GIUSEPPE DI GRAZIANO AND STEFANO GALLUCCIO 359second derivative of the claim respect to the asset,∂
(^2) Cs
∂S^2. This is analogous
to the term in equation (18.3) but, differently from that simplified case, our
result shows that the actual Gamma-induced error depends on the difference
of the squares of the volatility functions(t,S,v) andγ(t,S,v). In particu-
lar, different functional assumptions result in completely different hedging
errors. A second term is proportional to the option’s “Volga”, that is to the
second derivative with respect to volatility,∂
(^2) Cs
∂v^2. Finally, the third term is
proportional to the option’s “Vanna”, that is the mixed second derivative
with respect to asset and volatility,∂
(^2) Cs
∂S∂v. Each of these terms multiplies a
factor that, roughly speaking, depends on the difference between “true”
and “false” model. Thus, any such term can take both positive and negative
values. This last observation is crucial in the applications (section 18.4) since
it clearly indicates that if the characteristics,,!of the wrong model are
biased respect to those of the real oneγ,φ,θthen the total hedging error
assumes largely positive or negative values.
As an important remark, we notice that the first integrand in equation
(18.5) is proportional to the ratio of the option’s Vega∂∂vCs
(
∂Ds
∂v
)− 1
. Interest-
ingly, this provides a useful hint on how to optimally selectD: one should
better choose a claim whose Vega is close to that of the option to hedgeCso
thatYt,Tin equation (18.5) is not systematically biased.
As shown in Schoutens, Simons and Tistaert (2003) and in section 18.4,
the fact that the model is made consistent with the relevant set of market
option prices is not enough to identify the characteristics of the process
and therefore a residual hedging error is always present. In the next sec-
tion we perform (based on equation (18.5)) a number of empirical tests to
quantitatively assess the replication error in two benchmark scenarios.
18.4 Numerical results
In order to give a quantitative estimate of the hedging error in practical
situations we perform two independent empirical tests. In the first we aim
at measuring the impact ofmodel misspecificationon the replicating strategy,
while the latter is aimed at estimating the impact ofparameters misspecifi-
cation. We say that a model is misspecified if the functional form of the
characteristics,,!is different from that of the true setγ,φ,θ. A typical
example is that of a trader that uses the BS model to hedge a contingent
claim written on an asset that follows a stochastic volatility process. Simi-
larly, we say that model parameters are misspecified if the functional form
of the characteristics,,!is correct but the numerical coefficients are
different from the corresponding ones inγ,φ,θ. The latter situation arises
when, for instance, the trader has the right intuition and uses a stochastic