Advances in Risk Management

50 SENSITIVITY ANALYSIS OF PORTFOLIO VOLATILITY

3.2 SENSITIVITY ANALYSIS BACKGROUND

Recently, the activity in the scientific field of SA of Model Output has been

steadily growing, due to the increasing complexity of numerical models,

whereby SA has acquired a key role in testing the correctness and corrob-

orating the robustness of models in several disciplines. This has led to the

development and application of several new SAtechniques (Borgonovo and

Apostolakis, 2001a; Saltelli, 1997; Saltelli, 1999; Saltelli, Tarantola and Chan,

1999; Turany and Rabitz, 2000). Most of the recent literature in portfolio man-

agement has proposed SA approaches based on PDs (Drudi, Generale and

Majnoni, 1997; Gourieroux, Laurent and Scaillet, 2000; Manganelli, 2004;

McNeal and Frey, 2000). In the next paragraphs we present the Differen-

tial Importance Measure (D) and discuss in detail its relation to PDs and

Elasticity.

Let us consider the generic model output:

Y=f(x) (3.1)

wherex={xi,i=1, 2,...,n}is the set of the input parameters. Let also

dx=[dx 1 ,dx 2 ,...,dxn]T

denote the vector of changes.

Iff(x) is differentiable, then the differential importance ofxsatx^0 is

defined as (Borgonovo and Peccati, 2004):

Ds(x^0 ,dx)=

dfs(x^0 )

df(x^0 )

=

fs(x^0 )dxs

∑n

j= 1 fj(x

(^0) )dxj (3.2)
Dcan be interpreted as the ratio of the (infinitesimal) change inYcaused
by a change inxsand the total change inYcaused by a change in all
the parameters. Thus, Dis the normalized change inYprovoked by a
change in parameterxs. It can be shown that (Borgonovo and Apostolakis,
2001a; Borgonovo and Apostolakis, 2001b; Borgonovo and Peccati, 2004;
Borgonovo and Peccati, 2005):
A Dshares the additivity property with respect to the various inputs, for
example, the impact of the change in some set of parameters coincides
with the sum of the individual parameter impacts. More formally, let
S⊆{1, 2,...,n}identify some subset of interest of the input set. We have:
DS(x^0 ,dx)=
∑
s∈Sfs(x
(^0) )dxs
∑n
j= 1 fj(x
(^0) )dxj=
∑
S∈S
Ds(x^0 ,dx) (3.3)