Advances in Risk Management

188 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

9.4.3 Numerical examples

We are now illuminating the gains for our hedging problem by using
inflation-linked products via some numerical examples. Note that to per-
form our computations we need the following explicit forms of the hedging
errors in the above propositions:

The quadratic hedging errorH 1 corresponding toB* (trading in bond,
stock, inflation) is given by:

H 1 =Var(B)I(0)^2 exp((2(rN−rR+ν)+σ^221 +σ 222 )T)

+

(I(0)E(B)−x)^2

exp((θ′θ− 2 rN)T)

withθ=







μ−rN

σ 11

ν−rR

σ 22

−

μ−rN

σ 11

·

σ 21

σ 22







The quadratic hedging errorH 2 corresponding toBˆ* (“trading in bond
and stock”) is given by:

H 2 =Var(B)I(0)^2 exp((2(rN−rR+ν)+σ 212 +σ^222 )T)

+I(0)^2 exp((2(rN−rR+ν)+σ 212 )T)(E(B))^2 (exp(σ 222 T)−1)

+

(I(0)E(B) exp((−rR+ν−θσ 21 )T)−x)^2

exp((θ^2 − 2 rN)T)

withθ=

μ−rN

σ 11

It is now easiest to see the hedging effect of using inflation products by
considering a deterministicBwhich we therefore assume to equal 1. We will
further choose:

μ= 0 .1, σ 11 = 0 .3, rN= 0 .04, ν= 0 .01, σ 22 = 0. 04 I(0)= 100

and varyx,σ 21 ,rRand report the corresponding hedging errors in Table 9.1
where we chooserR=0.03 (panel A) andrR=0.05 (panel B) respectively.
What can be clearly seen is that with tradable inflation the hedging error
is smaller than if we can only use the stock for hedging inflation. The error
increases with decreasing covariance between inflation and the stock. Thus,
we have demonstrated that there are indeed investors who have advantages