Advances in Risk Management

122 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY

Table 6.3 Average monthly simulated values of firm’s equity

λ

β −1.5 − 1 −0.5 0 0.5 1 1.5

0.2 50.07 61.55 70.32 71.52 56.83 59.55 129.46

1 51.19 60.47 67.87 71.52 52.76 58.64 135.57

5 54.15 62.77 70.97 71.52 57.51 65.90 154.30

Table 6.4 Average monthly simulated values of path dependent

stochastic volatility in percent

λ

β −1.5 − 1 −0.5 0 0.5 1 1.5

0.2 85.32 68.82 45.45 27.22 50.20 71.45 97.46

1 84.53 73.00 51.80 27.22 58.28 75.32 100.10

5 82.96 74.39 54.19 27.22 56.56 74.97 100.26

Generally, equity is a non-monotonous function ofβandλparameters.
Equity increases for growingβ≤0, decreases inβ= 0 .5, and goes on increas-
ing for growingβ∈]0.5, 1.5]. Specifically, equity is a convex function ofλfor
|β|<1.5 with a minimum atλ∗=1. Finally, it becomes an increasing func-
tion ofλfor|β|=1.5. Hence, results show global volatility’s impact on both
firm’s equity and debt. Incidentally, existing literature has also shown that
firm’s global volatility impacts capital structure. Indeed, Leland and Toft
(1996) shows that the longer debt’s maturity is, the more sensitive are firm
and debt values to firm value’s global volatility. Moreover, the sensitivity of
debt relative to an increase in global volatility reacts in the opposite way to
equity’s sensitivity relative to the same increase in global volatility.
Briefly, we also display in Table 6.4 the conditional expected value of
average monthly stochastic volatility

σ ̄eV=E

Pˆ





√

1

τ

∫T

t

σV^2 (s,Is)ds

∣ ∣ ∣ ∣ ∣ ∣

Gt



=EPˆ

[√

σ ̄^2 V

∣

∣

∣

∣Gt

]

over the remaining time to maturity of debt, and under the minimal mar-
tingale measure. The stochastic integral composing firm value’s variance
is computed using the finite difference method, and average stochastic
volatility is computed always using Monte Carlo simulation methodology.
Whateverλ, the average stochastic volatility is a convex function ofβ
with a minimal value atβ∗=0. When− 1. 5 <β<0 andβ=1.5, the average
stochastic volatility is an increasing function ofλ. In contrast, whenβ=−1.5,
the average stochastic volatility decreases as a function ofλ. On the other