Advances in Risk Management

HAYETTE GATFAOUI 113

x,y∈{t,Vt,It}. Applying multivariate Ito’s lemma to the global variance of
firm value’s instantaneous return givesdR(t,Vt,It)=Trend dt+Vol 1 dWXt +
Vol 2 dWIt:

Trend=Rt(t,Vt,It)+RV(t,Vt,It)VtμV(t,Vt,It)+RI(t,Vt,It)μI(t,It)It

where

+

RVV(t,Vt,It)

2

σV^2 (t,Vt,It)Vt^2 +

RII(t,Vt,It)

2

σI^2 (t,It)I^2 t

+RVI(t,Vt,It)σV(t,Vt,It)Vtρ(t,Vt,It)σI(t,It)It

Vol 1 =RV(t,Vt,It)σV(t,Vt,It)Vtςβ

√

1 −ρ^2 (t,Vt,It)

Vol 2 =RV(t,Vt,It)σV(t,Vt,It)Vtρ(t,Vt,It)+RI(t,Vt,It)σI(t,It)It

Hence, the stochastic volatility framework becomes obvious. Indeed, the
dynamics of firm value and its global variance depend on two stochas-
tic parts which are correlated.^5 This setting has important implications for
Merton-type pricing models.
Following Merton (1974), the firm assets value is the sum of equity value
E(V,τ) and debt valueD(V,τ) such thatVt=E(Vt,τ)+D(Vt,τ) withτ=(T−
t) time to maturity, and following conditions:E(0,τ)=0,E(Vt,τ)=Vt−
D(Vt,τ)≥0,E(VT,0)=max(0,VT−B)=(VT−B)+. The option nature of
a firm’s balance sheet leads to consider equity as a European call on firm
value, with a strike equal to the promised payment (to debtholders) at firm
debt’s maturity. Hence, valuing risky debt requires pricing a European call.
However, as we lie in an incomplete market, there exists an infinity of equiv-
alent martingale measures allowing to price this European call (Mele and
Fornari, 2000). On the other hand, taking the risk-free asset as a numeraire,
the discount price of firm value becomes a semi-martingale under histor-
ical probabilityP. Nevertheless, among the set of equivalent martingale
measures compatible withV, there exists a unique equivalent martingale
measurePˆ, which minimizes the surrounding uncertainty or, equivalently,
the relative entropy measure (Delbaen and Schachermayer, 1996; Föllmer
and Schweizer, 1991; Gouriéroux, Laurent and Pham, 1998; Musiela and
Rutkowski, 1998). Similar to a Hull and White (1987) setting, fluctuations in
stochastic volatility generate a risk, which is not compensated. This feature

explains the existence ofPˆ, which is called the minimal equivalent martin-
gale measure. Probability measurePˆ is uniquely defined by its Girsanov
density (Karatzas and Shreve, 1991) and Karatzas (1996) as:

Lˆ(t)=d

Pˆ

dP

∣

∣

∣

∣

∣

Ft

=exp

{

−

∫t

0 α^1 (s,Vs,Is)dW

X

s −

∫t

0 α^2 (s,Vs,Is)dW

I

s

}

×exp

{

−^12

∫t

0 (s,Vs,Is)ds

}