Advances in Risk Management

116 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY

or, equivalently, under the minimal martingale measure:

dR(t,It)=

⌊

2 αβ^2 γ^2 t^2 α−^1 +^2 λ(ε−It)−^3 It

√

Itα 2 (t,It)

⌋

dt+^3 It

√

ItdWˆIt

withα 2 (t,It)=μVσ(Vt,(tIt,)It−)rρ(t,It);ρ(t,It)=σσVI((tt,,IItt));μV(t,It)=βμX(t)+μI(t,It)+
1
2 β(β−1)σ

2
X(t).
In the original universe, such a diffusion process behaves almost like a
mean reverting square-root process except that the random shocks affecting
its trend are higher in magnitude.^6 Moreover, when timettends towards
infinity, the stochastic variance of a firm’s value tends towardsR(t,It)=^2 It
such that global volatility reads

√

R(t,It)=

√
It. In the same way, the
previous diffusion asymptotically takes the new form:

dR(t,It)=λ

[

^2 ε−R(t,It)

]

dt+R(t,It)

√

R(t,It)dWtI

If variance is asymptotically zero, then its diffusion becomes
dR(t,It)=λ^2 εdt>0. Therefore, whenttends towards infinity and variance
is zero, the zero threshold becomes a reflecting barrier for firm value’s vari-
ance. Indeed, empirical features of equity volatility exhibit stationarity and
mean reversion patterns. In the asymptotic case,^2 εis the long-run mean
of firm value’s stochastic variance, andλis the velocity of mean reversion.
Our specification implies that the randomness in global variance comes only
from idiosyncratic risk factor. Indeed, stochastic volatility comes from the
non-observability of idiosyncratic risk factor. Namely, stochastic volatility
is due to the intrinsic risk of firm value because such a risk is non-tradable.
This setting gives some nice properties to firm’s debt pricing. Under this
framework, the next section undertakes some simulations ofItandR(t,It)
for given parameter values, and varyingβandλ.

6.3.2 Implication for debt pricing

The randomness of firm value’s variance depends only on idiosyncratic
risk factor (relation (6.9)). Moreover, correlation coefficient now expresses

ρ(t,It)= 

√

√ It

β^2 γ^2 t^2 α+^2 It

, and the firm value’s dynamic under the minimal

martingale measurePˆthen reads on time subset [t,T]:

ln

(

VT

Vt

)

=

(

r−

σ ̄^2 V

2

)

τ+

∫T

t

σV(s,Is)ςβ

√

1 −ρ^2 (s,Is)dWˆsX

+

∫T

t

σV(s,Is)ρ(s,Is)dWˆIs

whereσ ̄V^2 =^1 τ

∫T

t σ

2
V(s,Is)dsis the firm value’s average variance over the
time to maturity of debt (remaining life of the European call). Consider