MSc in Mathematical and Computational Finance
Credit Derivatives
Exercise Sheet 1
1. Modelling recovery: There are a few different ways of modelling recovery upon default. Let
φ(τ ) denote the amount recovered when there is a default at time τ . We assume that there
is a risk free rate process (rt )t and the market has a unique risk neutral measure Q.
Explain why the defaultable ZCB price is given by, for 0 ≤ t ≤ T ,
Pφ (t, T ) = EQ
Z
exp −
τ
Z
rs ds φ(τ )I{τ ≤T } + exp −
!
T
rs ds I{τ >T } |F t
!
.
t
t
We write P0 (t, T ) for the defaultable bond price with 0 recovery
(a) Recovery of par: We recover an amount V (t), the time t value of the claim, at a rate
π(t) if default occurs at t. Show that the time 0 price of a defaultable ZCB with
recovery of par is given by
Z
Prp (0, T ) = P0 (t, T ) +
T
π(t)V (t)λ(0, t)P0 (0, t)dt,
0
where λ(0, t) is a suitably defined hazard rate.
(b) Recovery of treasury: At default we recover c times the equivalent default free asset.
Show that
Prt (t, T ) = cP (t, T ) + (1 − c)P0 (t, T ).
2. In a reduced form model we model the dynamics of the interest rates by a Vasicek model
and similarly for the hazard rate
drt
= κ(θ − rt )dt + σdWt ,
dλt
= κ0 (θ0 − λt )dt + σ 0 dWt0
where κ, κ0 , θ, θ0 , σ, σ 0 are constants and W, W 0 are Brownian motions with constant correlation ρ in that dhW, W 0 i = ρdt.
(a) Using an affine term structure approach show that the survival probability
!!
Z
T
S(t, T ) = E exp −
λs ds
= exp (A(t, T ) − B(t, T )λt ) ,
t
can be found and give an explicit formula for B and an integral expression for A.
(b) Show that the price P0 (t, T ) of a defaultable ZCB with zero recovery in this model can
be written as
!!
Z
T
P0 (t, T ) = P (t, T )EQT
exp −
λs ds
,
t
where QT is the T -forward measure. Find the dynamics of the intensity process λ
under QT and hence find P0 (t, T ).
(c) Recall the price P (t, T ) of a ZCB in the Vasicek model and hence show that the implied
survival probability is
!
Z T
P0 (t, T )
0
(σ − σ)σρB(t, T )ds .
S(t, T ) =
exp −
P (t, T )
t
3. In a firm value model the firm value V evolves according to a geometric Brownian motion
dV = µV dt + σV dW,
where µ, σ are constants and W is a standard Brownian motion. We also assume that there
is a bank account which pays a constant rate r of interest.
The firm is regarded as composed of debt and equity. The bond price and share price are
traded and are functions of the firm value V .
(a) Determine the probability of default in the model where the debt payment of K is all
due at the fixed time T .
(b) Using a hedging argument where the bond of the firm is used as a hedge, derive the
equation satisfied by the equity price Et .
(c) Show that the probability that Brownian motion with drift b has not hit x < 0 by time
T is given by
x + bT
−x + bT
√
√
− e2bx N
,
P( inf Bt + bt ≥ x) = N
0≤t≤T
T
T
where N is the standard normal CDF.
(d) We now assume that the firm will default if the firm value hits a constant barrier K.
We assume that V0 > K. Calculate the probability of default by time T in this model.