Advances in Risk Management

HAYETTE GATFAOUI 115

6.3.1 Model specification

Targetingaconvenientdegreeofsimplification, wemaketwomajorassump-
tions. First, we assume that the volatility functional of the idiosyncratic risk
factor depends only onI. Second, we assume that the drift and volatil-
ity functionals of the systematic risk factor are deterministic functions of
time. Our assumptions are motivated by the empirical features exhibited
by equity volatility. To account for realistic features of equity volatility (see
Psychoyios, Skiadopoulos and Alexakis, 2003, and Phoa, 2003), the stochas-
tic global volatility of firm value has to be a stationary and mean reverting
process (to encompass some shock effects on volatility). We explain therein
how our assumptions fit the empirical characteristics.
Let constant real numbersωandδdepend on prevailing financial and
economic conditions, and assume that whatevert∈]0,T] forμX(t) andσX(t),
and whatevert∈[0,T] else:

μX(t)=ωtδ σX(t)=γtα μI(It)=λ

(

ε

It

− 1

)

σI(It)=

√

It

whereγ>0,α<0,λ>0,∈>0 and,>0 are constant parameters such that:

dXt

Xt

=ωtδdt+γtαdWtX dIt=λ(ε−It)dt+It

√

ItdWtI

We can also assume that debt is issued at timet 0 >0 and matures at
T=t 0 +τwhereτis the initial lifetime of debt. Moreover, we exclude the
caset=0 forμX(t) sinceδcan take negative values. We also assume that
μX(0)=μ 0 andσX(0)=σ 0 whereμ 0 ∈Randσ 0 >0 are bounded constant
values. Hence, the variance of instantaneous return of firm value writes
σV^2 (t,It)=β^2 σ^2 X(t)+σ^2 I(It)=β^2 γ^2 t^2 α+^2 It=R(t,It) whatevert∈]0,T], with
R(0,I 0 )=σV^2 (0,I 0 )=β^2 σ^20 +^2 I 0 being bounded. Our specification is con-
sistent with Andersen, Bollerslev, Diebold and Ebens (2001) who study
model-free measures of volatility and correlation of daily stock prices. The
authors analyse time-varying features of stock returns (see, Bekaert and
Wu, 2000; Bollerslev and Mikkelsen, 1999; Campbell, Lettau, Malkiel and
Xu, 2001; Christensen and Prabhala, 1998) and they find two main results.
First, variances exhibit a systematic common component in their evolution.
Second, an asymmetric relationship prevails between returns and volatility.
We then obtain:

RV(t,It)=RVV(t,It)=RII(t,It)=RIV(t,It)= 0 RI(t,It)=^2

Rt(t,It)= 2 αβ^2 γ^2 t^2 α−^1

In this case, the variance satisfies the following SDE in historical universe:

dR(t,It)=

[

2 αβ^2 γ^2 t^2 α−^1 +^2 λ(ε−It)

]

dt+^3 It

√

ItdWIt (6.9)