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CVA for CDS via SDM and Related IssuesAnalytical and Numerical Results

Alexander Lipton and Artur Sepp

Bank of America Merrill Lynch

March 2010

Lipton and A Sepp (Bank of America Merrill Lynch)

Credit Modeling 02/03 1 / 74
http://find/http://goback/

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Plan

A brief history of credit modeling.


http://find/

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Plan


How to model CDSs?



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Plan


How to model CDSs?

Reduced, structural, and hybrid single-name models.

Lipton and A Sepp (Bank of America Merrill Lynch) Credit Modeling 02/03 2 / 74

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Plan


How to model CDSs?

Reduced, structural, and hybrid single-name models.Examples.


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Plan


How to model CDSs?


Multi-name structural models.


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Plan


How to model CDSs?


Multi-name structural models.

Examples.


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Acknowledgements

We am also grateful to other Merrill Lynch colleagues, especially S.

Inglis, A. Rennie, D. Shelton, I. Savescu, and members of the CreditAnalytics Team and Structured Credit Team for all their help.


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References

Selected references:


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References


Reduced form model: Jarrow & Turnbull (2000), Hughston &Turnbull (2001), Lando (1998), Madan & Unal (1994).


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References



Structural model: Black & Cox (1976), Hilberink & Rogers (2002),Hyer et al (1998), Kiesel & Scherer (2007), Leland (1994), Longsta& Schwartz (1995), Lipton (2002), Merton (1974), Sepp (2004,

2006), Zhou (2001a), Zhou (2001a).


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References




2006), Zhou (2001a), Zhou (2001a).

Counterparty risk: Blanchet & Patras (2008), Brigo & Chourdakis(2008), Crepey, Jeanblanc & Zargari (2009), Leung & Kwok (2005),Haworth, Reisisnger & Shaw (2006), Hull & White (2000), (2001),

Misisrpashayev (2008), Turnbull (2005), Valuzis (2008).


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References




2006), Zhou (2001a), Zhou (2001a).

Counterparty risk: Blanchet & Patras (2008), Brigo & Chourdakis(2008), Crepey, Jeanblanc & Zargari (2009), Leung & Kwok (2005),Haworth, Reisisnger & Shaw (2006), Hull & White (2000), (2001),

Misisrpashayev (2008), Turnbull (2005), Valuzis (2008).Numerical methods: Andersen & Andreasen (2000), Boyarchenko &Levendorsky (2000), Cont & Tankov (2004), Cont & Voltchkova(2005), Clift & Forsyth (2008), dHalluin, Forsyth & Vetzal (2005),Feng & Linetsky (2008), Lipton (2003).


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Moodys number of defaults gure

All rated, Source Moody's

0

50

100

150

200

250

1920

1924

1928

1932

1936

1940

1944

1948

1952

1956

1960

1964

1968

1972

1976

1980

1984

1988

1992

1996

2000

2004


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Moodys table of recoveries

Source: Moody'sYear Sr. Secured Sr. Secured Sr. Unsecured Sr. SubordinaSubordinated Jr. SubordAll

Bank Loans Bonds Bonds Bonds Bonds Bonds Bonds

1982 NA $72.50 $35.79 $48.09 $29.99 NA $35.571983 NA $40.00 $52.72 $43.50 $40.54 NA $43.641984 NA NA $49.41 $67.88 $44.26 NA $45.491985 NA $83.63 $60.16 $30.88 $39.42 $48.50 $43.661986 NA $59.22 $52.60 $50.16 $42.58 NA $48.381987 NA $71.00 $62.73 $44.81 $46.89 NA $50.481988 NA $55.40 $45.24 $33.41 $33.77 $36.50 $38.981989 NA $46.54 $43.81 $34.57 $26.36 $16.85 $32.311990 $75.25 $33.81 $37.01 $25.64 $19.09 $10.70 $25.50

1991 $74.67 $48.39 $36.66 $41.82 $24.42 $7.79 $35.531992 $61.13 $62.05 $49.19 $49.40 $38.04 $13.50 $45.891993 $53.40 NA $37.13 $51.91 $44.15 NA $43.081994 $67.59 $69.25 $53.73 $29.61 $38.23 NA $45.571995 $75.44 $62.02 $47.60 $34.30 $41.54 NA $43.281996 $88.23 $47.58 $62.75 $43.75 $22.60 NA $41.541997 $78.75 $75.50 $56.10 $44.73 $35.96 $30.58 $49.391998 $51.40 $48.14 $41.63 $44.99 $18.19 $62.00 $39.651999 $75.82 $43.00 $38.04 $28.01 $35.64 NA $34.332000 $68.32 $39.23 $23.81 $20.75 $31.86 $15.50 $25.18

2001 $66.16 $37.98 $21.45 $19.82 $15.94 $47.00 $22.212002 $58.80 $48.37 $29.69 $23.21 $24.51 NA $30.182003 $73.43 $63.46 $41.87 $37.27 $12.31 NA $40.692004 $87.74 $73.25 $54.25 $46.54 $94.00 NA $59.122005 $82.07 $71.93 $54.88 $26.06 $51.25 NA $55.972006 $76.02 $74.63 $55.02 $41.41 $56.11 NA $55.022007 $67.74 $80.54 $51.02 $54.47 NA NA $53.53


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CDS gure

0

50

100

150

200

250

300

350

400

29/3/07 28/5/07 27/7/07 25/9/07 24/11/07 23/1/08 23/3/08

IBM 5Y GE 5Y MER 5Y DB 5Y DCX 5Y SIEM 5Y

Figure: Time series of 5Y CDS spreads (in bps) for 6 typical companies (IBM, GE,Merrill Lynch, Deutsche Bank, Daimler, and Siemens). Source: Merrill Lynch.


R d

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Recent trends

The volume of CDSs grew by a factor of 100 between 2001 and 2007.


R d

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Recent trends


Lately, CDS dealers reduced their market by 38% in one year.


R d

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Recent trends


Lately, CDS dealers reduced their market by 38% in one year.

According to the most recent ISDA survery published on April 22nd,the notional value of CDSs outstanding decreased to $38.6 trillion asof Dec 31st 2008 from $54.6 trillion mid-year and $62.2 trillion as ofDec 31st 2007.


CDS d li

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CDS modeling

First, we need to explain how to price CDSs.

A Lipton and A Sepp (Bank of America Merrill Lynch) Credit Modeling 02/03 9 / 74

CDS d li

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CDS modeling


Then we need to extend our theory to cover indices, tranches,baskets, etc.


CDS modeling

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CDS modeling



Three complementary approaches to pricing CDSs:


CDS modeling

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CDS modeling



Three complementary approaches to pricing CDSs:reduced form,


CDS modeling

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CDS modeling



Three complementary approaches to pricing CDSs:reduced form,structural,


CDS modeling

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CDS modeling



Three complementary approaches to pricing CDSs:reduced form,structural,hybrid


CDS modeling

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CDS modeling



Three complementary approaches to pricing CDSs:reduced form,structural,hybrid

All of them have pros and cons. Reduced form and structural

approaches dominate.


Simple non-risky model

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First, we consider non-risky bonds and introduce the short interestrate r(t) which is driven by the SDE:

dr(t) = f1 (t, r (t)) dt+ g1 (t, r(t)) dW1 (t) r(0) = r0



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Simple non risky model



Let D(t, T) be time t price of the non-risky zero-coupon bondmaturing at time T. This price can be written as follows:

D(

t, T) =

V1 (

t, r(

t))

D(0

, T) =

V1 (0

, r0)

where V1 (t, r) is the solution of the following backward Kolmogorovproblem

V1,t +12 g

21 V1,rr + f1V1,r rV1 = 0 V1 (T, r) = 1



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Simple non risky model



Let D(t, T) be time t price of the non-risky zero-coupon bondmaturing at time T. This price can be written as follows:

D(

t, T) =

V1 (

t, r(

t))

D(

0, T) =

V1 (

0, r0)

where V1 (t, r) is the solution of the following backward Kolmogorovproblem

V1,t +12 g

21 V1,rr + f1V1,r rV1 = 0 V1 (T, r) = 1

Functions f1, g1 are calibrated to the market to match the yield curveand some swaption volatilities. Alternatively,

D(t, T) = Et

exp

ZT

tr

t0

dt0

= exp

ZT

tr

t, t0

dt0

where r is the forward rate.Lipton and A Sepp (Bank of America Merrill Lynch) Credit Modeling 02/03 10 / 74

Reduced form model formulation

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The name defaults at the rst time a Cox process jumps from 0 to 1.SDE for the default intensity (hazard rate) X (t) is:

dX (t) = f(t, X (t)) dt+ g (t, X (t)) dW (t) + JdN(t) , X (0) = X0

where W (t) is a standard Wiener processes, N(t) is a Poissonprocess with intensity (t), and J is a positive jump distribution;W, N, J are mutually independent.



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We impose the following constraints

f(t, 0) 0, f(t,) < 0, g (t, 0) = 0



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We impose the following constraints

f(t, 0) 0, f(t,) < 0, g (t, 0) = 0For analytical convenience (rather than for deeper reasons) it is

customary to assume that X is governed by the square-root stochasticdierential equation (SDE):

dX (t) = ( (t) X (t)) dt+ q

X (t)dW (t) + JdN(t) , X (0) = X0

with exponential (or hyper-exponential) jump distribution.


Survival probability

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p y

For practical purposes it is more convenient to consider discrete jumpdistributions with jump values Jm > 0, 1

m

M, occurring with

probabilities m > 0; such distributions are more exible thanparametric ones because they allow one to place jumps where they areneeded.



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p y


m

M, occurring with

probabilities m > 0; such distributions are more exible thanparametric ones because they allow one to place jumps where they areneeded.In this framework, the survival probability of the name from time 0 totime T has the form

q(0, T) = E0n

eRT

0 X(t0)dt0

o= E0

neY(T)

o



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p y


m

M, occurring with


q(0, T) = E0n

eRT

0 X(t0)dt0

o= E0

neY(T)

oHere Y (t) is governed by the following degenerate SDE:

dY (t) = X (t) dt, Y (0) = 0



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m

M, occurring with


q(0, T) = E0n

eRT

0 X(t0)dt0

o= E0

neY(T)

oHere Y (t) is governed by the following degenerate SDE:

dY (t) = X (t) dt, Y (0) = 0

More generally, the survival probability from time t to time Tconditional on no default before time t has the form

q( t, TjX (t) , Y (t)) = eY(t)I(>t)Etn

eY(T)

X (t) , Y (t)o


Calculation of expectations

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Expectations of the form Et

neY(T)

X (t) , Y (t)

o, can be

computed by solving the following augmented partial dierentialequation (PDE)

(t + L) V (t, T, X, Y) + XVY (t, T, X, Y) = 0

V (T, T, X, Y) = eY

where

LV ( (t) X) VX + 12 2XVXX + m

m [V (X + Jm ) V (X)]


Calculation of expectations

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Expectations of the form Et

neY(T)

X (t) , Y (t)

o, can be

computed by solving the following augmented partial dierentialequation (PDE)

(t + L) V (t, T, X, Y) + XVY (t, T, X, Y) = 0

V (T, T, X, Y) = eY

where

LV ( (t) X) VX + 12 2XVXX + m

m [V (X + Jm ) V (X)]

Specically, the following relation holds

Et

neY(T)

X (t) , Y (t)o

= V (t, T, X (t) , Y (t))


Ane ansatz

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The corresponding solution can be written in the so-called ane form:

V (t, T, X, Y) = ea(t,T,)+b(t,T,)XY

where a, b are functions of time governed by the following system ofordinary dierential equations (ODEs):

8

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The corresponding solution can be written in the so-called ane form:

V (t, T, X, Y) = ea(t,T,)+b(t,T,)XY

where a, b are functions of time governed by the following system ofordinary dierential equations (ODEs):

8

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The value U of a credit default swap (CDS) paying an up-frontamount and a coupon s in exchange for receiving 1

R (where R is

the default recovery) on default as follows:

U = + V (0, X0)


CDS valuation 1

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R (where R is


U = + V (0, X0)Here V (t, X) solves the following pricing problem

(t + L) V (t, X) (r + X) V (t, X) = s (1 R) XV (T, X) = 0


CDS valuation 1

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R (where R is


U = + V (0, X0)Here V (t, X) solves the following pricing problem

(t + L) V (t, X) (r + X) V (t, X) = s (1 R) XV (T, X) = 0

Using Duhamels principle, we obtain the following expression for V:

V (t, X) = sZT

t D

t, t0

ea(t,t

0,1)+b(t,t

0,1)X

dt0

(1 R)ZT

tD

t, t0

dh

ea(t,t0,1)+b(t,t0,1)X

iwhere D(t, t0) is the discount factor between two times t, t0.


CDS valuation 2

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Accordingly,

U = sZT

0 D

0, t0

1 p0, t0 dt0 + (1 R) ZT

0 D

0, t0

dp

For a given up-front payment , we can represent the correspondingpar spread s (i.e. the spread which makes the value of thecorresponding CDS zero) as follows:

s(T) = + (1 R)R

T0 D(0, t0) dp(0, t0)RT

0 D(0, t0) (1 p(0, t0)) dt0


CDS valuation 2

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Accordingly,

U = sZT

0 D

0, t0

1 p0, t0 dt0 + (1 R) ZT

0 D

0, t0

dp


s(T) = + (1 R)R

T0 D(0, t0) dp(0, t0)RT0 D(0, t

0) (1 p(0, t0)) dt0Conversely, for a given spread we can represent the par up-frontpayment in the form

= s(T)ZT

0D

0, t0

1 p0, t0 dt0+ (1 R) ZT0

D

0, t0

dp

0, t0


CDS valuation 2

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Accordingly,

U = sZT

0 D

0, t0

1 p0, t0 dt0 + (1 R) ZT

0 D

0, t0

dp


s(T) = + (1 R)R

T0 D(0, t0) dp(0, t0)RT0 D(0, t

0) (1 p(0, t0)) dt0Conversely, for a given spread we can represent the par up-frontpayment in the form

= s(T)Z

T

0D

0, t0

1 p0, t0 dt0+ (1 R) ZT0

D

0, t0

dp

0, t0

In these formulas we implicitly assumed that the corresponding CDSis fully collateralized, so that in the event of default 1 R is readilyavailable.Lipton and A Sepp (Bank of America Merrill Lynch) Credit Modeling 02/03 16 / 74

Two name correlation 1

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It is very tempting to extend the above framework to cover severalcorrelated names. For example, consider two credits, A, B and assume

for simplicity that their default intensities coincide,

XA (t) = XB (t) = X (t)

and both names have the same recovery RA = RB = R.



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It is very tempting to extend the above framework to cover severalcorrelated names. For example, consider two credits, A, B and assume

for simplicity that their default intensities coincide,

XA (t) = XB (t) = X (t)

and both names have the same recovery RA = RB = R.

For a given maturity T, the default event correlation is dened asfollows

(0, T) =pAB (0, T) pA (0, T) pB (0, T)

ppA (0, T) (1 pA (0, T)) pB (0, T) (1 pB (0, T))

where A, B are the default times, andpA (0, T) = P(A T) , pB (0, T) = P(B T)

pAB (0, T) = P(A T, B T)



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It is clear that

pA (0, T) = pB (0, T) = p(0, T) = 1

ea(0,T,1)+b(0,T,1)X0



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It is clear that

pA (0, T) = pB (0, T) = p(0, T) = 1

ea(0,T,1)+b(0,T,1)X0

Simple calculation yields

pAB (0, T) = E0n

eRT

0 (XA (t0)+XB(t0))dt0

o+pA (0, T) +pB (0, T)1

=E

0ne2 R

T

0 X(t0)dt0o

+ 2p

(0, T

) 1



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It is clear that

pA (0, T) = pB (0, T) = p(0, T) = 1

ea(0,T,1)+b(0,T,1)X0


pAB (0, T) = E0n

eRT

0 (XA (t0)+XB(t0))dt0

o+pA (0, T) +pB (0, T)1

=E

0ne2 R

T

0 X(t0)dt0o

+ 2p

(0, T

) 1Thus

(0, T) =ea(0,T,2)+b(0,T,2)X0 e2a(0,T,1)+2b(0,T,1)X0

1 ea(0,T,1)+b(0,T,1)X0

ea(0,T,1)+b(0,T,1)X0



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It is clear that

pA (0, T) = pB (0, T) = p(0, T) = 1

ea(0,T,1)+b(0,T,1)X0


pAB (0, T) = E0n

eRT

0 (XA (t0)+XB(t0))dt0

o+pA (0, T) +pB (0, T)1

=E

0ne2 R

T

0 X(t0)dt0o

+ 2p

(0, T

) 1Thus

(0, T) =ea(0,T,2)+b(0,T,2)X0 e2a(0,T,1)+2b(0,T,1)X0

1 ea(0,T,1)+b(0,T,1)X0

ea(0,T,1)+b(0,T,1)X0

In the absence of jumps, the corresponding event correlation is verylow. If large positive jumps are added (while overall survivalprobability is preserved), then correlation can increase all the way toone. Assuming that T = 5y, = 0.5, = 7%, and J = 5.0, weillustrate this observation below.


Event correlation Figure

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0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00%

lambda

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

corr (left axis) theta (right axis)

Figure: Correlation and mean-reversion level = X0 as functions of jumpintensity . Other parameters are as follows: T = 5y, = 0.5, = 7%, andJ

= 5.0.Lipton and A Sepp (Bank of America Merrill Lynch) Credit Modeling 02/03 19 / 74

FTD-STD

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In the two-name portfolio, we can dene two types of CDSs which

depend on the correlation: (A) the rst-to-default swap (FTD); (B)the second-to-default swap (STD). The corresponding par spreads(assuming that there are no up-front payments) are

s1 (T) =

(1

R) RT0 D(0, t0) dh1 e

a(0,t0,2)+b(0,t0,2)X0iRT0 D(0, t

0) ea(0,t0,2)+b(0,t0,2)X0 dt0

s2 (T) =(1 R)

RT

0D(0, t0) d

h1

2ea(t

0,1)+b(t0,1)X0 ea(t0,2)+b(t0,2)X

RT0 D(0, t0)

2e

a(

t0,

1)+b

(t0,

1)X

0 ea

(t0,

2)+b

(t0,

2)X

0

dt0


FTD-STD

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In the two-name portfolio, we can dene two types of CDSs which

depend on the correlation: (A) the rst-to-default swap (FTD); (B)the second-to-default swap (STD). The corresponding par spreads(assuming that there are no up-front payments) are

s1 (T) =

(1

R) RT0 D(0, t0) dh1 e

a(0,t0,2)+b(0,t0,2)X0iRT0 D(0, t

0) ea(0,t0,2)+b(0,t0,2)X0 dt0

s2 (T) =(1 R)

RT

0D(0, t0) d

h1

2ea(t

0,1)+b(t0,1)X0 ea(t0,2)+b(t0,2)X

RT0 D(0, t0)

2e

a(

t0,

1)+b

(t0,

1)X

0 ea

(t0,

2)+b

(t0,

2)X

0

dt0It is clear that the relative values of s1, s2 very strongly depend onwhether or not jumps are present in the model.


FTD-STD Figure

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0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00%

lambda

s_1 s_2 s

Figure: FTD spread s1, STD spread s2, and single name CDS spread s asfunctions of jump intensity . Other parameters are the same as in Fig.1. It isclear that jumps are necessary to have s

1and s

2of similar magnitudes.


Counter-party eects

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An important application of the above model is to the evaluation ofcounter-party eects on fair CDS spreads. Let us assume that name

A has written a CDS on reference name B. It is clear that the pricingproblem for the value of the uncollateralized CDS V can be written asfollows

LV (t, X) (r + 2X) V (t, X)= s

(1

R) X

(RV+ (t, X) + V

(t, X)) X

where V is the value of a fully collateralized CDS on name B.


Counter-party eects

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LV (t, X) (r + 2X) V (t, X)= s

(1

R) X

(RV+ (t, X) + V

(t, X)) X

where V is the value of a fully collateralized CDS on name B.It is clear that the discount rate is increased from r + X to r + 2X,since there are two cases when the uncollateralized CDS can beterminated due to default: when the reference name B defaults; when

the issuer A defaults.


Counter-party eects

A i li i f h b d l i h l i f

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LV (t, X) (r + 2X) V (t, X)= s

(1

R) X

(RV+ (t, X) + V

(t, X)) X

where V is the value of a fully collateralized CDS on name B.It is clear that the discount rate is increased from r + X to r + 2X,since there are two cases when the uncollateralized CDS can beterminated due to default: when the reference name B defaults; when

the issuer A defaults.Although this equation is no longer analytically solvable, it can besolved numerically via, say, an appropriate modication of theclassical Crank-Nicholson method. It turns out that in the presence ofjumps the value of the fair par spread goes down dramatically.


Counter-party Figure

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0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00%

lambda

Fraction of Default Leg Value

Figure: Reduction in default leg value as function of jump intensity .Mean-reversion level = X0 is chosen in order to preserve survival probability.Other parameters are as follows: T = 5y, = 0.5, = 7%, and J = 5.0.


Typical structural model without jumps formulation

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A typical structural model without jumps for the evolution of thelog-value of the rm has the form:

dx = dt+ dW (t) , x(0) = x0

x(t) = ln

v(t)B(t)

= ln

v (t)

B(0) exp ((r d) t)

= 2

2


Typical structural model without jumps formulation

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A typical structural model without jumps for the evolution of thelog-value of the rm has the form:

dx = dt+ dW (t) , x(0) = x0

x(t) = ln

v(t)B(t)

= ln

v (t)

B(0) exp ((r d) t)

= 2

2

This value is governed by a Wiener process. The rm defaults if thevalue x(t) crosses zero barrier b(t) = 0.



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Assuming that all the relevant parameters are constant, thecorresponding formula for Q(t, T) is

Q(t, T) = I (0, t)

N

x(t) 12 2p

2

! ex(t)N

x(t) 12 2p

2

!!

where N is the cumulative normal distribution. It is easy to verify that

Q(0, 0) = 1, QT (0, 0) = 0

The latter fact is rather disconcerting, since it implies that the parshort term CDS spread vanishes when T ! 0.


Typical structural model with jumps formulation

A t ical st ct al odel ith j s fo the e ol tio of the

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A typical structural model with jumps for the evolution of thelog-value of the rm has the form:

dx = dt+ dW (t) +j+dN+ +jdN, x(0) = x0

x = ln v

B

= 2

2 +

+ 1 +

+ 1


Typical structural model with jumps formulation

A typical structural model with jumps for the evolution of the

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A typical structural model with jumps for the evolution of thelog-value of the rm has the form:

dx = dt+ dW (t) +j+dN+ +jdN, x(0) = x0

x = ln v

B

= 2

2 +

+ 1 +

+ 1This value is governed by a combination of a Wiener process and aPoisson process with exponentially distributed jumps. The rmdefaults if the value x(t) crosses zero barrier b(t) = 0. It was realized

early on that without jumps (or/and curvilinear or uncertain barriers)it is impossible to explain the short end of the CDS curve within thestructural framework. When all the relevant parameters are constant,the problem can be solved analytically via the Laplace transform.However, in general this approach does not work.


Unbounded PDF 1

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When the barrier is absent, can be found via a simple recursion

(knowing an analytical solution is useful for benchmarking purposes).Vis-a-vis the Gaussian distribution, has fat tails and a narrow peak.Let

=

p

t

= (x t) /pt0,0 (x) = n () /

pt

1,0 (x) = +P (,+)

0,

1 (x

) = P

(,)

where P (a, b) = exp

ab+ b2/2

N (a + b)


Unbounded PDF 2

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Then:

k,0 (x) =+ ((+ +)k1,0 + +k2,0)

k 10,l (x) =

(( )0,l1 + 0,l2)l

1

k,l (x) = +k1,l + k,l1

+ +

wk,l =e(++)t (+t)k (t)l

k!l!

(x) =

k=0,l=0

wk,lk,l (x)


No barrier PDF gure

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sigma=0.2, lambda=0.1, alpha=1.0, mu=0.0

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

-10.00 -8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00

X

PDF


Fokker-Planck equation

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We can solve the barrier problem by using the forward Fokker-Planckequation for the t.p.d.f. and putting probabilities below the barrier tozero. This equation has the form:

L

t

12

2xx + x +

Z0

(t, x +j) ejdj = 0

(0, x) = (x ) (t, x) = 0 if < 0


Fokker-Planck equation

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We can solve the barrier problem by using the forward Fokker-Planckequation for the t.p.d.f. and putting probabilities below the barrier tozero. This equation has the form:

L

t

12

2xx + x +

Z0

(t, x +j) ejdj = 0

(0, x) = (x ) (t, x) = 0 if < 0

When parameters are time-independent, we can nd via theLaplace transform.


Laplace transform

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The Laplace transform

1

22xxx ( + p) +

Z0 (p, x +j) ejdj = (x )

(p, 0) = 0


Laplace transform

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1

22xxx ( + p) +

Z0 (p, x +j) ejdj = (x )

(p, 0) = 0

Forward characteristic equation:

1

222 ( + p)

= 0


Laplace transform

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1

22xxx ( + p) +

Z0 (p, x +j) ejdj = (x )

(p, 0) = 0

Forward characteristic equation:

1

222 ( + p)

= 0

This equation has three roots of which two are positive and onenegative. We denote them as i.


Greens function construction 1

Hence the overall solution has the form:

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(p, x) = C3e3 (x); x

D1e

1(x

)

+ D2e

2(x

)

+ D3e

3(x

)

; x




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(p, x) = C3e3 (x); x

D1e

1(x

)

+ D2e

2(x

)

+ D3e

3(x

)

; x Matching conditions

D1 + D2 + (D3 C3) = 0

1D

1+

2D

2+

3(D

3 C

3) =

2

2D1e

1 + D2e2 + D3e

3 = 0




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(p, x) = C3e3 (x); x

D1e

1(x

)

+ D2e

2(x

)

+ D3e

3(x

)


D1 + D2 + (D3 C3) = 0

1D

1+

2D

2+

3(D

3 C

3) =

2

2D1e

1 + D2e2 + D3e

3 = 0

One more condition is obtained from the following observations:

L ei(x)H(x ) =

+ ie(x)H(

x)

L

ei(x)H( x)

= + i

e(x)H( x)




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(p, x) = C3e3 (x); x

D1e

1

(x

)

+ D2e

2

(x

)

+ D3e

3

(x

)


D1 + D2 + (D3 C3) = 0

1D

1+

2D

2+

3(D

3 C

3) =

2

2D1e

1 + D2e2 + D3e

3 = 0

One more condition is obtained from the following observations:

L ei(x)H(x ) =

+ ie(x)H(

x)

L

ei(x)H( x)

= + i

e(x)H( x)

Here L is the corresponding dierential operator and H(.) is theHeaviside function.



The corresponding prole has the form

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p g p

(p, x) =

C3e3 (x

)

; x D1e

1

e1 x e3 x + D2e2 e2 x e3x ; x

D1 =

2

2

( + 1)

(1 2) (1 3)D2 = 2

2( + 2)

(2 1) (2 3)

D3 =2

2" e

(13) ( + 1)

(1 2) (1 3)+

e(23 ) ( + 2)

(2 1) (2 3)#C3 = D1 + D2 + D3


Stehfest algorithm

Inverse Laplace transform yields (t, x). We use Stehfest algorithm

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to evaluate Q:

(t, x) = pN

k=1

(1)k StNk (kp, x)

p =ln 2

t


Stehfest algorithm

Inverse Laplace transform yields (t, x). We use Stehfest algorithmQ

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to evaluate Q:

(t, x) = pN

k=1

(1)k StNk (kp, x)

p =ln 2

t

Coecients StNk are very sti. We typically choose N = 20. For smallt inversion can be numerically unstable unless computation is carriedwith many signicant digits.

-15

-10

-5

0

5

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Hitting time density

The rst hitting time density f(t) has the form

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( )

f(t) = 2

2x (t, 0)




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f(t) = 2

2x (t, 0)

Its Laplace transform is given by

f(p) =

e2 ( + 2)

e1 ( + 1)

2 1




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f(t) = 2

2 x (t, 0)

Its Laplace transform is given by

f(p) =

e2 ( + 2)

e1 ( + 1)

2 1For the regular diusion we obtain

f(p) = e2

f(t) = 2

2x (t, 0) =

e(+t)2/22 t

p22t3


Hitting time density gure

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0.000

0.002

0.004

0.006

0.0080.010

0.012

0.014

0.016

0.018

0.020

0.00 5.00 10.00 15.00 20.00 25.00

Time

Prob

Jump-Diffusion Inversion Regular Diffusion Inversion Regular Diffusion Analytical


Greens function expansion

Perturbative calculation of Greens function in 1D. For brevity weassume that all the parameters are time independent

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assume that all the parameters are time independent.Time-dependent case can be solved in a similar way. To start with,we make the following transform

(t, x) = exp

2

22+

t+

2(x )

(t, x)




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(t, x) = exp

2

22+

t+

2(x )

(t, x)

The modied Greens function solves the following propagationproblem

t (t, x) 12 22 (t, x) Z

0 (t, x +j) ejdj = 0

(0, x) = (x

) (t, 0) = 0, (t,) = 0

where =

2

, = 12 2 + +1




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(t, x) = exp

2

22+

t+

2(x )

(t, x)

The modied Greens function solves the following propagation

problem

t (t, x) 12 22 (t, x) Z

0 (t, x +j) ejdj = 0

(0, x) = (x

) (t, 0) = 0, (t,) = 0

where =

2

, = 12 2 + +1We assume that 1 and represent as follows

(t, x) = (0) (t, x) + (1) (t, x) + ...


Zero order term

A simple calculation yields (with = 2t)

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A simple calculation yields (with = t)

(0) (t, x) =1p

n

x p

n

x + p


Zero order term


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A simple calculation yields (with t)

(0) (t, x) =1p

n

x p

n

x + p

First order rhs

H(1) (t, x) = Px p

,p Px + p

,p


Zero order term


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A simple calculation yields (with t)

(0) (t, x) =1p

n

x p

n

x + p

First order rhs

H(1) (t, x) = Px p

,p Px + p

,p

We use Duhamels principle and represent (1) as follows

(1) (t, x) =Zt

0

Z0(0) (t, x; s, 1) H(1) (s, 1) dsd1


First order term

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A very elaborate calculation yields

(1) (t, x) =

21

P

x p

,p

+xPx + p ,p (x ) Px + p , p

(x + ) eP xp

,p

+ (x ) eP

xp

, p


First order term

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A very elaborate calculation yields

(1) (t, x) =

21

P

x p

,p

+xPx + p ,p (x ) Px + p , p

(x + ) eP xp

,p

+ (x ) eP

xp

, p

The formula can be independently checked via the method of heatpotentials.


Quality of approximation gure

pdf vs. x

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0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.000 1.000 2.000 3.000 4.000 5.000 6.000

x

pdf_ana pdf_exp_0 pdf_exp_1 pdf_num

Figure: G (t, x) for t = 20, = 20%, = 2%, = 4, = 1.


Barrier PDF gure

sigma=0.2, lambda=0.1, alpha=1.0, mu=0.0, barrier=-2.0

sigma=0.2,lambda=0.1,

alpha=1 0

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s g a 0 , a bda 0 , a p a 0, u 0 0, ba e 0

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

-5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00

X

Pro

b

Series PDE Laplace

alpha=1.0,

mu=0.0


Hybrid model

Modeling of the value of the company is not always the best way ofd li i h d f l I i i h h h d li h k

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dealing with default. It is sometimes thought that modeling the stock

price instead is a better choice.


Hybrid model

Modeling of the value of the company is not always the best way ofd li ith d f lt It i ti th ht th t d li th t k

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price instead is a better choice.

A typical model of this kind can be written as

dS = r (t) Sdt+ (t, S) dW (t)

where (t, 0) 6= 0. From that angle, one can argue that the Bacheliermodel is actually better than Black-Scholes!!


Hybrid model

Modeling of the value of the company is not always the best way ofd li ith d f lt It i ti th ht th t d li th t k

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price instead is a better choice.A typical model of this kind can be written as

dS = r (t) Sdt+ (t, S) dW (t)

where (t, 0) 6= 0. From that angle, one can argue that the Bacheliermodel is actually better than Black-Scholes!!

Alternatively, one can add a terminal (Samuelson-style) jump todefault and model S as follows

dS = [r (t) + (S)] Sdt+ (t, S) dW (t) SdN(t)

Lipton and A Sepp (Bank of America Merrill L nch) Credit Modeling 02/03 42 / 74

Model Comparison

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Comparison of numerical and analytical t.p.d.f. as well as the solutionof the barrier problem with non-constant barrier is given below.

A Li t d A S (B k f A i M ill L h) C dit M d li 02/03 43 / 74

Model Comparison

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By bootstrapping the barrier, we can reproduce term structure ofCDS for most names. In addition, we can price equity derivatives and

produce a respectable volatility skew.


Model Comparison

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By bootstrapping the barrier, we can reproduce term structure ofCDS for most names. In addition, we can price equity derivatives and

produce a respectable volatility skew.The barrier can assumed to be stochastic as in Due and Lando(2001) or CreditGrades (2000). Reduction of the information setmakes reduced form and structural modeling almost identical, Jarrow,Protter, Yildirim (2004).


Survival probability gure

100%

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55%

70%

85%

0 1 2 3 4 5 6 7 8 9 10

RF MBC LFKY L AL

Figure: Typical survival probabilities Q(

0, T)

as functions of T (in years) for thereduced-form, Merton-Black-Cox, Lardy et al., Lipton, and Atlan-Lelanc models.Notice that QMBC (0, T) and QAL (0, T) are too at when T ! 0, whileQLFHY (0, 0) < 1. QRF (0, T) and QL (0, T) behave properly.

Li t d A S (B k f A i M ill L h) C dit M d li 02/03 44 / 74

Greens function without jumps

Without jumps, we need to nd Greens function for the correlated heatequation in the quarter-plane, i.e., to solve the following problem

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t 12

21x1 x1 + 212x1x2 + 22x2 x2

+ 1x1 + 2x2 = 0

(t, x1, 0) = 0, (t, 0, x2) = 0

(t, x1, x2)

! (x1

1) (x2

2)

Standard transform

= exp (t+ 1 (x1 1) + 2 (x2 2))

removes drift terms provided that

= , = 121


Transforms

A sequence of linear transforms allows us to rewrite the pricing problem as

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follows

(2)t

1

2

(2)

x(2)1 x

(2)1

+ (2)

x(2)2 x

(2)2

= 0

wt, x

(2)1 , 0

= 0, wt,x(2)2 /, x(2)2

= 0

w

t, x(2)

1 , x(2)2

!

x(

2)1 (2)1

x(2)

2 (2)2

where =p

12. We now have the standard heat equation in an angleand can use our previous results. This angle is formed by the horizontal

axis (2)

2 = 0 and a sloping line (2)

1 = (2)

2 /. It is acute when < 0and blunt otherwise. The size of this angle is =atan( /).


Eigenfunction expansionThe solution of the above problem via the method of images wasindependently introduced in nance by He et al., Zhou and Lipton. Wewant to nd the Greens function for the following parabolic equation

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want to nd the Green s function for the following parabolic equation

(2)t

1

2

(2)rr +

1

r

(2)r +

1

r2

(2)

supplied with the boundary conditions of the form

(2)

(t,

r,

) !r!0 C< , (2)

(t,

r,

) !r! 0, (2)

(t,

r,

0) = 0, (2)

and the initial condition

(2) (t, r, ) !t!0

(r r0) ( 0)r0

The fundamental solution has the form

t, r, j0, r0, 0 = 2e(r2 +r02)/2tt

n=1

In/

rr0

t

sin

n0

sin

n


Non-periodic Greens functionIt turns out that we can nd the fundamental solution via the method ofimages too. This approach seems to be new. To start with, we ndnon-periodic solution of the heat equation written in polar co-ordinates in

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non periodic solution of the heat equation written in polar co ordinates in

the positive half-plane 0 < r < , < < :

t, r, j0, r0, 0 = 1 t, r, j0, r0, 02 t, r, j0, r0, 0where = 0, s =sign(),

1

t, r, j0, r0, 0 = e(r2 +r02)/2t2t

(s+ + s)2

e(rr0/t) cos

2 t, r, j0, r0, 0 = e

(r2 +r02)/2t

22

tZ

0

hs+e

(rr0/t) cosh((+)) + se(rr0/t) cosh(())

i2 + 1

d


Graph of Greens function

0.025

0.030

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0.000

0.005

0.010

0.015

0.020

-20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00

0.05 0.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 4.6

Figure: Cross-section of non-periodic Greens function. Parameters are as followst = 5, r0 = 1.5, 0 = 0.


Method of images

Simple balancing of terms shows that

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p g

2

t, r, j0, r0, 0 = O 02Next, we represent the fundamental solution in the form

t, r, j0, r0, 0 = n=

t, r, j0, r0, 0 + 2n t, r, j0, r0,0Indeed, it is clear that the sum converges, every term solves the parabolicequation, only one term has a pole inside the angle, and

(0) = () = 0.


Correlated PDF gure

1.00

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0.

00

0.

50

1.

00

1.

50

2.

00

2.

50

3.

00

3.

50

4.

00

4.

50

5.

00

5.

50

6.

00

6.

50

7.

00

7.

50

8.

00

8.

50

9.

00

9.

50

10.

00

0.00

1.20

2.40

3.60

4.80

6.00

7.20

8.40

9.60

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

Figure: PDF in the positive quarter-plane


Greens function with jumps

With jumps, we need to nd Greens function for the correlatedjump-diusion equation in the quarter-plane. In the simplest case we have

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j p q q p p

to solve the following problem

t 12

21x1 x1 + 212x1 x2 + 22x2x2

+ 1x1 + 2x2

12Z

0

Z0 (x1 +j1, x2 +j2) e1

j12

j2 dj1dj2 + = 0

(t, x1, 0) = 0, (t, 0, x2) = 0

(t, x1, x2)

! (x1

1) (x2

2)

This problem has been recently solved (numerically) by Lipton and Sepp.Its analytical solution is not known (??).


Multi-name case

Consider N companies and assume that their asset values are drivenby the following SDEs

dVi (t) J

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dVi (t)

Vi (t) = (r di ii (t)) dt+ i (t) dWi (t) +

eJ

i 1 dNi (t)i = E

neJi 1

o


Multi-name case

Consider N companies and assume that their asset values are drivenby the following SDEs

dVi (t) J

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dVi (t)


eJ

i 1 dNi (t)i = E

neJi 1

oDefault boundaries are driven by the ODEs of the form

dBi (t)Bi (t)

= (r di) dt Bi (0) = bi


Multi-name caseConsider N companies and assume that their asset values are drivenby the following SDEs

dVi (t) J

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dVi (t)


eJ

i 1 dNi (t)i = E

neJi 1


dBi (t)Bi (t)

= (r di) dt Bi (0) = biIn log coordinates

xi (t) = lnVi (t)

Bi (

t)

dxi (t) = i (t) dt+ i (t) dWi (t) + JidNi (t) xi (0) = ln

vi

bi

= i

i = 12 2i iiLipton and A Sepp (Bank of America Merrill Lynch) Credit Modeling 02/03 53 / 74

Multi-name caseConsider N companies and assume that their asset values are drivenby the following SDEs

dVi (t) J

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dVi (t)


eJ

i 1 dNi (t)i = E

neJi 1


dBi (t)Bi (t)

= (r di) dt Bi (0) = biIn log coordinates

xi (t) = lnVi (t)

Bi (

t)

dxi (t) = i (t) dt+ i (t) dWi (t) + JidNi (t) xi (0) = ln

vi

bi

= i

i = 12 2i iiLipton and A Sepp (Bank of America Merrill Lynch) Credit Modeling 02/03 53 / 74

Correlation structure

We introduce correlation between diusions in the usual way andassume that

dWi (t) dWj (t) = (t) dt

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i ( ) j ( ) ij

( )




dWi (t) dWj (t) = ij (t) dt

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i ( ) j ( ) ij ( )

We introduce correlation between jumps following the Marshall-Olkinidea. Let (N) be the set of all subsets of N names except for theempty subset f?g, and its typical member. With every weassociate a Poisson process N (t) with intensity (t), and

represent Ni (t) as follows

Ni (t) = 2(N)

1fi2gN (t)

i (t) = 2(N) 1fi2g (t)




dWi (t) dWj (t) = ij (t) dt

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( ) j ( ) ij ( )

We introduce correlation between jumps following the Marshall-Olkinidea. Let (N) be the set of all subsets of N names except for theempty subset f?g, and its typical member. With every weassociate a Poisson process N (t) with intensity (t), and

represent Ni (t) as follows

Ni (t) = 2(N)

1fi2gN (t)

i (t) = 2(N) 1fi2g (t)

Thus, we assume that there are both common and idiosyncratic jumpsources.


Multi-name pricing problemWe now formulate a typical pricing equation in the positive cone

R(N)+ . We have

~(N)

~ ~

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tU(t,

x) + L( )

U(t,

x) = (t,

x)U(t,~x0,k) = 0,k (t,~y) , U(t,~x,k) = ,k (t,~y)

U(T,~x) = (~x)

where ~x0,k, ~x,k, ~yk are N and N 1 dimensional vectors,respectively,

~x0,k =

x1, ..., 0

k, ...xN

, ~x,k =

x1, ...,

k, ...xN

, ~yk = (x1, ..., x



R(N)+ . We have

~(N)

~ ~

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tU(t,

x) + L U(t, x) = (t, x)U(t,~x0,k) = 0,k (t,~y) , U(t,~x,k) = ,k (t,~y)

U(T,~x) = (~x)


~x0,k =

x1, ..., 0

k, ...xN

, ~x,k =

x1, ...,

k, ...xN

, ~yk = (x1, ..., x

The integro-dierential operator L(N) can be written asL(N)f(~x) = 12

i

2i 2i f(~x) + i,j,j>i

aijijf(~x) +i

iif(~x)

f(~x) + 2(N)

i2

Jif(~x)



R(N)+ . We have

~(N)

~ ~

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tU(t,

x) + L U(t, x) = (t, x)U(t,~x0,k) = 0,k (t,~y) , U(t,~x,k) = ,k (t,~y)

U(T,~x) = (~x)


~x0,k =

x1, ..., 0

k, ...xN

, ~x,k =

x1, ...,

k, ...xN

, ~yk = (x1, ..., x

The integro-dierential operator L(N) can be written asL(N)f(~x) = 12

i

2i 2i f(~x) + i,j,j>i

aijijf(~x) +i

iif(~x)

f(~x) + 2(N)

i2

Jif(~x)


Jump terms

In the case of negative exponential jumps,

0

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Jif(~x) =Zxi f(x

1, ..., xi +j, ...xN) i (j) dj

while in the case of discrete negative jumps

Jif(~x) = H (xi + Ji) f(x1, ..., xi + Ji, ...xN)

and H is the Heaviside function.


Jump terms

In the case of negative exponential jumps,

0

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Jif(~x) =Zxi f(x

1, ..., xi +j, ...xN) i (j) dj

while in the case of discrete negative jumps

Jif(~x) = H (xi + Ji) f(x1, ..., xi + Ji, ...xN)

and H is the Heaviside function.

We naturally split the operator L(N) into the local (dierential) andnon-local (integral) parts:

L(N)f(~x) = D(N)f(~x) +I(N)f(~x)


Adjoint operator

The corresponding adjoint operator is

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L(N)g(~x) = 12 i

2i 2i g (~x) +

i,j,j>i

aijijg (~x)i

iig(~x)

g (~x) + 2(N)

i2

Ji g(~x)

where

Ji g(~x) =Z

0g (x1, ..., xij, ...xN) i (j) dj

Ji g(~x) = g(x1, ..., xi

Ji, ...xN)


Greens formulaWe introduce Greens function (t,~x), or, more explicitly,

t0,~x; t,~

, such that

~

( )

~

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t0

t0,~xL(N) t0,~x = 0

t0,~x0k

= 0,

t0,~xk

= 0, (t,~x) = ~x~


Greens formulaWe introduce Greens function (t,~x), or, more explicitly,

t0,~x; t,~

, such that

~ L(N ) ~

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t0

t0,~xL(N) t0,~x = 0

t0,~x0k

= 0,

t0,~xk

= 0, (t,~x) = ~x~

It can be shown that

U

t,~

=Z

R(N)+

(~x)

T,~x; t,~

d~x

+k

ZTt

ZR

(N1)+

k

t0,~y 1

22kk

t0,~y; t,~

dt0d~y

ZT

tZ

R(N)+

t0,~x

t0,~x; t,~

dt0d~xIn other words, instead of solving the backward pricing problem withnonhomogeneous rhs and boundary conditions, we can solve theforward propagation problem for Greens function with homogeneous


Single-name caseWe now are in a position to describe the relevant operators in moredetail for N = 1, 2.

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Single-name caseWe now are in a position to describe the relevant operators in moredetail for N = 1, 2.When N = 1 we deal with the reference name alone. Since (1)

i f j l

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consists of just one element,(1) = ff1gg

L(1)f = 12 2121f+ 11f f+ 1J1f = D(1)f+I(1)f

L(1)g = 1

22

12

1g

1

1

g

g + 1J

1g =

D(1)g +

I(1)g

= +1

I(1)f(x1) = 1Z0x1

f(x1 +j1) e1j1 d(1j1) = 1

Zx10

f(y1) e1 (x1y1 )

I(1)

g(x1) = 1Z0 g (x1 j1) e1

j1 d(1j1) = 1

Zx1

g(y1) e1 (x

1y

1 )

I(1)f(x1) = 1H (x1 + J1) f(x1 + J1)I(1)f(x1) = 1f(x1 J1)


Two-name case 1When N = 2 we deal with the reference name 1 and protection seller2, say. Since (2) consists of three elements,

(2)

ff1g,f2g

,f1

,2gg

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= ff1g f2g f1 2ggwe have

L(2)f = 12 2121f+ 12 2222f+ 1212 12f+ 11f+ 21f f+1

J1f+ 2

J2f+ 1,2

J1

J2f

= D(2)f+I(2)f

L(2)g = 12 2121g + 12 2222g + 1212 12g11g21g g+1

J

1 g + 2

J

2 g + 1,2

J

1

J

2 g

= D(2)g +I(2)g = +1 + 2 + 1,2


Two-name case 2

Specically,

I(2)f(x1, x2) = 1 Z

x1

0

f(y1, x2) e1 (x1y1 )d(1y1)

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I Z0+2

Zx20

f(x1, y2) e2 (x2y2 )d(2y2)

+1,2

Zx10

Zx20

f(y1, y2) e1 (x1y1)2 (x2y2 )d(1y1

I(2)g(x1, x2) = 1Z

x1

g (y1, x2) e1 (x1y1 )d(1y1)

+2 Z

x2

g (x1, y2) e2 (x2y2 )d(2y2)

+1,2

Zx1

Zx2

g (y1, y2) e1 (x1y1 )2 (x2y2 )d(1y


Non-risky CDS pricing problemWe are now in a position to formulate the pricing problems of interest.

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Non-risky CDS pricing problemWe are now in a position to formulate the pricing problems of interest.

We start with a CDS on a reference credit 1 which is sold by anonrisky seller to a nonrisky buyer. Assuming for simplicity that the

coupon is paid continuously at the rate s and that the default

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coupon is paid continuously at the rate s, and that the defaultboundary is continuous, we write the pricing equation as follows

tU(1) (t, x1) +L(1)U(1) (t, x1) = (t, x1)

U(1)

(t, 0) = 0 (t) , U(1)

(t,) = (t)U(1) (T, x1) = 0

Here

(t, x1) = s

1 1 R1 e

1 x1 R1 =1R1

1 + 1

0 (t) = (1 R1) , (t) = sZT

tD

t, t0

dt0


Risky seller CDS pricing problem 1

For a CDS on a reference credit 1 which is sold by a risky seller 2 to anonrisky buyer we write the pricing equation as follows

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nonrisky buyer, we write the pricing equation as follows

tU(2) (t, x1, x2) + L(2)U(2) (t, x1, x2) = (t, x1, x2)

U(2) (t, 0, x2) = 0,

1

(t, x2) , U(2) (t,, x2) = ,

1

(t, x2)

U(2) (t, x1, 0) = 0,2 (t, x1) , U(2) (t, x1,) = ,2 (t, x1)

U(2) (T, x1, x2) = 0


Risky seller CDS pricing problem 2Here

(t, x1, x2) = s11 (t, x1, x2)22 (t, x1, x2)1,21,2 (t, x1, x2)1 (t x1 x2) =

1 R1

e1x1

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1 (t, x1, x2) =

1 R1 e x2 (t, x1, x2) = U

(1) (t, x1) e2x2

1,2 (t, x1, x2) =

1 R1

e1 x1

1 e2 x2

+Z0x1

U(1) (t, x1 +j1) e1j1 d(1j1) e2x2 +

1 R1 R2e1x12 x2U(1) (t, x1) = R2U

(1)+ (t, x1) + U

(1) (t, x1)

0,

1 (t,

x2) = 1 R1,

,

1 (t,

x2) = sZT

t D

t,

t0

dt0

0,2 (t, x1) = R2U(1)+ (t, x1) + U

(1) (t, x1) , ,2 (t, x1) = U

(1) (t, x1)


Illustrations

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s(0) L(0) R l(0) a(0) {disg {expg JPM 39.6 508 40% 203 243 0.1780 0.1780 5.6167 1.63

MS 29.31 534 40% 214 243 0.1286 0.1286 7.7759 2.41


Illustrations

CDS Spread Survival Prob Default Leg Annuity Leg

T JPM MS JPM MS JPM MS JPM

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T JPM MS JPM MS JPM MS JPM1y 0.0055 0.0155 0.9909 0.9745 0.0055 0.0153 0.9954 0.98

2y 0.0055 0.0155 0.9818 0.9496 0.0109 0.0302 1.9818 1.94

3y 0.0055 0.0151 0.9729 0.9275 0.0163 0.0435 2.9591 2.88

4y 0.0055 0.0146 0.9640 0.9076 0.0216 0.0555 3.9276 3.80

5y 0.0059 0.0137 0.9518 0.8928 0.0289 0.0643 4.8858 4.70

6y 0.0060 0.0126 0.9417 0.8829 0.0350 0.0703 5.8323 5.59

7y 0.0060 0.0117 0.9323 0.8741 0.0406 0.0756 6.7693 6.47

8y 0.0060 0.0113 0.9230 0.8613 0.0462 0.0832 7.6969 7.33

9y 0.0060 0.0112 0.9138 0.8475 0.0517 0.0915 8.6153 8.1910y 0.0060 0.0110 0.9048 0.8339 0.0571 0.0997 9.5246 9.03


Illustrations

JPM0.04

$\lambda^{dis}$

$\lambda^{exp}$

Spread

MS

0.06

0.07

$\lambda^{dis}$

$\lambda^{exp}$

Spread

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0.00

0.01

0.02

0.03

1y 2y 3y 4y 5y 6y 7y 8y 9y 10yT

0.00

0.01

0.02

0.03

0.04

0.05

1y 2y 3y 4y 5y 6y 7y 8y 9y 10yT

Figure: Calibrated intensity rates for JMP (lhs) and MS(rhs) in the models withENJs and DNJs, respectively.


Illustrations

JPM

0.25

0.30JPM

0.25

0.30

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0.00

0.05

0.10

0.15

0.20

0.00 0.10 0.20 0.30 0.40X

1y

5y

10y

0.00

0.05

0.10

0.15

0.20

0.00 0.10 0.20 0.30 0.40X

1y

5y

10y

Figure: PDF of the driver x(T) for JMP in the model with DNJs (lhs) and ENJs(rhs) for T = 1y, 5y, and 10 y.


Illustrations

MS

0.16

0.18MS

0.16

0.18

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0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 0.10 0.20 0.30 0.40X

1y

5y

10y

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 0.10 0.20 0.30 0.40X

1y

5y

10y

Figure: Same graphs as in Figure 3 but for MS.


Illustrations

JPM

50%

60% MS

50%

60%

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0%

10%

20%

30%

40%

80% 100% 120% 140% 160% 180%K/S

CDSImpliedVol

$\lambda^{dis}$

$\lambda^{exp}$

0%

10%

20%

30%

40%

80% 100% 120% 140% 160% 180%K/S

CDSImpliedVol

$\lambda^{dis}$

$\lambda^{exp}$

Figure: Log-normal credit default swaption volatility implied from model withTe = 1y, Tt = 5y as a function of the inverse moneyness K/c(Te, Te + Tt).


Illustrations

JPM

200%

250%

$\lambda^{dis}$

$\lambda^{exp}$

MS

200%

250%

$\lambda^{dis}$

$\lambda^{exp}$

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0%

50%

100%

150%

14% 41% 69% 96% 123%K/S

EquityImpliedVol

MarketVol

0%

50%

100%

150%

30% 59% 89% 118%K/S

EquityImpliedVol

MarketVol

Figure: Log-normal equity volatility implied by the model as a function of inversemoneyness K/s for put options with T = 0.5y.


Illustrations

rho=0.5

1.00

$\lambda^{dis}$

rho=0.99

1.00

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0.00

0.20

0.40

0.60

0.80

0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25 9.25T

Gaussian

Correlation

$\lambda^{dis}$

$\lambda^{exp}$

0.00

0.20

0.40

0.60

0.80

0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25 9.25T

Gaussian

Correlation

$\lambda^{dis}$

$\lambda^{exp}$

Figure: Implied Gaussian correlations for FTDSs.


Illustrations

146

156

InputSpread

Risky Seller, rho=0.75Risky Buyer, rho=0.75

Risky Seller, rho=0.055

60

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96

106

116

126

136

1 2 3 4 5 6 7 8 9 10

T, years

Spread,

bp

y ,

Risky Buyer, rho=0.0

Risky Seller, rho=-0.75

Risky Buyer, rho=-0.75

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10

T, years

Spread,

bp

InputSpread





Risky Seller, rho=-0.75

Risky Buyer, rho=-0.75

Figure: Equilibrium spread for protection buyer and protection seller of a CDS onMS with JPM as the counterparty (lhs); same for a CDS on JPM with MS as the

counterparty (rhs).


Conclusions

We gave a broad overview of the state of CDS modeling.

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g g


Conclusions


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We showed how to build meaningful structural default models andconnected them to reduced form models.


Conclusions


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We showed that even the simplest elementary building blocks of

credit universe are dicult to describe properly.


Conclusions

We gave a broad overview of the state of CDS modeling.W h d h b ild i f l l d f l d l d

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We showed that even the simplest elementary building blocks of

credit universe are dicult to describe properly.The opinions expressed in this talk are those of the speaker and donot necessarily reect the views or opinions of Bank of AmericaMerrill Lynch.


lip ton 2010

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