Advances in Risk Management

HAYETTE GATFAOUI 125

Table 6.6Average bounds of simulated

aggregate volatility in percent

λ

β 0 0.5 1 1.5

0.2 17.84 41.13 64.68 90.34

34.04 65.74 103.07 151.35

1 17.84 45.10 66.92 92.17

34.04 84.04 104.06 151.45

5 17.84 46.56 67.91 92.39

34.04 74.97 103.63 151.33

into account part of speculative grade corporate debt. Under convenient
assumptions about starting values and stochastic variables, a relevant choice
of stochastic functionals will describe some specific rating grades among
given speculative grade rating classes of corporate debt.
In our bounded volatility/bounded diffusion parameters setting, we
can establish bounds for credit spreads. Hence, we propose an alternative
approach to the one of Chen and Huang (2002). The authors give analytical
bounds to credit spread term structure in order to solve the problem of neg-
ative implied default probabilities. Such a problem arises when calibrating
credit models to empirical data, and comes from the no-arbitrage principle’s
violation. In the same way, we establish bounds for implied credit spreads.
We first give the average aggregate volatility’s bounds we get under our
framework. Briefly, we compute related boundsσlVandσuVfor ournsim
simulations ofVt, as well as the arithmetic mean of all obtainedσlVandσuV
in Table 6.6.
Whatever λ, average volatility’s bounds are constant when β=0.
When beta is non-zero, average aggregate volatility’s upper bounds
are concave functions of λ with a maximum value at λ∗=1. Dif-
ferently, average aggregate volatility’s lower bounds increase strictly
with λ. Moreover, our volatility’s bounds are increasing functions of
beta parameter. Consequently, formulae (6.10) and (6.11) allow to bound

debtDl(τ)<D(Vt,τ)<Du(τ) withDl(τ)=Vt−EPˆ[CBS(τ,r,Vt,B,σuV)|Ft] and

Du(τ)=Vt−EPˆ[CBS(τ,r,Vt,B,σVl)|Ft]. Thus, the risky yield-to-maturity

becomes bounded as^1 τln

(

B

Du(τ)

)

<y(τ)<^1 τln

(

B

Dl(τ)

)

. Hence, related credit

spread is bounded sinceSl(τ)<S(τ)<Su(τ) whereSl(τ)=^1 τln

(

B

Du(τ)

)

−r,

andSu(τ)=^1 τln

(

B

Dl(τ)

)
−r(see Table 6.7).
Average credit spread bounds behave like the average monthly credit
spreads reported in Table 6.5. Therefore, we can give an interval for possible