1. Pricing CDOs using the Intensity Gamma
Approach
Christelle Ho Hio Hen
Aaron Ipsa
Aloke Mukherjee
Dharmanshu Shah
Computing Methods
December 19, 2006

2. Table of Contents
Table of Contents.................................................................................................................2
Introduction..........................................................................................................................3
Advantages of Intensity Gamma Approach.....................................................................3
Default Correlation in Intensity Gamma..........................................................................4
Implementation Overview...............................................................................................4
Survival Curve.....................................................................................................................5
Construction of the Business Time Path..............................................................................9
Calculating IG forward intensities (ci)..............................................................................14
Getting the Default Times from the Business Time Paths.................................................16
CDO pricer.........................................................................................................................17
Validation...........................................................................................................................17
Roundtrip Test...............................................................................................................17
A Fast Analytic Price Approximation in the Intensity-Gamma Framework.................19
Calibration..........................................................................................................................21
Future work........................................................................................................................22
References..........................................................................................................................23

3. Introduction
It has become fashionable to attack David Li’s copula model [Li 2000] as an inadequate
tool for pricing CDO tranches and more generally any security exposed to default
correlation. This criticism belies the importance of the model. Despite its inadequacies it
has framed and motivated much of the current research.
Mark Joshi and Alan Stacey’s Intensity Gamma [Joshi 2005] model attempts to address
some of the issues of the standard Gaussian copula model by combining two important
concepts from past models in both option pricing and credit modeling: conditional
independence and business time. Conditional independence refers to the fact that events
are independent conditioned on some random variable. In copula models default times
are independent conditioning on a systematic factor. Business time refers to the idea that
changes in economic variables are driven by the arrival of information and not simply by
the passage of time. Both of these ideas underly the Variance Gamma model [Madan
1998] which the authors refer to and pay homage to in their naming of the model.
In a nutshell: “Conditional on the information arrival or business time process, (It),
defaults of different names are independent.”
Advantages of Intensity Gamma Approach
1) Market Does not believe in the Gaussian Copula: Different tranches of a CDO
have different base correlations. We observe an increasing base correlation with
increasing detachment point. Hence, the market implies that there is no constant
correlation amongst the names in the CDO, since this number keeps varying from
tranche to tranche. The Intensity Gamma approach does not need a base
correlation to price a CDO.
2) Pricing Non-Standard Tranches in CDOs: If the Gaussian Copula is to be used
to price a non-standard tranche such as 0-35%, we would need to extrapolate from
the market-observed base correlation curve of standard tranches. This can price a
0-34% tranche with a lower spread than the 0-35% tranche. The Intensity Gamma
approach does not suffer from this.
3) Pricing exotic credit derivatives: Portfolio credit derivatives with complicated
payoffs can be priced easily by Intensity Gamma. The Gaussian Copula needs a
base correlation to spit out default times, and hence is unfit for use in exotic credit
derivatives.
4) Time Homogeneity: The Intensity Gamma approach is not based on a particular
time frame. Once the method is calibrated, it can be used to price portfolio credit
derivatives of any maturity. In contrast the Gaussian Copula approach is
dependent on the time frame that we are working in. The base correlation for
pricing a 3 year CDO differs from that for a 5 year CDO, making it more difficult
to work with that Intensity Gamma approach. However, there is a caveat that
although the intensity gamma approach prices a lower base correlation with

4. increasing maturities, which is desired, the change is more pronounced than
observed in the market. Hence, it is not advisable to price products across
different maturities using the intensity gamma approach.
Default Correlation in Intensity Gamma
How does default correlation occur in such a model? Imagine a string marked at
uniformly distributed intervals, each marking representing a default. On average each
segment has about the same number of defaults. This represents the concept of
conditional independence: for a given business time path events occur independently.
Now imagine the string is kinked at various points, although the marks remain equally
spaced when we look at the string edgewise we see that the marks (defaults) have moved
closer together. This is illustrated in the diagram below.
The economic intuition is that information arrives in bursts. In periods of high
information arrival we can expect more events to occur than when no information has
arrived. The amount of business time which arrives between defaults is uniform but the
amount of calendar time is not: we clearly see periods of clustered defaults.
Implementation Overview
Implementation of the IG model consists of the following steps:
1. Survival curve construction
2. Calculating IG forward default intensities
Business time Calendar time

5. 3. Simulation of business time paths
4. Calculation of default times
5. Pricing CDO tranches
6. Calibration
Calibrating the model requires iterating steps 2 through 5 with different model parameters
in order to match model prices to market prices. The relationship between the blocks is
depicted schematically below:
The implementation was validated using the following tests:
1. Round trip test: verify CDS spreads are recovered
2. Homogeneous portfolio approximation
Survival Curve
a) Intuition
6mo 1y 2y .. 5y
name1 .
name2
.
.
name125
CDS spreads
Survival Curve
Construction
IG Default
Intensities
Calibration
Parameter guess
Business time
path generator
Default time
calculator
Tranche pricer
Objective function
0-3% …
3-6% …
6-9% …
.
.
Market tranche
quotes
Err<tol? NO
YES
Final parameters

6. A CDO is composed of a collection of CDSs and each CDS is a contract based on the
chances that a firm will survive to the maturity of the CDS. A survival curve is
essentially a mapping of the probability of survival of an entity, as time progresses. It is
essential to the pricing of a CDO since it speaks about how likely it is for a firm to
default.
Default is commonly modeled as Cox process, which is a doubly stochastic Poisson
process. Here λ, the forward intensity of a Poisson process is not constant in time. While
for a Coz process λ is stochastic, in our case we treat lambda as piecewise constant,
changing with time. The probability that a firm survives to time T is given by
Pr (τ > T) = exp[
0
( )
T
u duλ−∫ ]
Thus, instead of the actual survival probability X(0,T), the forward intensity curve λ(t)
gives the same information, i.e. both talk about the chances of a firm surviving in time.
Hence, in practice the λ(t) curve is referred to as the survival curve, and is integral to the
pricing of CDOs, CDSs and other credit products.
Survival Curve in terms of λ
0
0.001
0.002
0.003
0.004
0.005
0.006
0 1 2 3 4 5 6
Time
Intensity
b) CDS Pricing
A CDS is a contract where the “protection buyer” pays a running coupon c to the
“protection seller”, in return for protection against default on some firm. If default on the
specified firm takes place before the final CDS maturity, the protection seller pays the
protection buyer 1−R at the time of default. R is the observed recovery rate. At time of

7. default, coupon payments cease. For a CDS to fairly priced, the present values of the
fixed coupon leg and floating contingent payment at default must equal each other.
The present value of the protection buyers coupon payment stream considering a notional
of $1 is
PVcoup(0) = c
1
(0, )
N
i i
i
B Tδ
=
∑
Where B(0,Ti) is the price of a risky bond. A risky bond can be priced by multiplying a
risk free bond by the probability that the bond would survive till maturity.
B(0,Ti) = P(0,Ti) * X(0,Ti)
(Using notations and terminology as in Prof. Leif Andersen’s lecture notes at NYU)
The present value the protection seller’s payment is
PVprot(0) = C(0; 1 − R)
Where C(0; 1-R) represents the present value of the contingent payment of 1-R at time of
default.
Hence, the present value of the CDS can be expressed as the difference of the two legs
described above
PVCDS(0) = C(0; 1 − R) - c
1
(0, )
N
i i
i
B Tδ
=
∑
Equating the present value to zero, we obtain the fiar spread on the CDS as
Cpar =
1
(0,1 )
(0, )
N
i i
i
C R
B Tδ
=
−
∑
Calculation of Present Value of Contingent claim : C(0,1-R)
At the time of default, the protection seller pays the protection buyer 1-R. It is our goal to
find the present value of this contingent payment. Using elementary probability theory,
we know that the present value is the discounted value of 1-R multipled by the
probability of default. Also, we don’t know the exact time of default, and hence work
with expectations.
C(0;1-R) = E[(1-R) . exp(
0
( )r u du
τ
∫ ) . 1τ<T]
Where 1τ<T is 1 if τ<T and 0 otherwise.
Now, let us first find this value such that default takes place in an infinitesimal time
[z,z+dz]
C(0;1-R) = E[(1-R) . exp(-
0
( )
z
r u du∫ ) . 1τ Є[z,z+dz] ]
C(0;1-R) = E[(1-R) . λ(z) . exp(-
0
( ( ) ( ))
z
r u u duλ+∫ ) . dz]

8. Now adding up over the entire interval [0,T], we obtain the general form as
C(0;1-R) = E[
0
T
∫ (1-R) . λ(z) . exp(-
0
( ( ) ( ))
z
r u u duλ+∫ ) . dz]
But X(0,t) = exp(-
0
( )
t
u duλ∫ )
Hence we can write
dX
dt
= -λ . exp(-
0
( )
t
u duλ∫ )
Substituting in the equation for C(0;1-R), we obtain
C(0;1-R) = -
0
T
∫ (1-R) . P(0,z) .
(0, )dX s
dz
dz
Where P(0,z) is the price of a risk free bond of maturity z.
However, in reality, we cannot integrate continuously. It seems logical to discretize the
equation stated above, into time intervals equal to that between subsequent CDSs. Also,
we make a practical assumption that irrespective of exactly when default takes place, we
may treat it as taking place in the middle of that time period of contention.
Using these ground rules, the equation can be restated as
C(0;1-R) = (1-R) .
t T<=
∑ P(0, 1
2
j jt t− +
) [X(0,tj-1) – X(0,tj)]
c) Bootstrapping the Survival Curve
Bootstrapping procedure is commonly used to build a survival curve in practice. Let us
assume that we observe CDSs of maturities T1, T2, T3…TN in the market. The survival
curve is essentially a chart of λ versus time. It is customary to treat it piecewise constant,
although in some cases piecewise linear may make more sense. Bootstrapping works with
the following steps:
1) Assume λ(1) is the intensity for the survival curve from 0 to T1. Using this
assumption of λ(1), we calculate the fair price of a CDS of maturity T1.
2) If the calculated fair spread is same as the spread observed in the market, we
accept our guess of λ(1). Otherwise we need to make another guess of λ(1) so as
to find a fair price that matches the market. In MATLAB we can simply use the
fzero function to find this root.
3) After we have found λ(1), we make a guess for λ(2), which is the intensity from
T1 to T2. Using known λ(1), and guessed λ(2), we compute the fair spread for the
CDS with maturity T2. If the calculated fair price matches that seen in the market,
we accept our guess of λ(2) or else we use some root finding method to find the
value of λ(2) that matched the observed CDS price.
4) We keep doing this for subsequent λ’s.

9. Construction of the Business Time Path
The business time process was modeled as two gamma processes ( Γ) and a drift as
follows,
),(),( 221 1
λγλγ ttatIt Γ+Γ+=
A gamma process is a positive, pure jump process where the t∆ increments are
independently gamma distributed as ),;( λγ txf ∆ . The parameter γ controls the rate of
jump arrivals and λ inversely controls the size of the jumps. The naïve way to generate
a gamma process is to break up an interval into small subintervals and draw gamma
random variables ~ ),;( λγ txf ∆ for each period t∆ . However, this would be
computationally intensive since one would have to generate gamma random variables,
which are typically obtained via acceptance-rejection methods.
The gamma probability density function is defined,
)(
)(
),;(
1
γ
λλ
λγ
λγ
Γ
=
−− x
ex
xf
…where )(γΓ is the gamma function, which can be thought of as a function that “fills
in” the factorial function. If z is an integer, then,
!)1( zz =+Γ
Note that if γ =1, then the gamma density reduces to the exponential density with rate λ.
x
exf λ
λλ −
=),1;(
Furthermore, a gamma distributed random variable can be thought of as the sum of γ
exponential random variables, even though γ is not restricted to being a whole number.
From Cont and Tankov1
we have the following, more efficient, expression for the gamma
process as an infinite series:
∑
∑
=
∞
=
≤
Γ−−
=Γ
=
i
j
ii
i
TUit
T
VeX i
i
1
1
/1
1γ
λ
1
Rama Cont and Peter Tankov, Financial Modeling with Jump Processes (Chapman and Hall, 2003)

10. Ti, Vi – standard exponential random variable (contribute to determination of jump size)
Ui – standard uniform (determines the jump time)
Cont and Tankov suggest truncating the series at a point when Γi exceeds a threshold τ.
Joshi suggests that the remaining small jumps can then be approximated by a drift term.
The algorithm we employed is as follows, from Cont and Tankov, where Tenor is the
length in real time of our gamma process (5 years for a typical CDO, for example.)
Initialize k = 0;
REPEAT WHILE τ<∑=
k
i
iT
1
Set k = k+1;
Simulate kT , kV : standard exponentials.
Simulate kU : uniform [0,Tenor].
END
Set G = cumulative sum of T.
Sort U in ascending order – these can be thought of as the jumping times in a
compound Poisson process.
Sort G in the same order as U.
Then the gamma process increments are:
kie
V
X Tenor
G
i
ti
i
...1,*
==
−
γ
λ
Noticing that the maximum iG will be approximately equal toτ , the terms we are
truncating in this series will be of the order 




 −
Tenor*
exp
γ
τ
or less. What is the total
magnitude of this truncation error? We can express the residual series R as:
∑
∑
∞
+=
Γ−−
=
Γ−−
=
≡
>Γ=+=
1
/1
1
/1
1 };inf{,
ki
i
i
k
i
iT
VeR
ikRVeX
i
i
γ
γ
λ
τλ
The expected value of the residual can be computed. This is the magnitude of the drift
term. Given that Γk is the first point at which the sum of exponential variables exceeds

11. the threshold τ we can approximate it as τ+δ. What is the expected value of δ? In fact δ
is also exponentially distributed with mean 1. This is related to the memorylessness
property of exponential random variables: given that we have waited to a certain point in
time the expected time till the next arrival is unchanged. We therefore approximate Γk as
τ+1.
λ
γ
γ
γ
λ
λ
λ
λ
γτ
γτ
γγ
γ
γ
/)1(
1
/)1(1
1
//1
1
/1
1
/1
1
][
][
][][][
+−
∞
=
+−−
∞
=
−Γ−−
∞
+=
Γ−−
∞
+=
Γ−−
=






+
=
=
=
=
∑
∑
∑
∑
+
e
e
eEe
eE
VEeERE
i
i
i
i
T
ki
ki
i
kik
i
i
Recall that all of the random variables are independent which allows us to factorize the
expectations in the first and third lines. The fourth line follows by substituting into the
moment generating function of a standard exponential (1 – t/λ)-1
with t = 1/γ. The fifth
line is a standard result for geometric series.
What τ should we choose if we’d like to use a constant compensating drift b for all paths?
Equating E[R] to b gives the following expression:
1log −





−=
γ
λ
γτ
b
b
For generating gamma paths to an arbitrary time T, substitute γ with γT.
Business Time Sample Results
The function for generating Business Time Paths was tested by ensuring that the means
and variances were correct. Sample results are shown below for the following parameters:
1.,3.,1,5. 2121 ==== λλγγ , drift a = 1, Tenor = 5, number of paths = 100,000.
Sample Mean Std Err of the Mean Expected Mean
63.2672 0.0723 63.3333
Sample Var Expected Var
522.3126 527.7778

12. Sample Business Time Plots are shown in Figure 1 & Figure 2. Notice that the higherγ
in the second plot results in an increased number of recorded jumps (the actual number of
jumps is infinite, of course.)
Figure 1: Sample Business Time Path.

13. Figure 2: Sample Business Time Path
The truncation error was added back to the process in order to ensure the final Business
Time distribution had the correct mean. How much computation time was saved by this
implementation? As noted above, the truncation error was controlled by setting τ
appropriately. Table 1 shows that the speed enhancement due to this method was
approximately 20%.
Time to Generate All Paths
No Truncation Error Adjustment, set Error = 0.001 42 seconds
With Truncation Error Adjustment, set Error = 0.05 34 seconds
Table 1: The table shows the speed enhancement achieved by truncating the gamma process earlier and
adding in the truncation error correction term. Parameters: 1.,3.,1,5. 2121 ==== λλγγ , drift a = 1,
Tenor = 5, number of paths = 100,000. Note that setting the truncation error to 0.05 is relative to the
timescale of CDO Tenor, which is 5 in this case. So 0.05 represents 1% error.

14. Calculating IG forward intensities (ci)
As discussed above the survival curve construction process produces a set of piecewise
constant λ values for each name in the portfolio. These represent forward default
intensities or hazard rate for the company. With the standard assumption of Poisson
arrivals, survival probabilities for each name i are constructed using the expression:
∫
=
−
T
i dtt
eTX 0
)(
),0(
λ
One way of looking at this is that the survival probability of each company decays at a
rate proportional to time. That proportion is lambda. In the IG model survival
probabilities decay at a rate proportional to business time. The proportion is c. In IG
survival probabilities for each name i are constructed using the expression:
∫
=
−
T
ti dItc
eTX 0
)(
),0(
Here dIt represents the incremental arrival of information or business time. Calculating
the appropriate value for ci relies on the fact that these quantities must be equal. Put
another way the survival probability from the IG model must be consistent with the
survival probability implied by the market. If it were not so, the calibrating instruments,
single-name CDS, would not be priced consistently with the market: a fatal flaw for any
model.
Although we could calculate ci from the survival probability itself, it is much more
convenient to use the λ values already calculated. Since they are piecewise constant the
first integral above can be replaced with a summation and the survival probability is seen
as a product of conditional survival probabilities determined by the λ on each interval.
The IG ci values are assumed piecewise constant on the same intervals. This leads to the
following identity for c in terms of λ for an arbitrary interval from T1 to T2. To minimize
confusion λ̃ refers to the hazard rate and λi refers to the parameter of the ith gamma
process.

15. ∑
∏
=
=
−−−−






+
−=∴






+
==






−
≡
++=−
−≡
2
1
2
1
)(
~
221112
12
/1
1
log
~
][
)),((
),(),(
12
i i
i
i i
icaIIc
TT
c
ca
c
eeEe
t
GammaMGF
GammaGammaaII
TT
i
TT
λ
γλ
λ
λ
λ
λ
λγ
λτγλτγτ
τ
τγ
ττλ
γ
A subtle point noted by Joshi is that one of the parameters is in fact redundant. If the
parameter a is multiplied by m and each λi is divided by m this is equivalent to
multiplying each gamma path by m. Looking at the above we see that the resulting c
values will simply be divided through by m. This means that defaults will occur at the
same times and prices will be the same for the parameterization (a, γ1, λ1, γ2, λ2) as for
(ma, γ1, λ1/m, γ2, λ2/m). Thus a can be set to 1 without losing any flexibility in the model.
The identity above does not have a simple closed form for c and must be solved using a
zero-finding algorithm. The calculation must be done for each name and for each
piecewise constant interval. Since the ci values are a function of the IG parameters, this
inner calibration must be performed for each iteration of the outer calibration which
attempts to find IG parameters which consistently price different CDO tranches. Thus it
is important that it be done efficiently.
MATLAB’s fzero routine offers a simple solution for zero-finding. However it has the
limitation that it is not vectorized. Instead we implemented a parallel bisection method
which finds the zero for all names and intervals simultaneously. This is possible because
each of the calculations is completely independent. The algorithm is exactly identical to
standard bisection with the exception that each operation is performed on the entire array
of values therefore it has the performance of the longest individual iteration. This can be
compared to the serial method whose execution time is the sum of each individual
iteration. This method allows us to exploit MATLAB’s powerful matrix routines. The
main loop from ig_intensity2.m is shown below:
for k = 1:maxiter
cmid = (cmin + cmax)/2;
vmid = f(cmid);
rootfound = (vmid == 0);
cmin(rootfound) = cmid(rootfound);
cmax(rootfound) = cmid(rootfound);
% root is below midpoint
rootbelow = (vmax.*vmid > 0);
cmax(rootbelow) = cmid(rootbelow);
vmax(rootbelow) = vmid(rootbelow);
% root is above midpoint
rootabove = ~rootfound & ~rootbelow;
cmin(rootabove) = cmid(rootabove);
vmin(rootabove) = vmid(rootabove);

16. % global exit condition
if (max(max(cmax-cmin)) < tol)
% disp(sprintf('error < tol k = %gn', k));
break
end
end
c = (cmax + cmin)/2;
A similar technique is applied to speed up the calculation of the λ values. In that case all
values cannot be calculated simultaneously since the value of λ at a given time depends
all previous λ. The parallel bisection relies on the values being independent. Instead a
bootstrapping mechanism is employed in which we step forward through time calculating
the hazard rate for all names simultaneously. This code is implemented in
intensity_bisection.m and intensity_curve_test.m. It also serves to validate the results of
the survival curve generator discussed above since they should both produce identical
results.
Getting the Default Times from the Business Time Paths
In Joshi’s Intensity-Gamma Model, )(tci refers to the default rate for name i per unit of
information arrival. Therefore the probability of survival for a name i is:








−= ∫
T
o
ti dItctX )(exp)(
Since )(tci is piecewise constant with respect to t (real time), we can calculate default
times using the customary method of drawing a uniform random variable U~[0,1] and
setting it equal to the survival probability. Then, the default time iτ can be found by from
the following formula








<−= ∫
T
o
tiii dItcUT )()log(:minτ
Since the )(tci are piecewise constant, the integration must be performed in parts which
can take time, but the process was sped up by noting that most names will not default for
any given path. Thus, we first assume the )(tci are constant and assign each name its
maximum value. Then the earliest business time that a default could have occurred is:
max,
min,
)log(
i
i
i
c
U
I
−
=
If this is greater than the maximum business time, TenorI , then no default could have
occurred within the timeframe of the CDO.

17. CDO pricer
The CDO pricer code was adapted from C code written by Aloke Mukherjee for the class
Interest Rate and Credit Models implementing a Monte Carlo simulator of the one-factor
Gaussian copula model. The MATLAB port of this code is cdo.m. The default time
generation was removed and only the pricing portion of the code was retained in igcdo.m.
This is a misnomer since the code is not specific to IG: it takes the model-dependent
simulated default time scenarios as an argument.
Given this (sorted) set of times we simply simulate the effect of each firm’s default on the
fixed and floating leg cashflows. Default requires a payment on the fixed leg of the
notional associated with the affected fraction of the tranche. This payment must be
discounted to the present from the time of default. On the floating leg, default reduces
the amount of tranche notional on which the coupon is payed. Calculating the PV of the
fixed leg is done by applying the appropriate discount factor from default time to the
present weighted by the proportion of affected notional. For the floating leg we keep
track of remaining notional in each discretized time interval and make the simplifying
assumption that the coupon is payed on the average notional over each period. The break
even spread for the tranche is then calculated as the fixed spread which would make the
expected value of each leg equal: Cbe = E[Vfix]/(E[Vflt] / c) where c is the coupon
assumed in the valuation of the fixed leg.
Validation
Two important validations were performed. First we verify that we can reprice CDS
consistently with the input spreads. Secondly we use an analytic approximation
suggested in the paper to verify the Monte Carlo simulation.
Roundtrip Test
The model must reproduce the input CDS spreads. The roundtrip test (roundtriptest.m)
verifies this by assuming a CDS with a flat spread, recovering the associated (flat)
intensities and then using these to price the CDS using the intensity gamma model. A
CDS can be thought of as a CDO with only one firm and attachment and detachment
points of 0% and (1 – Recovery Rate)% respectively. Additionally, coupons are paid on
the entire notional until default at which point the coupons cease: this logic is triggered in
the pricer if the number of firms is one.
We verify that the output spread matches the input spread. In addition, the IG pricer
outputs a default time for each path. We can use these default times to construct a
survival curve simply by summing the number of defaults before a given time and
dividing by the total number of paths. This survival curve should be consistent with the
survival curve created when bootstrapping the default intensities. A sample run is shown

18. below as well as a graph of the survival curve. For comparison purposes the results for
the one-factor Gaussian copula model simulator are also shown.
EDU>> help roundtriptest
function pass = roundtriptest(spreadbps, paths);
Test pricer and intermediate steps by seeing whether we
recover the input spread.
spread -> lambda -> igprice -> spread
input:
spreadbps - a flat spread in bps - defaults to 40
paths - number of monte carlo paths to simulate - defaults to 100000
output:
pass - 1 if successful
Also produces a graph comparing the implied survival probabilities
and survival probabilities calculated from the generated default times.
2006 aloke mukherjee
EDU>> roundtriptest(100,100000);
closed form vfix = 0.0421812, vflt = 0.0421812
Gaussian vfix = 0.0422499, vflt = 0.0428865
IG vfix = 0.0429348, vflt = 0.0422907
input spread = 100, gaussian spread = 101.507, IG spread = 98.4998
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Implied survival curve
Gaussian copula
Intensity Gamma

19. A Fast Analytic Price Approximation in the Intensity-Gamma
Framework
Joshi and Stacey suggest a clever method for obtaining a fast, analytic price for a CDO
by making some approximations. The default intensities λi are assumed to be constant
over the period of the CDO and are calculated to match the survival probabilities at
maturity. For example, λi = (5 year CDS spread for name i) / (1-Recovery). The business
time default intensities ci were then derived in the regular way. The probability that a
specific name survives until the CDO’s maturity, given a business time IT, is
Ti Ic
e−
.
Then, the probability that there were exactly k defaults, given a business time IT, can be
calculated quickly. The method employed to accomplish this task was borrowed from the
Large Pool Model from CDO pricing.
Let Xi be the probability that firm i survives to business time IT, and P(k,n) be the
probability that k firms have defaulted by time IT, given that n firms existed at time zero.
• Start with 1 firm: P(k=0, n=1) = Xi
P(k=1, n=1) = (1-X1)
• Add 1 firm: P(k=0, n=2) = P(k=0,n=1) * X2
P(k=1, n=2) = P(k=0,n=1) * (1-X2) + P(k=1,n=1)*X2
P(k=2, n=2) = P(k=1,n=1) *(1-X2)
• And so on…
We assume that all defaults occur exactly at the midpoint of the CDO’s lifetime. Then we
can price the floating leg and the fixed leg of a CDO by integrating over all possible
business times as follows.
dIIIpIkKpkKvfltVfloat T
N
k
T )(*)|(*)(
0 1
=





=== ∫ ∑
∞
=
For a business time process consisting of one gamma process (and possibly a drift term),
the probability density of business times is trivially a gamma distribution. (What is the
PDF for two gamma processes?) We tested this method and compared the results to our
main Intensity Gamma CDO pricer in Table 2, Table 3, and Table 4. The results show
that the approximate prices are not terribly accurate. However, Joshi, suggest using it as a
control variate for performance improvement.
Table 3 is particularly informative, because it compares the two pricing methods while
using the same constant default intensities. Thus, the pricing differences are due solely to
the effect of assuming all defaults occur exactly in the middle of the CDO’s lifetime. It
seems that only the equity tranche is substantially affected by this.

20. Table 4 shows the results if, in the Approximate Intensity Gamma Pricer, that defaults are
evenly distributed over the CDO’s lifetime. The prices match almost exactly, as we
would expect.
TRANCHE Approximate Intensity-
Gamma Price (bps)
Complete Intensity Gamma
Price (bps)
0-3% 1429 1778
3-7% 135 187
7-10% 14 29
10-15% 1 5
15-30% 0 0
Table 2: Results from pricing CDO tranches using the Intensity Gamma method compared
to an approximate method. Actual default intensities curves were used for the
Complete Intensity Gamma pricing. The parameters used were: γ=.1, λ=.1, a=1.0, rfr = .05.
TRANCHE Approximate Intensity-
Gamma Price (bps)
Complete Intensity Gamma
Price (bps)
0-3% 1429 1573
3-7% 135 133
7-10% 14 13
10-15% 1 1
15-30% 0 0
Table 3. Results from pricing CDO tranches using the Intensity Gamma method compared
to an approximate method. In this case, the same approximate default intensities
were used in both runs, along with the same parameters: γ=.1, λ=.1, a=1.0, rfr = .05.
TRANCHE Approximate Intensity-
Gamma Price (bps)
Complete Intensity Gamma
Price (bps)
0-3% 1584 1573
3-7% 144 133
7-10% 14 13
10-15% 1 1
15-30% 0 0
Table 4:Results from pricing CDO tranches using the Intensity Gamma method compared
to an approximate method. In this case, the same approximate default intensities
were used in both runs, and the defaults were assume to be evenly distributed over the CDO’s
lifetime. The parameters were the same: γ=.1, λ=.1, a=1.0, rfr = .05.

21. Calibration
We chose to test the good fitting of correlation skew using two gamma processes. Thus
we tried to replicate market spreads with a two gamma processes CDO pricer.
There is a redundancy in the parameters needed to calibrate two gamma processes with
drift. Indeed when we multiply the multi gamma process drift a by some factor m and
dividing each lambda by the same m, is equivalent to multiplying the gamma paths by m.
Thus we also multiply all the individual names default rate by m and lose our calibration
of default rate. To avoid such a problem we set a = 1. Now there remain four parameters
to estimate. Our goal is to minimize the squared difference of market prices and
simulated prices. Mark Joshi advises to use a Downhill Simplex Method, however due to
the variability of results from our simulation we considered that a noisy optimization
method would be more suited. As we didn’t have any idea of the scale of parameters to
choose, we needed an optimization method that would allow a large search space.
Because of long computation times, genetic algorithm was not suited for this
optimization. We chose to use a Simulated Annealing algorithm, described in Wikipedia
as:
“The name and inspiration come from annealing in metallurgy, a technique involving
heating and controlled cooling of a material to increase the size of its crystals and reduce
their defects. The heat causes the atoms to become unstuck from their initial positions (a
local minimum of the internal energy) and wander randomly through states of higher
energy; the slow cooling gives them more chances of finding configurations with lower
internal energy than the initial one.
By analogy with this physical process, each step of the SA algorithm replaces the current
solution by a random "nearby" solution, chosen with a probability that depends on the
difference between the corresponding function values and on a global parameter T (called
the temperature), that is gradually decreased during the process. The dependency is such
that the current solution changes almost randomly when T is large, but increasingly
"downhill" as T goes to zero. The allowance for "uphill" moves saves the method from
becoming stuck at local minima—which are the bane of greedier methods.”
We first calibrated a one gamma process to fit our CDO spreads then used those
parameters as initial estimates for a two gamma processes optimization. Because of our
pricer computation time, the calibration took 48 hours to get a good approximation of a
standard 125 names 5 tranches 5 year maturity CDO. Mark Joshi claims that his C++
pricer takes 5 seconds to run a similar calibration. It might come from the fact that we use
a matlab pricer. However, it is hardly conceivable.
After getting implied base correlation from the resulting CDO spreads, we compare the
market and simulated base correlation skew.

22. Comparison of Base Correlations
0,00%
20,00%
40,00%
60,00%
80,00%
100,00%
120,00%
0-3% 3-7% 7-10% 10-15% 15-30%
Tranches
BaseCorrelation
Market
Base
Correlation
Simulated
Base
Correlation
The skews do not perfectly match. However, we do obtain a correlation skew with this
method. We might think that our calibration was not perfect or we need to add a third
gamma process. However in our current computation time, adding a third gamma process
is forbidden.
Future work
We successfully implemented the Intensity Gamma model and calibrated it to market
prices. The implementation was validated by repricing CDS and by checking against an
analytic approximation. There are a few avenues for improvement.
The performance of the current implementation could be sped up by using the
homogeneous portfolio approximation described above as a control variate. This along
with the application of quasi-random numbers are both identified in the paper as sources
for performance improvement. Additionally it is possible that there are improvements to
be made by vectorizing more of the implementation but it is already quite parallelized.
Two more advanced techniques to extend the model are also outlined in the paper.
Observing that using IG to price for products of different maturities than the calibrating
instruments produce a greater decrease in correlation than implied by the market, the
authors suggest introducing a random delay between the arrival of a sufficient amount of
information to trigger default and the default itself. Another enhancement to the model
would be to allow different gamma processes for modeling sector or geographic effects.

23. References
Li, David X. (2000) On Default Correlation: A Copula Function Approach. Journal of
Fixed Income, March 2000, pp 41-50.
Joshi, M. and Stacey, A. (2005) Intensity Gamma: A new approach to pricing
portfolio credit derivatives. www.quarchome.org/ig.pdf
D. Madan, P. Carr, E.C. Chang. (1998) The Variance Gamma process and option pricing,
European Finance Review 2 (1)