sults
In section 4.4 we have seen the expected (monetary) payoff for financial agents is given by (4.6). Furthermore we have seen this is a decreasing function in the number (N) of ILC’s the central bank is short. Hence when setting some strike K and some price p the demand N will be such that
p=e−ri(T−s)E˜(π(N)(T −s)−K(T −s))+ (4.8) holds. For the deterministic case equation (4.8) implicitly determines N as function of strike K and price p and from Corollary 4.3.6 we easily obtain
N = 1
d(T −s)max(0, aλk−K−e
ri(T−s)p). (4.9)
In the presence of a stochastic economy-wide shock there is no analytic formula available. However figure 4.2 provides an intuition - for simplicity we assumed a zero interest rate (i.e. ri = 0) and used the preference parameter b :=dN(T −s) on the
z-axis. Further for this plot (as for the remaining plots in this section) we used the following parameters: a = λ = k = 1 and T −s = 1. Since the central bank only sells these derivatives a negative value for b (that would be equivalent to a negative N) does not make sense. Therefore the function in the figure is bounded from below by zero. What is seen in the figure is that for low prices (i.e. p close to zero) b (and hence N) is very sensitive in the volatility. Intuitively this should be clear as for p = 0 and σecon > 0 the demand would be infinite. This is because in the face
of a shock there will be a chance that the derivative at maturity pays off something positive and by definition it never pays off something negative. This is consistent with standard arbitrage pricing theory (e.g. Black Scholes model) of plain vanilla options (e.g. European Call and Put) where the price is an increasing function of the volatility.
So far by equation (4.9) and Figure 4.2 we have seen that b is a function of K and p. On the other hand Lemma 4.3.5 and figure 4.1 respectively show that πe is a function of K and N (respectivelyb). Henceπe can also be considered as a function
of K and p - the actual tools the central bank holds in its hands. Again it is quite easy to find an analytical expression for the deterministic case
Corollary 4.5.1 In the absence of shocks optimal inflation π∗ and expected inflation
πe always coincide and
π∗ =πe=
(
aλk if p≥e−ri(T−s)(aλk−K)
K+eri(T−s)p if p≤e−ri(T−s)(aλk−K).
Also again in general there is no analytical expression available. However we can provide the reader with figure 4.3, mentioning the case of a standard normal shock ε.
Figure 4.2: The preference parameter b as a function of strikeK and pricep for the derivative. The upper surface is in the face of a standard normally distributed shock ε. The lower plain refers to the deterministic case.
Now we are in a position to address the final question of section 4.4 - i.e. which combination of price p and strike K minimizes ˜V1?
Again for the general case we can not answer this questions analytically. However it can be done for the less plausible deterministic scenario. First let’s consider the next little Corollary
Corollary 4.5.2 In a deterministic scenario the market price of risk must be zero - i.e. ri =µ. Hence the central bank and financial agents have the same deterministic
Figure 4.3: The expected inflation rate πe as a function of strike K and price p for
the derivative. This is done for a standard normal shock ε - For the deterministic case the picture would look similar but less smooth
measure and therefore we get:
˜ V1 = ( V2 = 12λk2+ 12(aλk)2 if p≥e−ri(T−s)(aλk−K) V2+dN(M(N)−p) = 12λk2+12 K+eri(T−s)p 2 +dN(M(N)−p) else. This give us the following (quantitative) result:
Theorem 4.5.3 In a deterministic scenario the optimal non negative price the cen-
tral bank should choose is p = −eri(T−s)K, where the strike K ∈ (−∞,0] has to be
some non positive number.
This is obvious by the previous Corollary when minimizing V2 resp. ˜V1 with respect to K and p under the constraintp≤e−ri(T−s)(aλk−K).
For the standard normal scenario the cost functions ˜V1 andV2 are shown in Figure 4.4. As in theorem 4.5.3 for the deterministic scenario we see thatV2 does not attain a single local Minimum, but so does ˜V1 for K ≈ −0.38 and p ≈ 0.44. Both curves are plotted using ri = 0 and for ˜ρ we have used −0.03.
Figure 4.4: The cost ˜V1 (upper plain) resp. V2 (lower plain) as a function of Strike K and price pfor a standard normal scenario.
From this we see that an increased volatility causes a shift of the optimal strike K to a negative number. This is due to the smoothing effect the volatility has on
the plot - i.e. V2 reaches its minimum value for less moderate values of K and p if the volatility is higher (see also Figure 4.4). To get some more intuition on how the increase of volatility shifts the optimal strike to the left see Figure 4.5.
Figure 4.5: The optimal choice of price p(green line) and Strike K (blue line). The higher the volatility the less moderate are the optimal parameters.
One last question should be the value of expected inflationπe in the ILC-trading
scenario. The following Corollary telling us it is always zero in the deterministic case does not generally hold in the face of uncertainty.
Corollary 4.5.4 In the absence of any shocks the optimal choice of p andK causes
π∗ =πe = 0 - no matter what the other parameters in V2 were!
Proof:
Just combine Corollary 4.5.1 and Theorem 4.5.3.
In the face of a shock the above Corollary does not hold true any more. Choosing the same parameters as in our plot of Figure 4.4 numerical results show that for optimal p and K the expected inflation πe ≈0.06. This is still above zero but much
lower than πe= 1 as rational expectation would indicate without trading the ILCs.