360 MODEL SELECTION AND HEDGING OF FINANCIAL DERIVATIVESvolatility model but he assumes that the Brownian motions are uncorre-
lated. Interestingly, a number of papers have recently appeared where the
assumption of zero correlation is essentially justified as it makes the model
analytically tractable (Andersen and Andreasen, 2000b; Piterbarg, 2003).
Our simple example below demonstrates that, even if a model with no
correlation is made consistent with the smile, the residual hedging error
can be significant if the actual dynamics is driven by correlated Brownian
motions.
We remind the reader that from a mathematical point of view there is no
difference between the two kinds of model risk, as correctly pointed out by
Cont (2005). However, our empirical tests show that the errors originating
from model misspecification are generally much larger and then, potentially,
much more dangerous.
To fix the ideas, we specialize equation (18.1) to the following caseγ(t,
St,vt)=Sxt
√
vt,φ(t,St,vt)=a(b−vt),θ(t,St,vt)=α√
vtwithx≥0. This model
corresponds to a Heston model with CEV-type local volatility and can be
analytically handled ifρ=0 (Di Graziano and Galluccio, 2005). In terms of
its market explicative power our framework is qualitatively similar to the
celebrated SABR model (Hagan, Kumar, Lesniewski and Woodward, 2002)
but has the advantage, over SABR, of possessing a mean-reverting instan-
taneous variance process, which is beneficial in the applications (Galluccio
and Le Cam, 2005). In any case, we point out that the qualitative picture
emerging from our empirical tests is not affected by the fine details of the
model specification equation (18.1).
As anticipated, equation (18.1) will be our benchmark “model”; that is,
it will be assumed to represent the actual asset dynamics. We thus generate
(by Monte-Carlo simulation) a set of benchmark call option prices using
equation (18.1) at different strikes that are meant to approximate to a good
degree of accuracy the actual smile observed in the market (for example the
S&P index).^3
In our first study we postulate that the trader believes that BS is the “right”
model and risk-manages his hedging portfolio accordingly. He then sells at
t=0 a 1 year call struck atK=0.975 whenS=1 and enters into a standard
delta-hedging self-financing strategy. We simulate the actual path followed
by the asset from equation (18.1) and, at expiryT=1, we record the hedging
errorYt,T. In Figure 18.1 we show the empirical probability distribution of
the random variableYt,T.
Two things are worth noticing. First, the average total error is positive.
This is mainly a consequence of the particular choice of the parameters
used to generate the “true” smile. A different choice of parameters (con-
sistent with the price of the call) can lead to the opposite bias. Second, the
result suggests that if one risk manages the portfolio using a simple BS delta
hedging argument, then there is a non-negligible probability to get a
total error that is a significant fraction of the total profit from the option.