3.2 Modelling risk factors
Moody's Monte-Carlo approach requires risk factors to be
modelled and simulated. The complexity of the task for rating DPPI products
comes from the fact that risk factors are numerous and can depend on each
other. We assume that the DPPI's portfolio is initially composed of long CDS
positions on N obligors. The 5Y CDS mid-spread and the rating of each obligor
n, n ? {1, .., N}, are known at the deal's inception and are equal to
(sn(0),Rn(0)).
3.2.1 Credit spread processes influenced by defaults and
ratings
Moody's assumes that individual 5Y CDS spread processes follow
a generalized Vasicek diffusion process specific to rating groups. The 21
available rating categories are grouped into 8 rating groups {Aaa, Aa, A, Baa,
Ba, B, Caa, Ca/C}. Let j ? {1, .., 8} denote the rating group's index. Then
(S(t) = (S1(t), .., S8(t)))t?[0,T] is the associated 8-dimensional spread
random process. We then give the stochastic differential
equation ruling the spread process (S(t)), where (á,
â, ã = 1, ó, ó, ADR, ADR, a, b, p)
are
historically calibrated parametres of the diffusion process and t0 is equal to
1 year:
?j ? {1, .., 8}, ?t ? [0, T], dSj(t) =
á(â - Sj(t))dt + min(ó,
óSãj (t))dW (j)
t
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where
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? ?????
?????
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â = â min (ADR,max (ADR, aADR(t) + b)) if t =
t0
PN n=1 Ln(Nn(t)--Nn(t--t0))
ADR(t) = PN if t = t0
n=1 An
â = â if t < t0
?(i,j) ? {1, .., 8}2, d (W(i),
W(j)) t = pijdt
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(3.1)
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and Sj(0) = Sj,0
Equation (3.1) is remarkable for several reasons. First, given
that ã = 1 and that the
spread process is continuous, it doesn't
allow credit spreads to be negative. Second,
it models a dependency between
the spread process itself and default events through
the parametres (ADR, ADR) and the variable ADR(t): the
idea is to make the current long term mean 3 depend on the portfolio's Average
Default Rate ADR(t) such that 3 is stressed for a limited time period equal to
t0 after any event of default and tightens in default-free environments. Third,
it accounts for a dependency between the obligor's rating and its spread
process: whenever an obligor's rating group changes, its associated spread
process is updated accordingly. Fourth, the random noise source (Wt = (W1(t),
.., W8(t))) is a correlated 8-dimensional brownian motion.
Such dependency relationships are far from being flawless
though: one could argue that they do not account for default events occuring
outside the portfolio's pool of obligors. One could also demonstrate that
individual spread processes are not driven by their belonging to a rating
group, but more by marketwide, firm or industry specific events that are not
necessarily reflected in a rating change.
In order to simulate a CDS spread for any tenor, Moody's
assumes that the term structure of spreads is deterministic, calibrated on
historical data and specific to each one of the 8 rating groups defined above.
Though such an assumption may seem highly questionable at first and lead to
obvious arbitrage opportunities, Moody's solves the issue by requiring stress
scenarios to be run with a flat term structure while preserving a stressed
Moody's Metric level.
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