Advances in Risk Management

HAYETTE GATFAOUI 117

Gt=Ft∪{Is,t≤s≤T} and compute the two first moments of probability dis-

tribution of ln

(

VT

Vt

)

conditional onGt:EPˆ

[

ln

(

VT

Vt

)∣

∣

∣Gt

]

=

(

r−σ ̄

(^2) V
2
)
τ, and
VarPˆ
[
ln
(
VT
Vt
)∣
∣
∣Gt
]
= ̄σV^2 τ.^7 Hence, conditional onGt, the firm value’s natu-
ral logarithm ln
(
VT
Vt
)
follows a normal law (firm value follows a lognormal
law) with volatility parameter
√
σ ̄V^2.
On the other hand, recall expression (6.8) of firm’s equity or, equi-
valently, the European call on firm value under the minimal martingale
measure. Applying the iterated expectations theorem, we getE(Vt,τ)=
EPˆ
⌊
EPˆ
[
e−rτ(VT−B)+
∣
∣Gt
]∣∣
∣Ft
⌋

. However, given the law of ln

(

VT

Vt

)

condi-

tionalonGt, equityvaluethenreadsE(Vt,τ)=EPˆ

[

CBS

(

τ,r,Vt,B,

√

σ ̄V^2

)∣

∣

∣

∣Ft

]

whereCBS

(

τ,r,Vt,B,

√

σ ̄^2 V

)

is the Black and Scholes (1973) price employed

with an average time-dependent volatility. Consequently, equity value is the
average Black and Scholes European call price over each possible volatility
path. Our deterministic systematic risk volatility assumption leads then to
an optimal Monte Carlo European call pricing. Indeed, we only need to
generate one Brownian motion, namely the randomness affecting stochastic
volatility (for example, idiosyncratic risk’s Brownian motion). This setting
allows a simple computation of debt value since:

D(Vt,τ)=Vt−EPˆ

[

CBS

(

τ,r,Vt,B,

√

σ ̄^2 V

)∣

∣

∣

∣Ft

]

(6.10)

Given current information set, we can price firm’s debt such that
uncertainty is minimized. Moreover, as functional diffusion parameters
are bounded on [0,T], stochastic volatility is also bounded whatever

t∈[0,T] sinceσlV<σV(t,It)

√
1 −ρ^2 (s,Is)<σuV, withσlV=β^2 σlX^2 +σlI^2 >0 and
σVu=β^2 σXu^2 +σuI^2. Therefore, the Black & Scholes call price is bounded by
(Frey and Sin, 1999):

CBS(τ,r,Vt,B,σVl)<CBS

(

τ,r,Vt,B,

√

σ ̄V^2

)

<CBS(τ,r,Vt,B,σVu) (6.11)

which implies that both firm equity and debt values are bounded. We are
then able to price corporate debt under the minimal martingale measure. We
can also establish debt bounds depending on the magnitude of variations
in firm value’s volatility. Under our assumptions, systematic risk drives
volatility’strendwhereasidiosyncraticriskaffectsthistrendthroughshocks.
Consequently, the magnitude of variations in firm value’s global volatility
is driven by the impact of both systematic and idiosyncratic risk factors.