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