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Rough Lecture Plan
• Lecture 1: Basic credit derivatives concepts;
reduced-form and structural models
• Lecture 2: Bonds, credit swaps, and survival
curve construction
• Lecture 3: Credit indices and option products
• Lecture 4: More options. Default co-dependence
and copulas.
• Lecture 5: CDOs and CDOˆ2s
• Lecture 6: More on CDOs and CDOˆ2s
• Lecture 7: Advanced topics and/or review
2-3 homeworks + one computer project
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Lecture 1: Basic credit derivativesconcepts
Leif Andersen
Banc of America Securities
Spring 2006
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Introduction
• Credit derivatives are financial securities that
facilitate transfer of credit risk from one agent
to another
• In a nutshell, one agent pays a fee (periodic or
upfront) in return for some type of structured
payment when either one or more reference
firms default
• The exact notion of what constitutes default
(outright bankruptcy, debt restructuring, fail-
ure to pay) is governed by fairly rigid legal lan-
guage. A precise definition is typicall not nec-
essary for pricing work, so we will not spend
any further time on this
• Suffice to think of default as a one-time event
that will “kill off” any firm sooner or later.
• A simple credit derivative: a coupon bond.
Bond holder is paid a coupon above the risk-
free rate in return for taking on the risk of
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the issuing firm defaulting on its bonds. The
higher this risk, the bigger the coupon, all else
equal. (A bond also has rates risk – so it’s a
hybrid derivative, really).
Simple Notions of Default Risk.
Transition Matrices
• Defaults are generally infrequent, low-probability
digital events, for which empirical data is some-
what thin. No direct empirical default data
(obviously) is available for still-alive firms.
• But the relative riskiness of firms can still be
gauged a number of ways: balance sheet in-
formation, debt rating, financing spreads, etc.
• For instance, the Altman Z-score is a standard
way to use financial statements to gauge credit
risk
• Most trading firms let ratings agencies do this
type of analysis, and rely on firms ratings as
rough-and-ready prediction of historical (notrisk risk-neutral) default probabilities
• A convenient way to work with ratings datais through published ratings transition matrix,typically spanning a period of 1 year. Hereis a typical example (consistent with Moody’sdata)
M(1) =
Grade Aaa Aa A Baa Ba B Default Aaa 91.027% 6.998% 1.003% 0.650% 0.238% 0.059% 0.025% Aa 7.003% 85.823% 5.997% 0.704% 0.266% 0.147% 0.060% A 2.000% 10.865% 80.251% 6.159% 0.397% 0.238% 0.090% Baa 0.299% 0.999% 3.798% 90.624% 3.680% 0.400% 0.200% Ba 0.151% 0.902% 3.701% 7.002% 72.855% 12.889% 2.500% B 0.007% 0.047% 0.217% 0.405% 8.898% 78.849% 11.576% Default 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 100.000%
• Notice that Default is an absorbing state: onceyou reach it, you will be stuck forever
• By the properties of transition matrices, wecan find the n-year transition matrix by matrixmultiplication:
M(n) = M(1)n
• By looking at the right-most column in theresulting matrix, we can get default estimatesat all annual horizons, for all ratings
Generator Matrix
• In a transition-matrix setting, it is often most
convenient to work with a fundamental transi-
tion matrix that is finer than the 1-year matrix
published by ratings agencies
• This is done by using a so-called generator ma-
trix G, defined as
M(T) = eGT = I +∞∑
k=1
GkTk
k!,
where I is the identity matrix. Notice that
M(1) = eG and M(n) = eGn = M(1)n, as
above
• Fitting a generator matrix to historical data
allows us to make statements about default
probabilities at all horizons, not just annual
ones
• The generator matrix used to produce the ta-
ble above is
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G =
Grade Aaa Aa A Baa Ba B Default Aaa -0.0971753 0.0788 0.0087 0.0065 0.0026 0.0004 0.000175 Aa 0.0788 -0.16071 0.072 0.0051 0.0029 0.00143 0.000482 A 0.0182 0.13035 -0.22666 0.0718 0.0031 0.0025 0.000705 Baa 0.0025 0.0083 0.0433 -0.10179 0.0452 0.001 0.001491 Ba 0.0009 0.0079 0.0463 0.0846 -0.32923 0.1711 0.018426 B 0 0 0 0 0.1182 -0.24746 0.129255 Default 0 0 0 0 0 0 0
• A generator matrix with risk-neutral probabil-
ities would imply much higher default proba-
bilites (risk aversion)
Structural Models
• In the dynamic description of the (random)
default time, there are two major classes of
models: structural and reduced-form. The lat-
ter is nearly always the model of choice in a
derivatives trading setting; the former is more
“fundamental” and finds applications in prop
trading or econometrics.
• Structural models are based on balance sheet
information and historically emerged first (they
originate with Robert Merton in the 70’s) –
and will do so as well in this class
Simple Merton Model
• A simple 1-period strutural model considers a
firm with assets V , debt B, and equity S. By
MM theorem
V (t) = B(t) + S(t).
• Consider now a horizon T , and assume (sim-
plistically) that the company is ”left alone”
until time T , at which point it is liquidated
and the proceeds handed out to debt and eq-
uity holders. Also assume that the firms debt
consists of T -maturity zero-coupon bonds with
notional D.
• At time T , the debt holders evidently receive
B(T) = min (D, V (T)) = V (T)− (V (T) − D)+ ,
where we use the notaton x+ = max(x,0).
The equity holders receive
S(T) = (V (T) − D)+ = V (T) − B(T).
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• In the Merton approach we assume that the
assets pay no dividends and satisfy a simple
risk-neutral diffusion
dV (t)/V (t) = rdt + σV dW (t).
• Then we get, from Black-Scholes,
S(0) = V (0)Φ(d+) − De−rTΦ(d−);
B(0) = V (0) − S(0),
where
d± =ln (V (0)/D) +
(r ± 1
2σ2V
)T
σV
√T
• We can use this expression in a number of
ways. First, if we know S(0) and B(0), we
can iterate to find the asset volatility σV . Sec-
ond, if we know σV , we can work backward to
find B(0) (and V (0)), assuming that only S(0)
is known.
• The first application is used in a number of
systems to create a functional link between
S(0) and B(0), used in risk-management ap-
plications (hedge bonds with equity, and vice-
versa). The second is the original idea of the
Merton model and gives us a way to price risky
debt B(0), even . Often we would estimate σV
from the volatility of the equity:
dS(t) = ..dt+σV V (t)dW (t) ≡ ..dt+σS(t)S(t)dW (t)
such that σV V (t) = σS(t)S(t), or
σV ≈ σSS(0)
V (0)≈ σS
S(0)
S(0) + D.
• In the Merton model, the risk-neutral default
probability over the horizon [0, T ] is
Pr (default) = Pr (V (T) < D) = Φ(−d−).
We can calibrate σV to hit this number if we
know it – or we can take it as a fundamental
“structural output” of the model.
Merton Extensions
• One problem with the basic Merton model is
the fact that it is a one-period model and can-
not tell us anything other than default proba-
bilities to a single fixed horizon
• An obvious extension is to introduce a contin-
uous barrier H and have the firm default the
moment the assets hit this critical level. To
describe this, let τ be the default time of the
firm. Then
τ = inf {t > 0 : V (t) ≤ H} .
• With the asset process following dV (t)/V (t) =
µdt + σV dW (t), it can be shown then that
Pr (τ < t) = Φ
(h − at
σV√
t
)+ e2ah/σ2
V Φ
(at + h
σV√
t
),
where a = µ − 0.5σ2V and h = ln (H/V (0)) .
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• While this extension allows for a full term struc-
ture of default probabilities, it is typically not
realistic. For instance, it can be demonstrated
(see Homework 1) that the credit spread term
structure generated by this model has collaps-
ing spreads for short maturities (contrary to
real markets).
• This is essentially a consequence of the fact
that defaults generated by diffusion processes
are “predictable” with slowly deteriorating firm
assets, rather than “surprising” (e.g, as in En-
ron or Parmalat).
More Extensions
• To overcome the problem with collapsing credit
spreads, there has been a number of attempts
to infuse more uncertainty into the barrier model
above. One approach (CreditGrades) assumes
that the barrier H is unobservable at time 0 –
instead we can only see its mean and standard
deviation.
• That means that we can be in default at time
0 (without knowing it). An instant later we
can then default.
• CreditGrades makes no dynamic sense, how-
ever, and credit spreads would be infinitely high
at t = 0 (and then collapse an instant later).
• Better, but more convoluted ideas, exist in the
literature – basically a matter of continuously
withholding information.
• In practice, these models have few, if any, ap-
plications
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• A different extension involves replacing the idea
of a constant barrier with a curved one. There
is no closed-form expression for default prob-
abilities in this approach, so numerical work
(in a grid) must be employed. The resulting
barrier H(t) looks something like this:
t
V(0)
H(t)
• Note that this model has poor stationarity prop-
erties: short-term default spreads will very likely
collapse once time passes
• Finally, some authors have enrichened the ba-
sic model with a more realistic model for lever-
age ratios. In the simple model above, the
leverage ratio of the firm in question is defined
as
L(t) =B(t)
V (t)=
B(t)
B(t) + S(t).
• In our simple models, this ratio can become
very high or very low, depending on how the
value of assets fluctuate. In reality, however,
firms tend to keep their leverage ratios quite
“sticky”: when assets (and stocks) are very
high, they tend to issue more debt; and vice
versa.
• See Goldstein and Colin-Dufresne (JOF, 2001)
for a mean-reverting model that tries to en-
compass this phenomenon. Basic idea is to
make the barrier H a random process, reflect-
ing the fact that debt levels change over time.
Reduced-Form Models
• We are done with our introduction to struc-
tural models,and shall only see them very briefly
again in this class
• We now turn to the type of models that are
used in credit derivatives trading environments:
reduced-form models. Understanding these mod-
els require us to have a good understanding of
Poisson processes and Cox processes, so we
proceed to this.
Poisson Processes
• So far in this class, we have only considered
continuous stochastic processes, most notably
Brownian motion
• We shall now look at a completely different
type of process, useful for processes with jumps
or – as our primary interest shall be – default
events8
• For this, we introduce the Poisson process N(t),
an increasing process that takes only integer
values 0,1,2, ...
• The sample paths of the Poisson process is
a stair-case, with unit-sized jumps at random
times τ1, τ2, ... N(t) 3 2 1 t 1τ 2τ 3τ
• Given that N(t) = n, the probability that N(t+dt) = n + 1 is λdt (and the probability thatN(t + dt) = n is 1 − λdt)
• Here, λ > 0 is the intensity of the Poissonprocess
• Notice that the Poisson process has indepen-dent increments and is Markovian
• It is not difficult to show that (with N(0) = 0)
Pr (N(t) = n) = e−λt(λt)n
n!,
and that
E (N(t)) = λt.
• The Poisson process is memory-less, in thesense that
Pr (N(t + δ) = n + m|N(t) = n) = Pr (N(δ) = m) .
• Let us consider the time of the first jump τ1.Its probability distribution is given by
Pr (τ1 > t) = e−λt
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with first moment
E (N(t)) = 1/λ.
• The density of τ1 can be found by differentia-
tion of the cumulative distribution:
Pr (τ1 ∈ [t, t + dt]) = λe−λt dt.
• The distributions of all interarrival time τ2−τ1,
τ3− τ2, ... are all identical to the distribution of
τ1 (due to the memory-less property).
Non-homogenous and Doubly-Stochastic
Poisson processes
• If the intensity of the Poisson process dependson time (a so-called non-homogeous Poissonprocess), we basically replace in previous re-sults λt with
∫ t0 λ(u)du. Key results are,
Pr (N(t) = n) = e−∫ t0 λ(u) du
(∫ t0 λ(u) du
)n
n!,
E(N(t)) =∫ t
0λ(u) du,
Pr (τ1 > t) = e−∫ t0 λ(u) du,
Pr (τ1 ∈ [t, t + dt]) = λ(t)e−∫ t0 λ(u) du dt
• We can also let the intensity be stochastic.The way this works is that a path of the in-tensity is drawn first, and then – conditionalon this path – we use the results for the non-homogenous Poisson process. For instance,
Pr (τ1 > t| {λ(u),0 ≤ u ≤ t}) = e−∫ t0 λ(u) du.
• By forming the expectation over all paths, weget the unconditional probability as
Pr (τ1 > t) = E(e−∫ t0 λ(u) du
).
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• This “doubly stochastic” non-homogenous Pois-
son process is often known as a Cox process
• Notice the close relation between the intensity
λ and the short rate r in an interest rate model.
Let X(0, T) be the time 0 survival probabil-
ity X(0, T) = Pr(τ > T) and let P(0, T) be
the time 0 discount bond maturing at time T .
Then
X(0, T) = E(e−∫ T0 λ(u) du
)
P(0, T) = E(e−∫ T0 r(u) du
)
• More generally, at time t,
X(t, T) = 1τ>tEt
(e−∫ Tt λ(u) du
)
P(t, T) = Et
(e−∫ Tt r(u) du
)
