Advances in Risk Management

HAYETTE GATFAOUI 111

dIt

It

=μI(t,It)dt+σI(t,It)dWIt (6.2)

where functionalsμX(t,Xt),σX(t,Xt),μI(t,It) andσI(t,It) are continuousFt-
measurable functions on [0,T]×R. To ensure strong solutions to previous
SDEs, we assume that these functionals are also bounded (Karatzas and
Shreve, 1991). For this purpose, we set whatevert∈[0,T] andXt,It∈R:
μlX<μX(t,Xt)<μuX,σlX<σX(t,Xt)<σXu,μlI<μI(t,It)<μuI,σlI<σI(t,It)<σuI
withμlX,μuX,σlX>0,σXu,μlI,μuI,σIl>0,σIuconstant values. As introduced in
Gatfaoui (2003), the dependence of firm assets valuevis-à-visthe two risk
factors is as follows:

Vt=XβtIt (6.3)

whereβis the beta of firm assets value (for example, a constant estimate over
our time horizon) as defined by the CAPM. Recall thatXrepresents market
conditions as well as business cycle, andIrepresents firm-specific features
such as default and liquidity risk. Moreover, beta parameter is commonly
thought as a systematic risk measure.As in Gatfaoui (2003), observing simul-
taneously systematic risk factorXand idiosyncratic risk factorIis equivalent
to observe simultaneously firm assets valueVand its specific risk factorI.
Moreover, applying generalized Ito’s lemma leads to the next expression for
firm value under original probabilityP:^3

dVt

Vt

=μV(t,Vt,It)dt+

[

βσX(t,Xt)dWtX+σI(t,It)dWIt

]

(6.4)

with^4 μV(t,Vt,It)=βμX

(

t,

(

Vt

It

) (^1) β)
+μI(t,It)+^12 β(β−1)σX^2
(
t,
(
Vt
It
) (^1) β)
.
Settingd〈V,I〉t=ρ(t,Vt,It)dt, the instantaneous (stochastic) correlation
between firm value and its idiosyncratic risk factor is then:
ρ(t,Vt,It)=
σI(t,It)
σV(t,Vt,It)
(6.5)
whereσV(t,Vt,It)=
√
β^2 σX^2
(
t,
(
Vt
It
) 1
β
)
+σI^2 (t,It)istheglobalvolatilityofthe
instantaneous return of a firm’s assets value. This global stochastic volatility
depends on the beta parameter, and the respective volatilities of the two risk
factors affecting firm value. Since our diffusions’ functionals are bounded,
global volatility is therefore bounded as a continuous function of these func-
tionals. Our specification follows the results of Campbell, Lettau, Malkiel
and Xu (2001) who show that the global volatility of any financial asset has
both a systematic component (systematic volatility) and an idiosyncratic
component (a specific volatility). Specifically, unsystematic volatility allows
for accounting for security- as well as event-specific factors (for example,