Advances in Risk Management

358 MODEL SELECTION AND HEDGING OF FINANCIAL DERIVATIVES

The main idea of the proof is the following. At any timet≤T, build a

portfolioVtconsisting of an amountψ
(1)
t of stocks,ψ

(2)
t units in the money
market account and a quantityψ
(3)
t of another optionDt(which is used
to complete the market). At inception, the trader observes a set of market
option prices and calibrates what he believes to be the right model (18.2) to
those prices. We assume, for the sake of simplicity, that all relevant option
prices are well reproduced by the model (18.2). Assume that the option price
function associated to the trader’s “wrong” model is given byCt=C(St,vt,

t). He will then choose coefficientsψ(ti)inVtso as to make the processYt:

Yt=ψ

(1)

t St+ψ

(2)

t βt+ψ

(3)

t Dt−Ct

vanish at any time whenStandvtsatisfy equation (18.2). In particular, the
amount of stocks and options held in the portfolio will be given by:

ψ

(1)

t =

∂C

∂S

−

∂C

∂v

(

∂D

∂v

)− 1

∂D

∂S

, ψ

(3)

t =

∂C

∂v

(

∂D

∂v

)− 1

(18.4)

respectively. Note that the hedging ratios involve only price sensitivities cal-
culated by using the “wrong” model. If the real asset dynamics where given
by (18.2), thenYtwould be identically zero. In general this is not true since
equation (18.1) (as opposed to equation (18.2)) describes the “true” asset
dynamics. By applying Ito’s formula toYt, by using the correct asset dynam-
ics (18.1) forStandvt, and by remembering that option prices generated with
model (18.1) are matched by model (18.2), we finally arrive at the following
expression of the total hedging error (Di Graziano and Galluccio, 2005):

Yt,T=−

∫T

t

βT−s

∂CS

∂v

(

∂DS

∂v

)− 1 [

1

2

(^2 s−γs^2 )

∂^2 Ds

∂S^2

+

1

2

(!^2 s−θ^2 s)

∂^2 Ds

∂v^2

+(ρ′s!s−ργsθs)

∂^2 Ds

∂S∂v

]

ds

+

∫T

t

βT−s

[

1

2

(^2 s−γs^2 )

∂^2 Cs

∂S^2

+

1

2

(!^2 s−θ^2 s)

∂^2 Cs

∂v^2

+(ρ′s!s−ργsθs)

∂^2 Cs

∂S∂v

]

ds (18.5)

This formula is the extension of equation (18.3) in a set-up given by equations
(18.1) and (18.2).
To shed some light on our result, we first notice that the total hedging
error is made of two contributions: one integral is associated to the option to
hedgeC, the other to the additional claimDin the replicating portfolio. Each
of them is in turn composed of three additive terms that can be identified
as follows. One term is proportional to the Gamma of the option, i.e. the