Advances in Risk Management

GIUSEPPE DI GRAZIANO AND STEFANO GALLUCCIO 363

We immediately notice that large errors are now less likely when compared
to the BS delta hedging strategy. In addition, the largest contribution to the
total error comes from the two Vanna terms since they are the most affected
by the wrong assumption on correlation.
As a second example, we consider the situation whereD, likeC,isan
option struck atK=0.975 but it expires one year later, i.e.T=2. Results
are shown in Figure 18.3. We now see that the largest contribution to the
total error comes from the Gamma terms since the two options have very
different Gamma in this case. However, all errors are still much smaller than
in the simple BS case.

CONCLUSION

In this chapter we have examined the errors arising from hedging a con-
tingent claim written on a tradable asset that follows a generic stochastic
volatility model when traders have a bad assessment of the real dynamics.
We provide a general formula for the total hedging error extending some
knownresultstothecaseofstatevariablesdrivenbystochasticvolatilitypro-
cesses. We have numerically shown that errors due to a bad representation of
the whole dynamics are significantly larger than those arising from just a bad
estimation of model parameters, in general. However, even if the trader uses
a model that is formally equivalent to the true one, errors due to parameters
misspecification can still be quite large. This in particular should generate
some concern when hedging is performed with a model that assumes no
correlation between Wiener noises, a framework that has recently gained
some favor in the market due to its mathematical tractability.

NOTES

1. It must be noticed, however, that in general one needs to artificially assume strong
   time-dependency of model parameters to achieve a good model “calibration”, i.e. to
   ensure that market vs. model errors are within the bid–ask spread (Galluccio and Le
   Cam, 2005). Despite this, at least theoretically it is possible to well-approximate any
   smile shape by properly adjusting the postulated dynamics if coefficients are allowed
   to take arbitrary values.


2. We will assume throughout the paper that interest rates are deterministic. Extending
   the present approach to include the effect of stochastic rates is possible but results are
   essentially unaffected by this choice in the range of options expiry we consider.


3. We recall that we do not address here the question of whether equation (18.1) is the
   correct representation of reality, thus we do not need to calibrate our “benchmark”
   model to the S&P market. Instead,we assumethat equation (18.1) is the true model and
   study the replication error induced by taking equation (18.2) as a good approximation
   of the market, described by equation (18.1).