Advances in Risk Management

48 SENSITIVITY ANALYSIS OF PORTFOLIO VOLATILITY

Such models are usually categorized in the literature as Stochastic Volatility
models^1 and autoregressive models, namely ARCH, GARCH and their gen-
eralizations (Bollerslev, 1986; Bollerslev and Engle, 1993; Engle, 1982). The
rapid development of the computation technology has enabled the utiliza-
tion of increasingly complex models. At the same time, some recent studies
have shown that, as the use of these models becomes widespread, it is felt
the need for the development of appropriate SA techniques capable of pro-
viding analysts with tools that fully exploit the information embedded in the
models (Drudi, Generale and Majnoni, 1997; Manganelli, 2004; Manganelli,
Ceci and Vecchiato, 2002; McNeal and Frey, 2000; Saltelli, 2003).
In a recent paper, Saltelli (2003) demonstrates how SA can be thought of
as an essential ingredient in portfolio management. McNeal and Frey (2000)
and Gourieroux, Laurent and Scaillet (2000) use partial derivatives (PD) to
study the sensitivity of the Value at Risk (VaR) models. These authors derive
analytically the expressions for the first and second derivatives of the VaR,
and explain how they can be used to simplify statistical inference and to
perform a local analysis of the VaR. A similar application of this technique
can be found in Drudi, Generale and Majnoni (1997), where the sensitivity of
risk assessment is tested with respect to the number of factors employed, the
measures of volatility (conditional versus unconditional) and correlations
(stable versus unstable), and the linearization of non-linear payoffs.
Manganelli, Ceci and Vecchiato (2002) propose a tool based on the cal-
culation of the PDs ofσpestimated via the GARCH model to help “risk
managers to find out what the major sources of risk are, or allow them to
evaluate the impact on the portfolio variance of a certain transaction.” In a
more recent paper by Manganelli the implications of the approach in asset
allocation are discussed (Manganelli, 2004).
Recent studies in the SAliterature have highlighted that PD-based SAsuf-
fers of several limitations when used for parameter impact evaluation and
risk management purposes (Borgonovo and Apostolakis, 2001a; Borgonovo
and Apostolakis, 2001b; Borgonovo and Peccati, 2004; Borgonovo and Pec-
cati, 2005; Cheok, ParryandSherry, 1998). Moreprecisely, thesestudiesshow
that utilizing a PD-based SA to evaluate the impact of parameter changes
with respect to the generic model output:

1 is equivalent to neglecting the relative parameter changes, or, equiv-

alently, to impose that all the parameters are varied in the same

way;

2 does not allow the appreciation of the model sensitivity to changes in

groups of parameters.

One could think of replacing the pure PD with the parameter Elasticity
(E) (Simon and Blume, 1994). In this case Limitation 2 would still be in place,