Advances in Risk Management

114 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY

where α 1 (t,Vt,It)=α(t,Vt,It)ςβ

√

1 −ρ^2 (t,Vt,It); α 2 (t,Vt,It)=α(t,Vt,It)

ρ(t,Vt,It);α(t,Vt,It)=μVσV(t(,tV,Vt,tI,tI)t−)r.

α(t,Vt,It) is the global market risk premium due to the global (aggregate)
risk borne by firm value whereasα 1 (t,Vt,It) andα 2 (t,Vt,It) are the market
risk premia related respectively to the systematic and idiosyncratic risk fac-
tors affecting firm value. Consequently, the dynamic of ln(V) (for example,

firm value’s dynamic) under the minimal martingale measurePˆwrites after
applying generalized Ito’s lemma:

dln (Vt)=

[

r−

σV^2 (t,Vt,It)

2

]

dt+σV(t,Vt,It)ςβ

√

1 −ρ^2 (t,Vt,It)dWˆtX

+σV(t,Vt,It)ρ(t,Vt,It)dWˆIt (6.7)

wheredWˆtX=α 1 (t,Vt,It)dt+dWXt anddWˆtI=α 2 (t,Vt,It)dt+dWtI are two

independent Ft-adapted Pˆ-Brownian motions. Under the incomplete-
market assumption, the no-arbitrage principle and minimal martingale
measure allow us to price the European call on firm valueV(see Hofmann,
Platen and Schweizer (1992) for option pricing in an incomplete market,
and El Karoui, Jeanblanc and Shreve (1998) for properties of the Black and
Scholes formula). Indeed, the European call’s current value (firm’s equity)
is the expected discount value of its terminal payoff:

E(Vt,τ)=E

Pˆ⌊

e−rτ(VT−B)+

∣

∣Ft⌋ (6.8)

Using a Monte Carlo method (Jäckel, 2002) allows us to compute this
expectation, and finally estimate debt value since we haveD(Vt,τ) =

Vt−E(Vt,τ)=Vt−EPˆe−rτ(VT−B)+

∣
∣Ft.
The stochastic volatility framework has nice properties since it adds flex-
ibility to asset pricing, and then can improve Merton’s debt valuation.
However, the computational cost may be high since we need to simulate
two Brownian motion paths. Nevertheless, such a setting may be extremely
simple in some cases and highly useful for debt pricing. We focus on some
useful and optimal simplification for Merton’s debt pricing.

6.3 A STOCHASTIC VOLATILITY MODEL

In this section, we price risky debt under our previous stochastic volatility
framework. We start from a general case, and then concentrate on a particu-
lar case while specifying our functionals. Our simplified framework allows
for a tractable and easy computation of a firm’s debt.