TARAS BELETSKI AND RALF KORN 1790 5 10 15
t20 25 302202001801601401201008060I (t)Figure 9.1Simulated path of an inflation index as a
geometric Brownian motioni=1, 2, ... , 30, as well as the reference inflation index oft 0 =100. The opti-
mizationisdonefortheHARAutilityfunctionwithγ=0.5. Thepathsshown
in Figure 9.2 of course correspond to the simulated path of the inflation index
of Figure 9.1.
As already stated, the figure illustrates the fact if an inflation-linked
bond without deflation protection is used the optimal portfolio process for
problem (P) is constant. For the other two inflation-bonds that are deflation-
protected at par one can observe that the optimal portfolio of the zero
inflation-bond is always higher than the one of the inflation-bond contain-
ing coupon payments, which are not deflation-protected, due to the fact that
the zero inflation-bond’s structure is totally option-like, where as including
coupon payments creates a mixture of the inflation index (stock) and an
option-like inflation-bond.
Further, the opposite movements of the inflation index and the optimal
portfolio processes of inflation-linked bonds with deflation protection can be
detected. The optimal pure fraction of the bond and inflation index (taking
over the role of the stock) in the portfolio is given by the solution of the basic
problem (P) and coincides with the optimal portfolio of the inflation-linked
bond without deflation protection. An inflation-linked bond with deflation
protection can be seen as the combination of the inflation index (replicating
strategy (9.20)) and the conventional bond (the remaining part of (9.5)), for
example, can be replicated by the inflation index and this conventional bond.