GIUSEPPE DI GRAZIANO AND STEFANO GALLUCCIO 355“information” to uniquely identify the model and its associated risk-neutral
pricing measure. However, the market prices of European vanilla options
do not contain any information about the conditional distributions of the
underlying stochastic process. As the above argument suggests, unique
model identification is an impossible task to achieve and for any model
choice there exists an associatedmodel risk. Although many different defini-
tions of model risk are possible (see Cont, 2005, for a review) here we refer
to model risk as the residual uncertainty on the price of financial derivatives
and on their risk-management once thatallrelevant market information has
been properly included in the pricing model (through model calibration).
In this chapter we examine the impact of model choice on the hedging of
financial derivatives in a simplified set-up. In particular, we will assume that
the true dynamics of the underlying asset (the stock) follows a given stochas-
tic volatility process while market agents, lacking this information, trade
according to a different model. In order to guarantee market consistency,
we will further require that all available market information be captured
by the “wrong” model. This will be achieved by calibrating the model to
all option market prices, which we generate using the “right” model. We
shall address the hedging error problem both analytically and numerically
and show that using the “wrong” model for risk management purposes can
generate significant replication errors even for short maturity options and
despite the fact that the market smile is almost perfectly matched by trader’s
model. Needless to say, we do not expect to provide a good representation
of real world asset dynamics; our study is meant to gather some intuition
about model errors in risk management. In particular, our results suggest
that model selection must take into account historical information to obtain
a proper representation of the real asset dynamics.
The rest of the chapter is organized as follows. Section 18.2 introduces the
model and the notation, while section 18.3 is devoted to the computation of
the total hedging error in models with stochastic volatility. Numerical tests
are performed in section 18.4. Finally, section 18.5 draws some conclusions
and perspectives for future research.
18.2 Model and Mathematical setup
We shall assume that the market consists of a single (non-dividend-paying)
underlying assetStwhoseP-dynamics are driven by a two dimensional
Brownian motionB=(B^1 ,B^2 ) defined on a probability space (,F,P), where
F=(Ft)t≥ 0 is the natural filtration ofBandPthe “physical” probability mea-
sure. In addition, we postulate the existence of a money market account
βt=exp(rt), whererindicates the constant riskless interest rate.^2 Stochastic
volatility models, like the ones considered in this paper, are intrinsically
incomplete. It is however possible to “complete” the market by adding