Advances in Risk Management

126 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY

Table 6.7 Average bounds of simulated credit spreads in basis points

λ

β −1.5 − 1 −0.5 0 0.5 1 1.5

0.2 3481.72 1349.20 159.84 0.06 205.49 1493.86 3779.83

5220.18 2316.99 373.59 20.93 614.73 2521.02 5686.73

1 2583.16 1479.75 268.96 0.06 305.11 2263.50 3549.02

3636.47 2394.72 665.56 20.93 1173.51 3260.99 5517.20

5 2537.73 1630.65 377.01 0.06 473.15 2422.85 3114.64

3231.33 2545.06 781.39 20.93 1215.32 3362.97 5178.89

variation of credit spread at each time between debt’s issue date and matu-
rity. Evolutions of credit spread bounds over time can be viewed as extreme
scenarios describing credit spread’s evolution (best and worst possible
situations).

6.5 CONCLUSION

We focused on the credit risk valuation of Gatfaoui (2003) whose model-
ing proposes to value corporate debt in a Merton framework, and accounts
for systematic and idiosyncratic risk. Specifically, the option nature of debt
allows the author to price corporate debt through a call on firm value con-
sistently with the constant parameter-based dynamics of systematic and
idiosyncratic risk factors. Our work addressed the extension of such a setting
in two key points.
First, we considered the stochastic parameter-based dynamics of the two
previous risk factors. Under regularity conditions, this setting is equivalent
to a stochastic volatility option pricing (stochastic credit pricing) model.
Namely, we consider two risk sources affecting firm value, whereas we only
observe firm value. Consequently, we lie in an incomplete market where
incompleteness is due to the unobservable idiosyncratic part of firm value
(incompleteness engenders a stochastic volatility for firm value). Hence,
the no-arbitrage principle and minimal martingale measure allow to price
firm’s equity and therefore debt. Such a valuation becomes possible in the
historical universe as well as the minimal martingale measure’s universe
under a bounded volatility assumption. The equivalent minimal martingale
measure is useful insofar as it reduces global risk to its minimal component,
namely intrinsic (idiosyncratic) risk.
Second, we illustrated such a framework while specifying the stochastic
parameters of the diffusions under consideration. We undertook corporate
debt pricing in a bounded volatility case. Under our functional assumptions,