up to the actual value of the shock. Hence the whole problem reduces to one question: What is the optimal first move (i.e. setting price p and strike K) for the central bank?
central bank sets price pand strike K
? 9 XX XX XXXz H H H H H H H j 6
financial agents buy N ILC’s to match an equilibrium
monetary obligation for the central bank
decrease in private agents’ expectation
central bank chooses actual inflation rate
π∗
economy shock ε observed by the central bank
To avoid any confusion for the reader we should definitely emphasis the following remark:
Remark 4.4.2 In our argument we said the central bank sets the strike K and the price p to minimize its cost function V˜1. In particular this means the price set by the
bank is not due to a utility indifference argument. The latter would mean the bank lets financial agents participate on the gain coming along with the higher credibility as a conservative central bank. However there is no reason why the central bank should let financial agents participate. Furthermore, as we should see later, the surprising story of this chapter is that the central bank can increaseits over all utility by issuing these ILC’s. This is in contrast to utility indifference pricing, where the over all utility is unchanged. In fact, this explains why the central bank would be extremely keen to sell ILC’s, as it is utility improving.