CREDIT RISK: Lecture IV
Lina ElJahel
2006
1 Introduction to Jump Processes
Why Poisson Processes?
1. Ultimate goal: A mathematical model of defaults that is realistic and
tractable and useful for pricing and hedging.
2. Defaults are
(a) sudden, usually unexpected
(b) rare (hopefully :)
(c) cause large, discontinuous price changes.
3. Require from the mathematical model the same properties.
4. Furthermore: The probability of default in a short time interval is
approximately proportional to the length of the interval.
2 What is a Poisson Process?
N
(
t
) = value of the process at time t.
1. Starts at zero:
N
(0) = 0
2. Integervalued:
N
(
t
) = 0
,
1
,
2
,...
3. Increasing or constant
4. Main use: marking points in time
T
1
,T
2
,...
the jump times of
N
5. Here Default: time of the ﬁrst jump of
N
,
τ
=
T
1
6. Jump probability over small intervals proportional to that interval.
7. Proportionality factor =
λ
1
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View Full Document3 Discretetime approximation:
1. divide [0
,T
] in
n
intervals of equal length Δ
t
=
T/n
2. Make the jump probability in each interval [
t
i
,t
i
+ Δ
t
] proportional to
Δ
t
:
p
:=
P
[
N
(
t
i
+ Δ
t
)

N
(
t
i
) = 1] =
λ
Δ
t
(1)
3. more exact approximation:
p
= 1

e

λ
Δ
t
4. Let
n
→ ∞
or Δ
t
→
0
4 Important Properties
Homogeneous Poisson process with intensity
λ
. Jump Probabilities over
interval [
t,T
]:
1. No jump:
P
[
N
(
T
) =
N
(
t
)] = exp
{
(
T

t
)
λ
}
(2)
2.
n
jumps:
P
[
N
(
T
) =
N
(
t
) +
n
] =
1
n
!
(
T

t
)
n
λ
n
e

(
T

t
)
λ
(3)
3. Interarrival times
P
[(
T
n
+1

T
n
)
∈
tdt
] =
λe

λt
dt
4. Expectation (locally)
E
[
dN
] =
λdt
5 Distribution of the Time of the ﬁrst Jump
T
1
time of ﬁrst jump. Distribution:
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 Spring '05
 wqeww
 Poisson Distribution, Poisson process, poisson processes, Cox process, Jump Processes

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