112 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITYfinancial and corporate events), which are known to impact asset returns’
volatility (Hilliard and Savickas, 2002). However, though idiosyncratic risk
plays an increasing role, given market history, global volatility remains
driven by its systematic component (market volatility). Recall that global
volatility is the global risk level’s target of the firm in accordance with its
shareholders’ interests (financial and dividend policies). Moreover, volatil-
ity is also considered as a proxy of liquidity risk. Indeed, Karpoff (1987),
Lamoureux and Lastrapes (1990) and Schwert (1989) show that volatility is
correlated with trading volume.
Moreover, considering expression (6.5) of the correlation coefficient and
the firm’s value dynamic (6.4), the diffusion of the firm’s assets value takes
a new form in the historical universe:
dVt
Vt=μV(t,Vt,It)dt+σV(t,Vt,It)[
ςβ√
1 −ρ^2 (t,Vt,It)dWtX+ρ(t,Vt,It)dWIt]
(6.6)whereζβ=sign(β) represents the sign of beta (for example,ζβ=1ifβ≥ 0
andζβ=−1ifβ<0). Therefore, describing the firm value’s dynamic with
relations (6.1), (6.2) and (6.3) is equivalent to characterizing the firm’s assets
value with relations (6.4) or, equivalently, (6.6) and (6.2). As this specifica-
tion introduces two risk factors whilst we only observe firm assets value, we
therefore lie in an incomplete market. Such a setting appears to be equiv-
alent to a stochastic volatility framework provided that global volatility
σV(t,Vt,It) is non-zero whatever (t,Vt,It)∈[0,T]×R^2.
6.2.2 Stochastic volatility and Merton’s pricing
Indeed, relations (6.6) and (6.2) are similar to the state-diffusion and stochas-
tic volatility model of Hofmann, Platen and Schweizer (1992). In this case,
we have more risk factors (systematic and idiosyncratic risk) than exist-
ing or, equivalently, primary assets (firm value). Consequently, we are
unable to give a unique price to any contingent claim on firm assets value.
At best, we can define bounds for such a price (Frey and Sin, 1999) or
minimize the uncertainty while computing a price. We address these two
points therein. First, to shed light on the stochastic volatility analogy, we
assume that firm value’s global volatilityσV(t,Vt,It)isaC1,2([0,T]×R^2 )
function (continuous, once derivable relative to time, and twice derivable
relative to its two last arguments). It is sufficient to assume thatσX(t,Xt)
andσI(t,It) are twoC1,2([0,T]×R^2 ) functions. Consider the firm value’s
global varianceR(t,Vt,It)=σV^2 (t,Vt,It), and letRx(t,Vt,It)=∂R(t,Vt,It)/
∂x, Rxx(t,Vt,It)=∂^2 R(t,Vt,It)/∂x^2 andRxy(t,Vt,It)=∂^2 R(t,Vt,It)/∂x∂yfor