Advances in Risk Management

172 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

nominal and the real interest rate we can produce models of different com-
plexity out of relation (9.2). Note also that we have a mean-reverting drift
with the real rate being the mean reversion level. Especially, with this geo-
metric Brownian motion based model we are able to derive option pricing
formulae of Black–Scholes type. To understand their derivation note that
our situation can be viewed as an alternative to the pricing of a foreign
exchange option. The inflation index allows us to switch between an invest-
ment in the nominal and in the real currency (see also Korn and Kruse, 2004,
for the formal argument):

Proposition 1 Under the assumptions of deterministic real and nominal

interest rates in equation (9.2) the fair price of a European call option on

the inflation indexI(T) at timetwith strike priceKand maturityTis

given by:

CI(t,I(t))=I(t) exp

(

−

∫T

t

rR(s)ds

)

N(d(t))

−Kexp

(

−

∫T

t

rN(s)ds

)

N(d(t)−σI

√

T−t) (9.3)

whereNis the cumulative distribution function of the standard normal

distribution and

d(t)=

ln

(

I(t)

K

)

+

∫T

t (rN(s)−rR(s))ds+

1

2 σ

2

I(T−t)

σI

√

T−t

(9.4)

Of course, much more natural products than call options on a consumer
price index such as an inflation index are inflation linked bonds. A typical
such example is a coupon bond with coupons protected against inflation
and the final payment of the notional being protected against inflation and
deflation, for example, it consists of payments:

Ci

I(ti)

I(t 0 )

, at timesti,i=1,...,n, (9.5)

max

{

F

I(tn)

I(t 0 )

,F

}

, at timetn=T (9.6)

Under the above assumptions of deterministic interest rates we can again
derive a closed formula for its price (Korn and Kruse, 2004):

Proposition 2 Under the assumptions of deterministic real and nominal

interest rates in equation (2.2) the fair price of an inflation-linkedT-bond at

timetwith a reference datet 0 ≤t≤t 1 , face valueFand coupon payments