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Irreducible Risks of Hedging a Bond with a Default Swap
Vivek Kapoor Volaris Capital Management LLC1
Tis against some men’s principle to pay interest, and seems against others interest to pay the principle.
Benjamin Franklin
As long as the music is playing, you’ve got to get up and dance.
Charles Prince
Professor Larry Summers chides financial valuation as being a limited pursuit with little to offer
beyond the conclusion that one 16-oz ketchup bottle can’t be priced at anything other than the
price of two 8-oz ketchup bottles ([1]). Not impressed by the implied proximity of 16 – (8+8)
with zero, he lamented the lack of any insight that had to offer on what governed the unit price of
ketchup.
Now if one purchased empty 16-oz bottles and squeezed the 8-oz bottles into them, one would
encounter challenges associated with stickiness of the ketchup and the occasional spill and the
uncertainty in being able to redeem the two emptied bottles at the exact price as that of the larger
one. The new bottle would also need a new marketing wrap…. Conducting a ketchup
economics experiment - with sincerity - will involve learning more than zero about ketchup.
Professor Summers’ chide about ketchup economics is still relevant in the world of derivatives.
Analysis of derivatives remains entrenched in a make-belief risk-neutral world. An assumption
of immaculate replication (i.e., zero residual risk) generally precedes analysis of derivatives that
do not address irreducible risks associated with attempted replication. An alternative framework
is demonstrated here that does not start with the zero residual-risk assumption. I present the
problem of hedging a bond with a default swap and highlight the conditions where perfect
replication does not work and irreducible residual risk shows itself while attempting replication.
The resulting framework for calculating residual risks around break-even averages of attempted
replication opens the door to behavioral interpretations of derivative prices that are informed of
both the power and the limitations of attempted replication.
1 Volaris Capital Management LLC is a registered investment adviser. Information presented is for educational purposes only and does not intend to make an offer or solicitation for the sale or purchase of any specific securities, investments, or investment strategies. Investments involve risk and, unless otherwise stated, are not guaranteed. Be sure to first consult with a qualified financial adviser and/or tax professional before implementing any strategy discussed herein. Past performance is not indicative of future performance.
2
1. Introduction
Changing credit quality and interest rates can result in bonds trading at a significant discount or
premium from par. Recoveries in cohorts of similar subordination vary significantly, making it
impossible to have perfect foresight about recovery while entering into a swap agreement. The
variance of the change in wealth of a portfolio of a defaultable bond and purchased default swap
is minimized in a static hedging framework, accounting for recovery uncertainty and differences
between market value and notional value of the bond. The swap-notional that minimizes the
variance of wealth change is assessed along with the zero-mean wealth change based break-even
premium of the default swap. The irreducible risks associated with attempting to hedge a bond
with a default swap are quantified.
To focus on the random default time and the associated random recovery, I simplify other
parameters of the problem. I do not consider interest rate uncertainty. For simplicity of
presentation I do not consider the term structure of interest rates and default probability –
however the hedging framework described here lends itself readily to computations with interest
rate and default probability term structures. The approach outlined here is applicable regardless
of the parameterization for default time and recovery uncertainty. Correlated recovery and
default time is handled in the static hedging framework computationally. For a primer on Credit
Derivatives the reader is referred to Douglas [2007] (reference [2]).
2. Change in Wealth of a Portfolio of a Defaultable Bond and a Default Swap
The formulation below does not rely on any specific probabilistic description of default and
recovery. It provides a framework to assess the optimal hedge, associated break-even average
spread, and it enables assessing residual risks. In contrast, the risk-neutral approach involves
taking expectations of discounted cashflows under a risk-neutral-measure that is fit to observed
prices – a tautology that can’t entertain the notion of irreducible risks.
A risky bond with a par value of n promises to pay a continuous coupon of c dollars per notional
dollar per unit time over the time interval [0, T]. The market value of the defaultable bond is p.
The notional of the CDS is ns and the premium paid to purchase default protection is s dollars per
CDS notional dollar per unit time. The non-defaultable CDS counterparty pays (1 - R)×ns in the
event of default of the bond (Figure 1).
Figure 1. Portfolio of a defaultable bond and a credit default swap (CDS)
-p, c, n
Trading Book -p, c, R x n
Bond
CDS
s
(1- R) ns
3
p initial (t = 0) price of bond being hedged with a credit default swap
dt
time at which bond issuer defaults on its obligations
T time to maturity of bond
TtH d Heaviside function taking a value of 0 if Ttd and 1 otherwise
n notional value of bond
r no-default rate of interest
c annual coupon rate
s annual swap rate
R fraction of bond notional paid by obligor in the event of Ttd
ns credit default swap notional
The change in wealth of the purchaser of the bond follows
d
d
t
rrt
d
T
rrT
db decnRneTtHdecnneTtHpW00
1 (1)
The product of the term involving recovery and time-to-default dictates that with all else equal a
bond investor is better off with a low or negative correlation between time-to-default and
recovery. With positive correlations between these variables, early defaults, in addition to
shutting off the coupon stream, expose the bond investor to lower than average recoveries. I
explore hedging such a defaultable bond by purchasing a CDS.
The wealth change incurred by purchasing CDS protection follows
d
d
t
r
s
rt
sd
T
r
sds desnenRTtHdesnTtHW00
)1(1 (2)
A bond holder attempting to hedge with a CDS incurs wealth change that combines (1) and (2)
d
d
t
r
s
rt
sd
T
r
s
rT
dsb
desncnenRRnTtH
desncnneTtHpWWW
0
0
)1(1
(3)
The time-integral term arises from assessing the present value of ongoing cashflows of coupons
netted with CDS premium payments. That integral term can be explicitly assessed to get
4
r
esncnenRRnTtH
r
esncnneTtHpW
d
d
rt
s
rt
sd
rT
s
rT
d
1)1(1
1
(4)
Par Bond Case
Setting the CDS notional to be equal to the bond notional (ns = n) results in a wealth change that
is independent of bond recovery in (4). On further inspection of (4) it is apparent that when p =
n (i.e., par bond) setting rsc results in 0W for any td and R:
Rtr
ernneTtH
r
ernneTtHnW
d
rtrt
d
rTrT
d
d
d , 01
1
1
(5)
For the par-bond case the CDS with a swap spread of rcs and ns = n enables a zero-risk
portfolio for a bond holder and the default-recovery uncertainty have no bearing on that fact.
General Case
Grouping terms in (4) involving the bond price, bond notional, CDS notional, and product of
CDS notional and spread, yields
ss snCnBnApW (6)
r
ecR eTtH
r
eceTtHA
d
d
rtrt
d
rTrT
d
11
1
drt
d eRTtH B
)1(1
r
eTtH
r
eTtHC
drt
d
rT
d
11
1
It follows from (6) for the average change of wealth
ss snCnBnApW (7)
5
In (7) and elsewhere, quantities with overbars are their statistical averages. For the average
wealth change to be zero the product of the swap spread and swap notional can be expressed as
follows:
C
nBnApsnW s
s
0 (8)
Substituting this into the wealth change expression (i.e., enforcing zero mean) gives for the
deviations of wealth change around the mean
C
nBnApCnBnApW s
s (9)
Grouping terms in (9), involving the bond price, bond notional, and swap notional, separately,
yields
C
CBBn
C
CAAn
C
CpW s1 (10)
We can write (10) succinctly:
snanapaW 321 (11)
C
CBBa
C
CAAa
C
Ca 321 ; ;1
The squared wealth change perturbations around the mean follow:
sss nnaapnaapnaananapaW 323121
22
3
22
2
22
1
2222 (12)
Taking ensemble averages gives the wealth change variance
sssW nnaapnaapnaananapa 323121
22
3
22
2
22
1
2 222 (13)
The rate of change of the wealth change variance with the default swap notional follows
naapaanadn
dsW
s
3231
2
3
2 222 (14)
The wealth change variance extremizing default swap notional is found by setting this to zero
6
2
3
32312 0a
naapaan
dn
dsW
s
(15)
That this extremum is indeed a wealth change variance minimum is ascertained by the second
derivative:
02 2
3
2
2
2
adn
dW
s
(16)
3. Default-Recovery Probabilistic Model
Providing examples of the analysis of Section 2 requires prescribing objective default and
recovery probabilities – and their joint density:
default totimeyProbabilit
default totimeyProbabilitd
d
d
t
t
F
df (17)
RF
dRRRRRf
rec
rec
default on recovery yProbabilit
default on recovery yProbabilitd
(18)
RRF
dRRR
ddRRf
rect
rect
d
d
default on recovery
default totimeyProbabilit,
default on recovery
default totimeyProbabilitd,
(19)
I employ the popular and convenient exponential parameterization for the default time pdf
eF
ef
d
d
t
t
1 (20)
In this model, the mean time to default and the standard deviation of the time to default are the
inverse of the default hazard rate
/1dtdt . (21)
I adopt a beta distribution for recovery
7
,;
111
RRF
RRRf
rec
rec
(22)
where the gamma function is defined as
0
1 dxexz xz and the incomplete beta function is
defined as
R
dxxxR0
11 1,; . The relationship between recovery model parameters
and its mean and standard deviation follows:
1 ;
2
2
RR (23)
1
12
R
RRR
;
1
11
2
R
RRR
Berd &Kapoor [2003] (reference [3]) employed a beta distribution for recovery in addressing
pricing of Digital Default Swaps (DDS) that pay fixed recovery and therefore expose protection
buyers directly to recovery uncertainty. They employed the non-linear dependence of break-
even spread and assumed recovery for a standard default swap when hedging a par-bond to argue
for recovery uncertainty as a risk-premium embedded in digital default swaps. This work
provides a more direct line of attack of the DDS problem addressed based on direct enforcement
of break-even averages, hedge error minimization and residual risk quantification.
There is a case for a positive correlation between time-to-default and recovery – i.e., earlier than
expected defaults to have lower recovery rates compared to later than expected – as discussed by
Altman and coworkers in [4]. In this work the joint density between time-to-default and
recovery is prescribed through the contrivance of a Normal Copula. In this approach standard
correlated Normal Variates u and v with zero mean and unit standard deviation are simulated
with an input correlation coefficient . This can be accomplished based on standard
independent normal variates u and w and prescribing v = wu 21 . Associated with
these correlated variates are the corresponding univariate standard normal cumulative density
functions u and v where
x h
dhex 2
2
2
1
(24)
The corresponding copulated variates of interest are the simulated time to default,
uFdt
1
and recovery vFR rec 1
.
8
4. Sample Results on Hedging Bond With CDS
I present sample calculations to illustrate the approach outlined above, that is based on statistical
modeling and optimization analysis without assuming perfect replication.
Bond Maturity (years) 20
Risk-free rate (1/year) 0.02
Coupon Rate (1/year) 0.06
Price ($) 95
Notional ($) 100
Default Hazard Rate (1/year) 1/40
Mean Recovery Rate 0.4
Recovery Standard Deviation 0.25
Table 1. Parameters of long bond position to be hedged by a default swap.
It is recognized that the default and recovery parameters have to be gleaned from history and
issuer balance sheet information. Unfortunately, there is no free lunch – i.e., if your eyes are
wide open to residual risks then the characteristics of the security underlying the derivative
contract can matter.
This example is used for specificity while I demonstrate my approach to derivatives as a
stochastic modeling and optimization problem with explicit quantification of residual risks. I
show sensitivity to a range of correlation between the time-to-default and recovery and that is
followed by a sensitivity to the bond price.
Long Bond Wealth Change Statistics
The Copula correlation range [-1, +1] maps into a somewhat smaller range of correlations
between time-to-default and recovery: Rtd [-0.83,0.92], as shown in Figure 2. The
asymmetry of the time-do-default probability density and the recovery probability density and
the finite range of recoveries results in an asymmetric and incomplete range of correlations
between these variables.
It is instructive to start with a brief documentation of the change in wealth distribution of the
long bond position holder, including its dependence on the correlation between time to default
and recovery.
9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
de
fau
lt-t
ime
re
cove
ry c
orr
ela
tio
n
copula correlation
Figure 2. Time-to-Default and Recovery Correlation
0
10
20
30
40
50
60
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
% o
f b
on
d n
oti
on
al
default time & recovery correlation
expected change in wealth
std dev of change in wealth
Figure 3. Expectation and Standard Deviation of Change in Wealth vs Correlation for Long
Bond Position (Table 1).
10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-100 -80 -60 -40 -20 0 20 40 60 80 100
(a) 63.0 ;75.0 Rtd
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-100 -80 -60 -40 -20 0 20 40 60 80 100
(b) 00.0 ;00.0 Rtd
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-100 -80 -60 -40 -20 0 20 40 60 80 100
(c) 68.0 ;75.0 Rtd
Figure 4. Probability density (vertical axis) of change in wealth (hor. axis % of notional) of long
bond position (Table 1).
11
The correlation between time-to-default and recovery upon default directly determines the
expectation and standard deviation of wealth change (all else being equal). The higher the
correlation is between time-to-default and recovery upon default is, the lower is the expected
wealth change and the higher is the wealth change variance (Figure 3 and Figure 4). If the
cognition of this correlation is embedded in the market of long bond risk takers then it may be
surmised that the bond price and coupon reflect the effect of such a correlation between time to
default and recovery. While in this work the correlation between time-to-default and recovery is
being imposed at a single bond level, it would be unrealistic to consider this to be a fully
diversifiable risk factor, for it can be argued that epochs of high default rate are concomitant with
low recovery values due to the excess supply of distressed bonds.
Approach to Optimality
0
1
2
3
4
5
6
90 92 94 96 98 100 102 104 106 108 110
std
. dev
. of
hed
ger
chag
e o
f w
ealt
h(%
of
bo
nd
no
tio
nal
)
CDS hedge notional (% bond notional)
irreducible hedging error
Figure 5. Hedged Wealth Change Standard Deviation versus CDS hedge notional ( = +0.75)
The hedge notional that minimizes the bond hedger’s variance in change in wealth is directly
found in equation (15). The approach to optimality can be explored by evaluation of the wealth
change variance in equation (13) for a range of CDS notional, as shown in Figure 5. That the
bond being hedged is not a par bond has precluded a perfect hedge. The optimal risk minimizing
(as opposed to immaculate risk eliminating) hedging notional, the zero-average wealth change
spread, and the residual risks are shown in Figure 6.
The entity holding an optimally hedged bond position (long or short) using a CDS (bought or
sold) requires risk capital that can be quantified by the approach to derivatives shown here.
Despite the inability to eliminate uncertainty completely, the reduction in uncertainty is palpable.
For the case shown in Figure 5, the unhedged long bond had a wealth change standard deviation
of 53.5% of its notional, that was reduced to 62 basis points!
12
Optimally Hedged Bond Wealth Change Statistics
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
irre
du
cib
le w
ealt
h c
ha
nge
std
. d
ev
(% b
on
d n
oti
on
al)
default time and recovery correlation
(a) Irreducible standard deviation of change in wealth (% of bond notional)
91
92
93
94
95
96
97
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
vari
an
ce o
pti
ma
l CD
S h
edge
no
tio
na
l
(% o
f b
on
d n
oti
on
al)
default time and recovery correlation
(b) Variance Optimal CDS hedge notional (% bond notional)
0.0445
0.045
0.0455
0.046
0.0465
0.047
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
CD
S sp
rea
d fo
r ze
ro a
vera
ge c
ha
ge i
n
wea
lth
(1
/yr)
default time and recovery correlation
(c) CDS spread for zero average wealth change
Figure 6. Characteristics of an optimally hedged long bond position with a CDS
13
0
2
4
6
8
10
12
14
16
18
20
-5 -4 -3 -2 -1 0 1 2 3 4 5
(a) 63.0 ;75.0 Rtd
0
2
4
6
8
10
12
14
16
18
20
-5 -4 -3 -2 -1 0 1 2 3 4 5
(b) 00.0 ;00.0 Rtd
0
2
4
6
8
10
12
14
16
18
20
-5 -4 -3 -2 -1 0 1 2 3 4 5
(c) 68.0 ;75.0 Rtd
Figure 7. Probability density (vertical axis) of change in wealth (hor. axis % of notional) of a
variance optimally hedged long bond position with CDS.
14
Sensitivity to Deviations From Par Bond
0
1
2
3
4
5
6
90 92 94 96 98 100 102 104 106 108 110std.
dev
. of
hedg
er c
hang
e in
wea
lth
(% o
f b
on
d n
oti
on
al)
CDS hedge notional (% of bond notional)
bond price 95% of notional
bond price 100 % of notional
Figure 8. Hedged Wealth Change Standard Deviation versus CDS hedge notional for discount
and par bond ( = +0.75)
The variance optimal CDS notional was found to be closer to the price of the bond rather than its
notional value (Figure 8, Figure 9a). The irreducible hedging error increases linearly with the
absolute deviation of price from par (Figure 9b). The corresponding CDS spread, that renders
the hedged portfolio to be zero expected wealth change, varies inversely with the optimal CDS
hedge notional (Figure 9c). In the face of irreducible risks, it can be expected that the market
CDS spreads will reflect demand supply dynamics and the irreducible risks. If there is an excess
of natural demand for a default swap than there are natural suppliers, the supplier’s aspiration for
return on their risk-capital2 will be an important input into the pricing dynamics.
In Appendix-A of their work on irreducible risks of CDOs, Kapoor and co-workers [5] have
provided analytical expressions for the variance optimal CDS hedge notional, the CDS spread
corresponding to zero average wealth-change, and the residual risk for the uncorrelated recovery
and time-to-default case. That work replicates the classical risk neutral results for the par bond
case – but also quantifies irreducible variance of change in wealth for non-par bonds amidst
recovery uncertainty. This paper extends [5] to account for correlation between recovery and
time to default, and to Digital Default Swaps in the next section.
2Lack of assessment of risk-capital can be consequential – as seen in the financial crisis of 2007-2008.
15
95
96
97
98
99
100
101
102
103
104
105
94 96 98 100 102 104 106
Var
ian
ce O
pti
mal
CD
S H
ed
ge
No
tio
nal
(% o
f bo
nd
no
tio
nal
)
bond price (% of bond notional)
(a) Variance Optimal CDS Hedge Notional (% of bond notional)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
94 96 98 100 102 104 106
Irre
du
cib
le S
td D
ev
of C
han
ge in
W
eal
th (%
bo
nd
no
tio
nal
)
bond price (% of bond notional)
(b) Irreducible Wealth Change Standard Deviation (% of bond notional)
0.034
0.036
0.038
0.04
0.042
0.044
0.046
94 96 98 100 102 104 106CD
S Sp
read
for Z
ero
Ave
rage
Ch
ange
in
We
alth
(1/y
r)
bond price (% of bond notional)
(c) CDS Spread for Zero Average Change in Wealth (1/yr)
Figure 9. Sensitivity of optimally hedged bond (Table 1) to price of bond ( = +0.75)
16
5. Digital Default Swap
While a non-par bond can’t be perfectly hedged by a CDS that only has running swap payments,
the sensitivity to recovery is muted because the CDS contract pays the floating loss. If the
discount from par was paid by the bond holder to the CDS protection seller then a risk-free
contract could be constructed – as indicated by the wealth balance equation (4). As such
recovery uncertainty plays a secondary role in that setting.
A type of default swap has a fixed payoff of the swap notional – a Digital Default Swap (DDS).
Instead of paying out floating losses relative to notional values, a DDS pays a fixed amount,
equal to the swap notional.
Figure 8. Portfolio of a defaultable bond and a digital default swap (DDS)
The formulation made in Section 2 is directly applicable to hedging a bond with a DDS – by
simply replacing (1-R)ns by ns in describing the payoff of the swap under default. By simply
replacing (1-R) by 1 in the formulation of the intermediate variable B in equation (6), it is
directly applicable to a DDS. Results are presented for the bond described in Table 1. Results
for a discount bond and par bond and for 0 and 0.75 copula correlation to correlate time-to-
default and recovery are shown below:
Item/Casep = $100
ρ = 0.00
p = $100
ρ = 0.75
p = $95
ρ = 0.00
p = $95
ρ = 0.75
Optimal Hedge Notional
(% of bond notional)60 81.09 56.63 77.72
Irreducible Weal Change std. dev.
(% of bond notional)13.25 8.38 13.27 8.08
Zero Average Wealth Change DDS Spread
(1/yr)
0.0667 0.0499 0.0758 0.0558
Table 2. Sample Results for DDS: Impact of Deviation from Par and Correlation
-p, c , n
Trading Book -p, c, Rxn
DDS
s
ns
Bond
17
0
5
10
15
20
25
30
35
40
50 60 70 80 90 100 110 120
std
. de
v. o
f h
ed
ge
r ch
an
ge
in
we
alt
h (
% o
f b
on
d n
oti
on
al)
DDS hedge notional (% bond notional)
p = $100; ρ = 0.00
p = $100; ρ = 0.75
p = $95; ρ = 0.00
p = $95; ρ = 0.75
Figure 10. Hedged Wealth Change Standard Deviation versus DDS hedge notional
Table 2 and Figure 10 show that the residual risk while hedging with a digital default swap is
much larger than that found using a CDS. For the uncorrelated recovery and time-to-default case
the variance optimal hedge DDS notional sn̂ is close to the bond price p multiplied by R1 .
For positively correlated recovery and time-to-default the variance optimal hedge DDS notional
sn̂ can be much larger than p multiplied by R1 . This is because for a fixed maturity, not
much larger than the average time-to-default, (20 years in sample bond of Table 1, with an
average time-to-default of 40 years) one expects to see smaller than average recoveries for
pertinent defaults (i.e., time-to-default less than bond life) if there is a significant positive
correlation between recovery and time-to-default.
The DDS spread that results in a zero-mean-wealth change is approximately snnrc ˆ/)( . The
large residual risk in the case of DDS indicates that actual pricing will be strongly a function of
demand supply dynamics, and can be far away from the breakeven average spreads.
18
6. Summary
Increasing the correlation between time-to-default and recovery results in a less attractive bond
(all else being equal) by lowering the expected wealth change and increasing its uncertainty.
One can surmise that in a developed bond market it is likely that the bond price and coupon
jointly reflect the correlation between time-to-default and recovery. Increasing the time-to-
default and recovery correlation was found to result in lower irreducible risks associated with
hedging a non-par bond.
The variance optimal hedge ratio for a credit-default-swap (CDS) hedging a bond was found to
be closer to the bond price rather than its notional value. The idealized par bond case results are
a subset of the more general framework presented here. The digital default swap (DDS) exposes
the hedger to higher irreducible risks and a higher sensitivity to recovery-time-to-default
correlation.
Variance optimal hedging was pursued here. Its analytical formulation is simpler compared to
minimizing other hedge error metrics – that can be pursued computationally. My purpose is to
demonstrate a framework of examining hedge errors, that have remained undocumented in risk
neutral analysis that simply equates cashflow averages under a risk-neutral measure fit to
derivative prices. Given that the wealth change distribution of a long bond is highly asymmetric
and that even the residual wealth change distribution is not necessarily symmetric, other hedge
error metrics should also be pursued (e.g., downside deviation, confidence level specific losses,
etc.). I believe assessing residual risks of attempted replications schemes should be the central
tenet of analysis of derivatives – in addition to assessing the average cost of attempted
replication. Bouchaud and co-workers (see reference [4]) have pioneered such analysis of
options in multi-period settings.
The downside to having a derivative trading culture that is lacking in assessing risk capital,
especially of the purported replication schemes, are now well known from the experience of the
great financial crisis. Accountants and quants need to stop taking comfort in the precision and
perfection of their risk-neutral “valuation models” that simply paper over irreducible risks.
Exchange listed products are associated with transparency in pricing. The exchanges are also
required to learn how to successfully impose margin requirements commensurate with risks.
Modeled values of derivative securities are no substitute for exchanges and electronic auctions.
Much more needs to done to understand the proclivities of market participants and their risk
preferences. Human psychology and risk preference expression mechanics are an integral part of
markets – including derivative markets. By hiding residual-risks, the risk-neutral paradigm has
failed to spur any meaningful research in this arena – hence the critical importance of approaches
that highlight residual risks of attempted replication. Understanding and interpreting derivative
prices in the real-world involves understanding market dynamics and real-world statistics and
human reactions and their ways of decision making under uncertainty.
19
References
[1] Lawrence H. Summers. “On Economics and Finance,” The Journal of Finance, Volume
40, Issue 3, 633-635, July 1985
[2] Rohan Douglas. Credit Derivative Strategies: New Thinking on Managing Risk and
Return, Jun 2007.
[3] Arthur M. Berd and Vivek Kapoor. “Digital Premium,” The Journal of Derivatives,
Spring 2003
[4] Edward I. Altman, Brooks Brady, Andrea Resti, and Andrea Sironi. “The Link between
Default and Recovery Rates: Theory, Empirical Evidence, and Implications,” Journal of
Business vol. 78, no. 6: 2203–27. November 2005
[5] Andrea Petrelli, Olivia Siu, Jun Zhang, and Vivek Kapoor. Optimal Static Hedging of
Defaults in CDOs, DefaultRisk.com 2006
[6] J-P Bouchaud and M. Potters. Theory of Financial Risk and Derivative Pricing, From
Statistical Physics to Risk Management, Cambridge University Press, Cambridge 2003
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