110 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITYIn the light of such results, we extend the work of Gatfaoui (2003) to
price risky debt in a Merton framework with stochastic volatility. For this
purpose, our chapter is organized as follows: section 6.2 states the basis for
the stochastic functional-based credit pricing model, then we underline the
link with stochastic volatility models and introduce our pricing methodol-
ogy. After specifying our stochastic functionals, we formalize our stochastic
volatility model in section 6.3. Then, section 6.4 undertakes a simulation
study to assess the impact of stochastic volatility both on risky debt val-
uation and credit spreads, and finally section 6.5 draws some concluding
remarks.
6.2 THE GENERAL MODEL
Weintroducethedynamicpricingofafirm’sriskydebtwhilevaluingitstotal
assets(forexample, firmassetsvalue). Themathematicalbackgroundaswell
as pricing methodology is introduced along with the setting proposed by
Gatfaoui (2003).
6.2.1 Basic setting
Consider a probability space (,F,P) with a natural filtrationFt=σ(ws,0≤
s≤t) wherew′t=(WtX,WtI). (WtX) and (WtI) are two independentP-Brownian
motions and represent the public information set at current timet. Let
F=(Ft)t∈[0,T]be theP-augmentation ofFtwithT<∞. We set the assump-
tions prevailing in the Black and Scholes (1973) and Merton (1974) worlds
except that diffusion parameters are rather stochastic than constant (for
example, incompleteness of financial market). Briefly, there is no arbitrage
opportunity and the spot risk free interest rateris constant.
Consider a firm whose assets value at current timetisVt, which is an
Ft-adapted process. This firm is supposed to issue two kinds of financial
assets, namely a risky debt represented by a discount bond maturing at time
Twith terminal valueB(for example, promised payment to debtholders),
and no-dividend-paying equity. The firm’s potential default can only occur
at timeT. LetXtandItbe the systematic and idiosyncratic risk factors
respectively, describing any financial asset in the market, and therefore firm
value. Namely,Xtrepresents that part of firm value, which depends purely
on market conditions, andItrepresents that part of firm value, which results
from firm-specific patterns. These two risk factors areFt-adapted processes
whose dynamics are:
dXt
Xt=μX(t,Xt)dt+σX(t,Xt)dWtX (6.1)