3.5 DPPI: any hidden pricing issue?
So far, we have not raised any pricing nor hedging issue
pertaining to DPPI products. We shall briefly address both points hereafter.
3.5.1 From the investor's perspective
Initially, the return served to the investor can be split
into two separate pieces: the LIBOR/EURIBOR component results from the
structure being long a risk-free bond paying an interbank 3-month rate, while
the extra risky spread of 100 bps is generated by a long leveraged CDS
portfolio. Later on though, the structure may have to realize MtM losses due to
a deleveraging signal sent by the portfolio investment rules: in that case, the
posted collateral must be partly sold in order to pay for the potential loss on
the unwound CDS positions. As a result, the structure is exposed to two major
market risks:
· one is the credit risk inherent in holding long CDS
positions;
· the other is the interest rate risk created by the
potential selling of the collateral before its due term; similarly to what is
being done for CSO structures, such a risk is hedged by entering into 3-month
forward rate agreements rolled every quarter.
As a result, the DPPI's NPV boils down to being equal to the
sum of the par-value of the posted collateral and of the MtM of the long CDS
positions. Moreover, from an investor's perspective, it is sensitive only to
underlying CDS spread levels and not to interest rates, although the
structure's rating will depend on the interest rate curve through its sizing
effect on the Reserve R(t).
3.5.2 From the investment bank's perspective
Unlike CSO tranches, which require delta-hedges to be adapted
dynamically, DPPI
products do not need any hedging as such. The only market
risk that is not passed to
the investor and that is kept within the bank's
book is the so-called «gap» risk: in case
of a cash-out event, the bank guarantees the bond floor value
to the investor while the structure's NPV may have jumped well below that value
due to overnight volatile market conditions. In other words, pricing that
initial risk GR(0) (for notation convenience) requires to evaluate quantities
such as:
GR(0) = E [(BF(ô) - NPV
(ô))1{ô<ô, NPV
(ô)<BF(ô)}B(0,ô)] (3.3)
Given that default processes are pure jump processes, NPV (.)
is not continuous either: consequently, GR(0) is not equal to 0. Writing a
payoff function such as in equation (3.3) is easy but remains theoretical
though: in practise, one would have to assess dependency relationships between
interest rate risk factors, which mainly impact the structure's bond floor
BF(t), and credit spread levels, which drive the structure's NPV. In other
words, pricing each payoff component of the DPPI is a challenging task that can
only be addressed in simplified frameworks such as the one described in [25]
(time-continuous processes with discrete trading dates).
Another way of evaluating that gap risk is to look at the
potential MtM impact of names defaulting in the portfolio when the NPV gets
close to the Bond Floor BF: by construction, the function TM(t)R(t) has a
mitigating impact on that risk: as the Reserve R(t) shrinks, TM(t)R(t) starts
driving down the Target Notional Exposure TNE(t), hence reducing the MtM impact
of any potential default. More generally, assuming all N names in portfolio are
equally weighted with identical deterministic recovery rates RR and that
readjustment bounds lu and ld are set to 0, we can express the MtM
impact of one default ÄMtM(1) as:
1 - RR
ÄMtM(1) = N · TNE(t)
We then want to give a lower bound for ND(t) defined as the
number of defaults the structure can sustain before its reserve R(t) is fully
wiped out:
R(t)
?t ? [0,T], ND(t) : = ÄMtM(1)
R(t)
TNE(t)
N
= 1-RR ·
= N
1 - RR(DR + SR(t) · D(t))
N
= 1-RR · DR
This last term does not depend on t. Assuming RR = 40%, N =
135 and DR = 1%, we find that ?t ? [0, T], ND(t) = 2.25. In other words, at any
point in time, the structure can stand 2 simultaneous defaults before breaking
the bond floor in the worst case scenario.
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