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Intensity (and Variants) Gamma
Alan Stacey
(joint work with Mark Joshi and carried out in large
part at the Quantitative Research Centre (QuaRC),
Royal Bank of Scotland)
Credit Risk Under Lévy Models
Edinburgh, September 22nd 2006
The one-factor Gaussian copula
 The joint distribution of default times is determined from
marginal distributions via a Gaussian copula.
 In the one-factor model, conditional on a single Gaussian random
variable, Z, the default times are independent
 A single correlation number, , determines how much the
default times are determined by the value of Z.
 If we restrict our attention to equity tranches, the map from  to
price is strictly decreasing.
 So given the price of the 0 to x% equity tranche, there is a unique
correlation (x%) which gives rise to this price. This is known as
base correlation.
 The map, x→ (x), is the base correlation smile
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A standard but not a model
 This has become the market standard for quoting correlation. The
price of a tranche can be quoted as a spread, or as the value of 
which would imply that spread.
 However, it is very hard to infer new prices within the Gaussian
copula framework. Even arbitrage-free interpolation of the base
correlation curve is very difficult. In practice fairly sophisticated
interpolation and mapping methodologies have been developed to
obtain prices.
 The model is not based on any financial explanation of why
defaults are correlated – it just correlates default times in a naïve
way.
 No dynamics.
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Desiderata of a correlation model
 Calibrates (more or less exactly) to relevant liquid market
instruments:
 single name products (e.g. credit default swaps)
 quoted tranches of standardized baskets (iTraxx, CDX etc.)
 Deduce arbitrage-free prices of non-liquid instruments reasonably
painlessly including
 tranches of standardized baskets with non-standard attachment
points.
 bespoke CDOs with similar characteristics to an index
 hybrid CDOs, e.g. mix of regions or credit quality
 more general portfolio credit derivatives, e.g., CDO2
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Desiderata (2)
 Realistic internal dynamics.
 Stable Greeks with good P&L explanatory power leading to
good hedges.
 Important market changes (e.g. spread widening) taking
place with the model (and hence within-the-model Greek
calculation)
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The basic Intensity Gamma model
 Based on stochastic or business time – the flow of information. If
a lot happens in a given year then each firm has an increased
chance of defaulting.
 One has an increasing business time process, t.
 Name i defaults with a constant hazard rate, ci, but defaults are
driven by the business time process (common to all names) t, not
calendar time.
 So conditional on the process (t), then for S ≤ T, the probability
that a name survives to time T, given that it has survived to S, is
exp(−ci(T− S)).
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The gamma process
 We will take t to be stationary with independent increments. An
increasing process with this property is known as a subordinator.
 The most well-known subordinator is the gamma process. Then t
has a gamma distribution, with parameters t and  (for some ≥0,
>0). This has density function
t
 t 1  x

x e ;
( t )
x0
 It is helpful to think of this as the sum of t independent copies of
an exponential random variable with parameter  (mean 1/). Of
course, this is only strictly true when t is an integer.
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Calibrating to individual default probabilities
 The unconditional survival probability to T of a name which
defaults with business time hazard rate c is just a Laplace
transform: E(−cT).
 If we take business time to be a gamma process, this is just
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(1  c /  )  T
 So calibrating each ci to a survival probability for name i is
immediate.
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Refining the very basic model
 For each name, we wish to match specified survival probabilities
at a few different times.
 We take ci to be a piecewise constant function of calendar time.
 Note that if  in the specification of the gamma process changes,
then each ci changes by the same factor. Effectively we have only
one free parameter for the gamma process.
 We need more flexibility in our model for business time. For
i=0,1, take (it) to be a gamma process with parameters i and i,
the two processes being independent. Then set
t = t0 + t1 + at
for some constant drift a.
 We call this a multigamma process.
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Pricing with intensity gamma
 Given a choice of multigamma parameters we then rapidly
calibrate each ci to the survival probabilities for name i at a small
number of times. These are inferred from CDS prices (and a
recovery rate assumption).
 A product whose payoff is determined by the default times of a
basket of names can then be priced by Monte Carlo.
 Draw a random path for business time, (t).
 Conditional on (t), the default times for each name are
independent. Draw each of these.
 Compute the payoff and discount, assuming deterministic
interest rates.
 Average over many paths.
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Matching the correlation market
 We aim to match the quoted prices of a single index. Prices are
typically quoted for four or five tranches e.g. with detachment
points 3%, 6%, 9%, 12%, 22%.
 Given multigamma parameters, we can price each tranche. We
then find multigamma parameters which best match the market
prices using an optimizer.
 Having chosen the multigamma parameters to match quoted
tranche prices, we then use the same multigamma process for
 non-standard attachment/detachment points
 bespoke baskets with similar properties to the index to which
we have calibrated: same region and similar levels of credit
quality and diversity
 Significantly different maturities turn out to be more difficult.
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Matching an investment grade curve
5-year ITraxx
70%
base correlation
60%
50%
40%
Market
Intensity Gamma
30%
20%
10%
0%
0%
5%
10%
15%
detachment point
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20%
25%
North American High-Yield index
5-year High-Yield
50%
45%
base correlation
40%
35%
30%
market
25%
Intensity Gamma
20%
15%
10%
5%
0%
0%
10%
20%
30%
detachment point
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40%
Extensions to the model (1)
 Some products depend on a basket of names corresponding to
different indices, e.g.,
 High Yield/Investment Grade hybrids
 different regions.
 Divide the names of interest into sub-baskets corresponding to
different indices.
 Calibrate a different multigamma processes to each index.
 The defaults of each sub-basket are driven by the corresponding
multigamma process.
 One needs a way to make the different multigamma processes
strongly correlated.
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Extensions to the model (2)
 Can introduce a random time lag between information arrival and
default. This is more realistic. One way to do this is to have
information arrive as a multigamma process, (t), as before, with
the impact of the information spread out in a way that decays
exponentially with parameter α. We then have a positive residual
information process (Rt) satisfying
dRt  dt  Rt dt
and then an impact process, (It) driving defaults as before
dI t  Rt dt
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Extensions to the model (3)
Retains tractability and rapid calibration to individual names with
benefits including
 no longer have simultaneous defaults, although if there is a big
jump in business time one will get a lot of defaults in a short
space of time
 better matching of the market across different time horizons
 credit spread widening (one-factor only) within the model
 Can use a different class of subordinators, e.g. tempered stable
processes.
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Summary of strengths
 Provided one can match the index prices, one can obtain
arbitrage-free prices for products whose payoffs depend upon the
default times of a basket of names. These are consistent with
 Single-name survival probabilities (typically derived from
CDSs)
 Tranche prices for the corresponding CDO index (and, to some
extent, multiple indices where appropriate).
 Once calibrated to the appropriate index, pricing is rapid and
straightforward. No ad hoc interpolation or curve-mapping
techniques are required.
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Some limitations
 Intensity Gamma is only a default model. It does not model the
dynamics of credit spread movements. Within the model, credit
spreads are deterministic. (In the time-lag extension, however,
systemic movements of spreads do occur.)
 Hedging of spread movements must be outside-the-model.
 Similarly, the multigamma parameters are fixed, but if the index
tranche prices move then they must be re-calibrated.
 Not capable of matching the market prices of correlation products
with different maturities.
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Disclaimer
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