296 Frequently Asked Questions In Quantitative Finance
whereDis the amount of the debt, to be paid back at
timeT.
If we can hedge the debt with a dynamically changing
quantity of equity then the Black–Scholes hedging argu-
ment applies and we find that the current value of the
debt,V,satisfies
∂V
∂t
+^12 σ^2 A^2
∂^2 V
∂A^2
+rA
∂V
∂A
−rA= 0
subject to
V(A,T)=min(D,A)
and exactly the same partial differential equation for the
equity of the firmSbut with
S(A,T)=max(A−D,0).
The problem forSis exactly that for a call option, but
now we haveSinstead of the option value, the under-
lying variable is the asset valueAand the strike is
D, the debt. The formula for the equity value is the
Black–Scholes value for a call option.
Reduced form
The more popular approach to the modelling of credit
risk is to use an instantaneous risk of default or hazard
rate,p. This means that if at timetthe company has not
defaulted then the probability of default between timest
andt+dtispdt. This is just the same Poisson process
seen in jump-diffusion models. Ifpis constant then this
results in the probability of a company still being in
existence at timeT, assuming that it wasn’t bankrupt at
timet,beingsimply
e−p(T−t).
