Advances in Risk Management

HAYETTE GATFAOUI 123

Table 6.5 Average monthly simulated values of credit spreads in basis
points

λ

β −1.5 − 1 −0.5 0 0.5 1 1.5

0.2 4265.33 1921.20 345.89 2.86 507.29 2486.03 5285.82
1 2945.94 2194.89 566.89 2.86 862.23 2908.93 5156.47

5 2748.30 2348.05 743.01 2.86 938.75 2982.52 5030.70

hand, whenβ=0.5 orβ=1, the average stochastic volatility is a concave
function ofλ. Moreover, the average stochastic volatility is constant what-
everλwhenβ=0. Such a behavior of stochastic volatility has some impact
on the term structure of corporate credit spreads.

6.4.2 Credit spread

We extend our study while assessing the impact of stochastic volatility on
credit spreads. In particular, we focus on the term structure of credit spreads.
Lety(τ) be the yield-to-maturity of firm’s risky debt (for example, the
default risky yield). Such a yield is linked with the current value of firm’s

debt such thatD(Vt,τ)=e−y(τ)τB, which implies thaty(τ)=−τ^1 ln

(

D(Vt,τ)

B

)

.

Hence, related credit spreads (for example, yield spreads against gov-

ernment bonds) writeS(τ)=y(τ)−r=−τ^1 ln

(

D(Vt,τ)

B

)

−r. For the sake of

simplicity, we assume a flat risk-free term structure here. Then, simulated
debt values allow to compute monthly related credit spreads with varying
moneyness and time to maturity. Results are first displayed in Table 6.5.
Second, part of these results is summarized in Figure 6.4, which plots credit
spreads when lambda is 5.
Whateverλ, credit spreads are a convex function ofβwith a minimum
atβ∗=0. Moreover, credit spread behaviors relative toλare mitigated. For
|β|=1.5, credit spreads are decreasing functions ofλwhereas the reverse
is true for|β|=0.5 or 1. Finally, credit spreads are constant whenβ=0 due
to the independence of debt relative toλunder the minimal martingale
measure.
First, the higher the absolute value of beta is, the wider the related credit
spread becomes for a given level of lambda. Second, the credit spread’s
level related to a given negative value of beta lies slightly under the credit
spread’s level related to the corresponding positive value of beta. Finally,
credit spreads are convex decreasing functions of time to maturity. Equiv-
alently, credit spreads are convex increasing functions of debt’s maturity.
Indeed, when time to maturityτdecreases fromτ 0 =10 years to 0, maturity