Download Spread Curves and Intensity Models

Spread Curves and Intensity Models
Philipp J. Schönbucher
ETH Zürich
London, February 2002
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
The Basic Payoff Tree
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
The Basic Model
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Setup of the Basic Model
• N discrete points in time
0 = T0, T1, T2, . . . , TN ;
δk := Tk+1 − Tk
convergence to continuous-time when distances δk → 0
• τ : time of default
• In each interval [Tk , Tk+1], a default can happen:
Branch “down”.
• . . . or the obligor survives over [Tk , Tk+1]:
Branch “across”.
• Default is absorbing (“one-way”): there is no way back up.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Arrow-Debreu Securities: the Price Building Blocks
For Payoffs independent of default (possibly stochastic)
• Building block: Bk(t)
Default-free zero coupon bond with maturity Tk
? B(t, Tk ) or Bk (t): price at time t
? 1: payoff at time Tk (in every node of the tree)
For Payoffs in survival (possibly stochastic)
• Building block: Bk(t)
Defaultable zero coupon bond with maturity Tk
? B(t, Tk ) or B k (t): price at time t
? 1: payoff at time Tk only in the survival-node of the tree.
? 0: payoff otherwise: the building block has zero recovery
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Payoffs at default
• for example recoveries
• building block:
Default-securities ek (t):
? ek (t): price at time t
? 1: payoff at Tk if and only if default in [Tk−1, Tk ]
? 0: payoff otherwise.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Pricing Defaultable Securities
• Identify the payoffs in events of survival and default
• Payoffs in default:
? recovery rate π
? take care specifying the claim size in default (notional, accrued interest,
amortisation)
? observe seniority
• For calibration securities this should be straightforward.
• Model price = weighted sum of building-block prices
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Pricing a Defaultable Coupon Bond
$ %'& ( ) ( * +
!
,- . * %/+ 0
"
0
54 0
31 2 54 0 6
#
132 54 0 6
132 54 0 6
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Pricing a Credit Default Swap
% &(' ) * ) + ,
0
!
-. / + (& , 0
#"
21 436587 0
$
21 436587 21 436587
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Pricing with Building Blocks
Fixed Coupon Bonds: coupon c δk−1, payable at Tk , recovery π
PN
k=1 c
δk−1B k
| {z }
coupons
+B N
+π
| {z } |
+principal
PN
k=1 ek
{z
+recovery
}
Credit Default Swap: fee s δk−1, payable at Tk , payoff at default 1 − π
−
PN
k=1 s
|
δk−1B k
{z
}
−fee
+(1 − π)
PN
k=1 ek
{z
}
|
+default payoff
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Pricing Using the Tree
We used the tree as a convenient way of illustrating the events / states of
nature, in which different payoffs take place.
But we can also use a tree as a (very simple) pricing tool.
For this we need to specify:
• the branching probabilities: “across” and “down” at all nodes
• the default-free interest rates: Bk (t)
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
One-Step Survival Probabilities and Default Hazard Rates
We denote the one-step survival probabilities with pk (t):
pk (t) := P [ “across” at node from Tk to Tk+1 | Ft ] .
and the default hazard rate over [Tk , Tk+1] with Hk (t):
[default probability over [Tk , Tk+1]]
1 − pk (t)
δk Hk (t) :=
=
.
[survival probability over [Tk , Tk+1]]
pk (t)
equivalently:
pk (t) =
1
.
1 + δk Hk (t)
Hk is also known as odds-ratio.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Longer-term Survival Probabilities
The survival probability from 0 to TK is
PK (t) = p0 · p1 · · · pK−1 =
K−1
Y
pk .
k=0
Then we have of course
PK (t) =
K−1
Y
k=0
1
1 + δk Hk (t)
and
1
Hk (t) =
δk
µ
¶
Pk+1(t)
−1 .
Pk (t)
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Pricing Defaultable Zerobonds
The tree-value of a defaultable zero coupon bond with maturity TK is
B K (t) = BK (t)
K−1
Y
pk (t)
k=0
B K (t) = BK (t)
K−1
Y
k=0
1
1 + δk Hk (t)
B K (t) = BK (t)PK (t).
Conversely, if we have a full term-structure of B k (t), then we have all
branching probabilities pk (t).
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Survival Probabilities
Thus we can characterize the survival probabilities:
B k (t)
Pk (t) =
.
Bk (t)
The conditional survival probabilities are:
P (t; Tk , Tl) =
Pl(t)
Pk (t)
= probability of surviving from Tk to Tl conditional on surviving until Tk . (k < l)
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Pricing the Building Blocks ek (t)
Payoff of ek = 1 at Tk if and only if default in [Tk−1, Tk ]
• Probability of default in [Tk−1, Tk ]
= Pk−1(t) − Pk (t) = δk−1Hk−1(t)Pk (t)
• Discounting until Tk using Bk (t) yields
ek (t) = δk−1Hk−1(t)B k (t)
(remember Pk Bk = B k )
Here we used independence. Pricing is more complicated otherwise. (See
approximative solutions later.)
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
CDS Revisited
Initially, s is chosen such that the CDS has zero value:
N
X
s δk−1B k = (1 − π)
k=1
N
X
ek
k=1
PN
thus
or
s = (1 − π) PN
k=1 ek
k=1 δk−1 B k
s = (1 − π)
N
X
wk Hk−1,
PN
= (1 − π)
k=1 Hk−1 δk−1 B k
PN
k=1 δk−1 B k
δk−1B k
where wk = PN
k=1
n=1 δn−1 B n
.
The CDS-rate equals [LIED]× [a weighted average of the Hk ]
Quick-and-Dirty Rule: Hk = s/(1 − π).
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
CDS Valuation at later Dates
A CDS that was entered earlier at a CDS-rate s0 as protection buyer is now
worth
N
X
(s0 − s)
δk−1B k .
k=1
where s is the now current CDS rate. (Notional = 1)
Reasons:
• Economically:
Close the position: You will receive sδk−1 at each Tk until default or maturity.
You have to pay s0δk−1 at the same dates. The difference is the value of the
position.
• Mathematically:
Substitute the valuation equations for CDS and transform.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Spread Curve Construction
1. Select a set of calibration instruments
• defaultable (all affected by the same default)
• “standard”: can be priced in tree (e.g. straight bonds, CDS)
• liquid: They have market prices which contain information.
2. Choose a default-free interest-rate curve. (Usually Libor/Swapcurve.)
This will give us the Bk (0)
3. Assume average recovery rates π for the instruments. (Must be equal for
CDS and deliverable bonds.)
4. Choose the Hk such that the “tree” prices of the calibration instruments
match the observed market prices.
5. If there are too many solutions, add “smoothness” criteria to remove
indeterminacy.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
What is Calibration?
We are looking for a probability distribution for the time of default which is
consistent with observed market prices.
There are many such possible probability distributions.
Here, the modeller’s / user’s intuition is needed, to eliminate implausible
default probability distributions.
Therefore we need a setup where it is easy to build an intuition. A setup where
the model quantities have a direct relationship to real-world quantities that are
known and on which we have experience (and therefore also an opinion).
Mathematically, we are choosing a martingale measure in an incomplete
market.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
What is a good fit?
Approximate using basis functions gn,k :
Hk = α1g1,k + α2g2,k + . . . + αN gN,k
Spread Curve Construction:
find the weights αn that fit best to the bond prices.
By choosing the basis functions we can restrict the set of possible spread
curves.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Good Choices of Basis Functions:
• Close to Principal Components
g1 ’level’; g2 ’slope’; g3 ’curvature’
should be orthogonal functions
• Localised Basis Functions: (like Kernel smoothing)
gn > 0 only around Tn,
’hat’ or cubic (or 4th order) spline functions
improved speed for optimisation, no structure is pre-defined, need many
bonds
may want to add smoothness criteria to optimisation
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
The Optimisation Problem
²k : Price error of bond k assuming weights α = (α1, . . . , αN ):
Difference of actual price and price resulting from α
Find the weights α that minimize the squared sum of the price errors:
min
α
K
X
²2k
k=1
• Can achieve perfect pricing when
N (number of weights) = K (number of bonds).
• Sensible to have N < K. Need at least N ≤ K.
• Can give weights to the individual bonds (for liquidity etc.)
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Filling Gaps in the Curve
① collect similar bonds
• issuers of similar credit rating, industry, region
• use weights for degree of similarity
② pre-process the (few) bonds by the issuer in question (’real’ bonds vs. similar
bonds)
③ modify the optimisation problem:
• minimize the weighted sum of the pricing errors of the similar bonds
• weights = weights of similarity
• constraint: such that the ’real’ bonds are priced exactly.
④ may adjust weights by sign of the pricing error:
if a similar bond has a lower rating, only give weight to errors that would
underprice this bond.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Of course, all this is fudging and has no theoretical justification. Therefore it is
essential to check whether the resulting Hk make sense as local default
intensities.
If there are not enough traded assets, compare the calibration output to
a fundamental credit risk model.
If you trust your fundamental model, get the prices of B k directly from
the model.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Forward Rates
define the default-free forward rate Fk (t), the defaultable forward rate F k (t):
µ
¶
1
Bk (t)
Fk (t) =
−1 .
δk Bk+1(t)
µ
¶
1
B k (t)
F k (t) =
−1 .
δk B k+1(t)
¶
µ
1
Pk (t)
−1 .
Hk (t) =
δk Pk+1(t)
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Relation to Forward Spreads
the conditional probability of default over [Tk , Tk+1] is given by:
F k (t) − Fk (t)
1 def
Hk (t)
P (t, Tk , Tk+1) =
=
δk
1 + δk Hk (t)
1 + δk F k (t)
[Default Probability]
=[Length of time interval] × [Spread of forward rates]
× [Discounting with defaultable forward rate]
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Default Probabilities as δk → 0
use continuously compounded forward rates:
∂
ln B(t, T )
∆t &0
∂T
∂
f (t, T ) = lim F (t, T, T + ∆t ) = −
ln B(t, T )
∆t &0
∂T
f (t, T ) = lim F (t, T, T + ∆t ) = −
The marginal probability of default at time T is:
P def(t, T, T + ∆t )
lim
= f (t, T ) − f (t, T )
∆t &0
∆t
The probability of default in [T, T + ∆t ] is approximately proportional to the
length of the interval [T, T + ∆t ]
with proportionality factor (f (t, T ) − f (t, T ))
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
What have we reached?
• Prices of standard defaultable securities are given in terms of the building
blocks Bk , B k and ek
• The prices of the building blocks can be represented in terms of “forwardrates” Fk and Hk
• Fitting default-free forward rates Fk is standard.
• Can adapt forward rate - fitting methods to fit Hk
• Connection between hazard rates of default and credit spreads.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Outlook: Generalisation to Dynamic Models
we can view the tree model as “snapshot” of a more complicated model where
the survival probabilities pk can change over time.
Then we have:
1 − pk (t) = P [ τ ∈ [Tk−1, Tk ] | Ft ∧ τ ≥ Tk−1 ]
pk (t) is the best guess at time t
of the survival probability over [Tk−1, Tk ], given survival until Tk−1.
Example:
Assume some model gives us defaultable bond prices B k (t) for all maturities
Tk . We can still set up a tree and price within the tree.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
General Implied Survival Probabilities
Value: = discounted expected payoff
B(t, T ) = E [ βt,T · 1 ] = E [ βt,T ] .
βt,T is the discount factor over [t, T ].
Z
βt,T = exp{−
T
r(s)ds}
t
where r(s) is the continuously compounded short rate at time t.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Defaultable Zerobonds:
Payoff = 1{τ >T }
(
1 if default after T , i.e. τ > T ,
=
0 if default before T , i.e. τ ≤ T .
Value = discounted expected payoff
£
¤
B(t, T ) = E βt,T 1{τ >T } .
1{A} is the indicator function of event A:
1{A} = 1 if A is true, and 1{A} = 0 if A is false.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
If there is no correlation:
£
¤
B(t, T ) = E βt,T 1{τ >T }
£
¤
= E [ βt,T ] E 1{τ >T }
£
¤
= B(t, T )E 1{τ >T }
= B(t, T )P (t, T )
P (t, T ) is the implied probability of survival in [t, T ].
(βt,T is the discount-factor over [t, T ].)
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
The Implied Survival Probability
the implied survival probability is the ratio of the ZCB prices:
B(t, T )
P (t, T ) =
B(t, T )
• Default Probability = 1- Survival Probabilities
• initially at one
P (t, t) = 1
• eventually there is a default:
P (t, ∞) = 0
• P (t, T ) is decreasing in T .
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Intensity-Based Models
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Why Poisson Processes?
Ultimate goal:
A mathematical model of defaults that is realistic and tractable and useful for
pricing and hedging.
Defaults are
• sudden, usually unexpected
• rare (hopefully :-)
• cause large, discontinuous price changes.
Require from the mathematical model the same properties.
Furthermore: Previous section
The probability of default in a short time interval is approximately
proportional to the length of the interval.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
What is a Poisson Process?
N (t) = value of the process at time t.
•
•
•
•
Starts at zero: N (0) = 0
Integer-valued: N (t) = 0, 1, 2, . . .
Increasing or constant
Main use: marking points in time
T1, T2, . . . the jump times of N
• Here Default: time of the first jump of N
τ = T1
• Jump probability over small intervals proportional to that interval.
• Proportionality factor = λ
BTW: Except for the last two points, the same notation and properties apply to
Point Processes, too.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Discrete-time approximation:
• divide [0, T ] in n intervals of equal length
∆t = T /n
• Make the jump probability in each interval
[ti, ti + ∆t] proportional to ∆t:
p := P [ N (ti + ∆t) − N (ti) = 1 ] = λ∆t.
• more exact approximation: p = 1 − e−λ∆t
• Let n → ∞ or ∆t → 0.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Important Properties
Homogeneous Poisson process with intensity λ
Jump Probabilities over interval [t, T ]:
• No jump:
P [ N (T ) = N (t) ] = exp{−(T − t)λ}
• n jumps:
1
(T − t)nλne−(T −t)λ.
n!
• Inter-arrival times P [ (Tn+1 − Tn) ∈ t dt ] = λe−λtdt.
• Expectation (locally) E [ dN ] = λdt.
P [ N (T ) = N (t) + n ] =
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Distribution of the Time of the first Jump
T1 time of first jump.
Distribution: F (t) := P [ T1 ≤ t ]
Know probability of no jump until T :
P [ N (T ) = 0 ] = e−λT
= probability of T1 > T . Thus
1 − F (t) = e−λt
F (t) = 1 − e−λt
F 0(t) = f (t) = λe−λt.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
➭ T1 is exponentially distributed with parameter λ.
➭ This is also the distribution of the next jump, given that there have been k
jumps so far.
➭ Independently of how much time has passed so far:
It never is ’about time a jump happened’
or ’nothing has happened, I don’t think anything will happen any more. . . ’
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Inhomogeneous Poisson Process
Imhomogeneous = with time-dependent intensity function λ(t)
Probability of no jumps (survival):
( Z
P [ NT = Nt ] = exp −
)
T
λ(s)ds .
t
Probability of n jumps:
1
P [ NT − Nt = n ] =
n!
ÃZ
T
!n
λ(s)ds
RT
− t λ(s)ds
e
.
t
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Density of the time of the first jump:
Z
b
P [ T1 ∈ [a, b] ] =
f (t, u)du
Z
a
b
=
Ru
− t λ(s)ds
du.
λ(u)e
a
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Compound Poisson Process:
there is another random variable Y that is drawn at jump times.
For us:
T1 time of default
Y recovery rate
Marker Y , distributed like K(dy). f function of X =
P
Yi :
df = ∆f = (f (X + Y ) − f (X))dN
Z
E [ dX ] = yK(dy)λdt = y eλdt
Z
E [ df (X) ] = (f (X + y) − f (X))K(dy)λdt.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Cox Processes
default rate = Intensity of PP = credit spread.
Credit spreads are stochastic. ☞
Need stochastic intensity
• define a stochastic intensity process λ, e.g.
dλ = µλdt + σλdW
• λ(t)∆t : default probability over the next time-interval [t, t + ∆t ].
(That’s all we need to know at t.)
• at t + ∆t : Intensity has changed, λ(t + ∆t ) = λ(t) + dλ is new (local) default
probability.
• Conditional on the realisation of the intensity process, the Cox process is an
inhomogeneous Poisson process.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Cox Processes: The Conditioning-Trick
The Gods are gambling in a certain sequence:
• First, the full path of the intensity λ(t) is drawn from all possible paths for
λ(t).
• Then they take this λ(t) and use it as intensity for an inhomogeneous
Poisson process N .
They draw the jumps of N (t) according to this distribution.
• Then the information is revealed to the mortals:
At time t they may only know λ(s) and N (s) for s up to t.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
The Conditioning Trick
First, pretend you knew the path of λ, what would the price be? (Depending
on λ, of course.)
Then average over the possible paths of λ.
Let X(N ) be a payoff that we want to value. It depends on the question
whether a default occurred (N = 1) or not (N = 0).
E [ X(N ) ] = E [ E [ X(N ) | λ(t) ∀t ] ]
The inner expectation is easily calculated treating N as an inhomogeneous
Poisson Process.
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WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Properties of Cox Processes
Probability of no jumps (survival):
"
( Z
exp −
P [ NT = Nt ] = E
)#
T
λ(s)ds
.
t
Probability of n jumps:
"
1
n!
P [ NT − Nt = n ] = E
ÃZ
!n
T
λ(s)ds
RT
− t λ(s)ds
e
#
.
t
Density of the time of the first jump:
Z
b
P [ T1 ∈ [a, b] ] =
f (t, u)du
a
47
➠ ➡ ➡ ➠ ■ ?
WBS Training: Mathematics of Credit Derivatives: Unit 2
Z
b
=
h
E λ(u)e−
(c) Philipp J. Schönbucher
Ru
t λ(s)ds
i
du.
a
The expectations always only refer to the realisation of λ.
48
➠ ➡ ➡ ➠ ■ ?
WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
Simulation
Simulation of a default time τ with intensity process λ(t)
1. Draw a full realisation of λ(t).
2. Draw (independently) a random variable U , uniformly distributed on [0, 1]
3. Define τ through
Z
τ
U = exp{−
λ(s)ds}.
0
Then the survival probability up to time T is
"
P[ τ ≥ T ] = P
#
T
U ≤ exp{−
"
=E
Z
λ(s)ds}
0
¯
##
"
Z T
¯
T
¯
= E exp{−
λ(s)ds
P U ≤ exp{−
λ(s)ds} ¯ {λ(t)}t≥0
¯
0
0
"
Z
49
➠ ➡ ➡ ➠ ■ ?
WBS Training: Mathematics of Credit Derivatives: Unit 2
(c) Philipp J. Schönbucher
1
g
0,95
0,9
0,85
U
0,8
l
0,75
0,7
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
50