118 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY6.4 SIMULATION STUDY
We use Monte Carlo accelerators to examine behaviors of debt pricing as
well as its related credit spread in a stochastic volatility setting. We study
our pricing framework as a function of a systematic risk measureβand
velocityλ. For statistical investigation purposes, the range of values we
consider can be larger than the realistic range of values that describes the
real world.
6.4.1 Volatility and debt
We simulate stochastic volatility and its impact on a firm’s debt and equity.
Then, we plot the paths obtained for these random variables or display their
average values (arithmetic means of simulated data). Assuming that the
initial debt’s time to maturity isτ=T−t=10 years, we setα=−^14 ,ε=0.5,
γ=1,It=3.5,=
√
0. 2
ε, andR(t,It)=β(^2) t−^12 + 0. 4 It. Specifically, we assume
that debt is issued at timet>0 and matures atT=t+τ. We also assume
that the starting value of the remaining life of firm’s debt isτ=τ 0 =10 years.
Daily values of global volatility
√
R(t,It) (equation (6.9)) are computed for
different values of beta and lambda parameters (for example,β=0, 0.5, 1,
1.5 andλ=0.2, 1, 5) withtrunning fromT-ttoT(see Figures 6.1 to 6.3).
Recall thatVt=Itwhenβ=0. Moreover, onlyβ^2 intervenes in our global
volatility.
The higher lambda is, the more stable are the evolutions and convergence
to long-run means of stochastic volatility
√
R(t,It), idiosyncratic factorIt
6
5
4
Volatility (10 years)
3
2
1
(^00)
(^1112223334445556667778889991110122113321443155416651776)
Time (days)
Beta 0 Beta 0.5 Beta 1 Beta 1.5
1887199821092220233124422553266427752885299731083219333034413552
Figure 6.1Simulated volatility whenλ=0.2