1) Counterparty default risk 2) Modelling aspects 3) Pricing of credit instruments 4) Analytical Methods 5) FFT based methods 6) PDE based methods 7) Illustrations

1. Quantitative Methods for Counterparty Risk
Artur Sepp
Joint work with Alex Lipton
Bank of America Merrill Lynch
Quantitative Finance Workshop
Technical University of Helsinki
September 2, 2009
1

2. Plan of the presentation
1) Counterparty risk
2) Modelling aspects
3) Pricing of credit instruments
4) Analytical Methods
5) FFT based methods
6) PDE based methods
7) Illustrations
2

3. References for technical details
1) Lipton, A., Sepp, A. (2009) Credit Value Adjustment for Credit
Default Swaps via the Structural Default Model, The Journal of Credit
Risk, 5(2), 127-150
http://ssrn.com/abstract=2150669
2) Lipton, A. and Sepp, A. (2011). Credit value adjustment in the
extended structural default model. Forthcoming in The Oxford Hand-book
of Credit Derivatives, Oxford University Working paper
http://ukcatalogue.oup.com/product/9780199669486.do
3) Inglis, S., Lipton, A., Savescu, I., Sepp, A. (2008) Dynamic credit
models, Statistics and its Interface, 1(2), 211-227
http://intlpress.com/site/pub/files/_fulltext/journals/sii/2008/0001/
0002/SII-2008-0001-0002-a001.pdf
4) Sepp, A. (2006) Extended credit grades model with stochastic
volatility and jumps, Wilmott Magazine September, 50-62
http://ssrn.com/abstract=1412327
3

4. Simple Example of a CDS contract, I
The reference name defaults at random time 1
Contract maturity is T, the spread is c
The pay-o for the protection buyer :
V Buy =
(
c; 1 T
1; 1 T
(1)
The pay-o for the protection seller:
V Sell =
(
c; 1 T
1; 1 T
(2)
4

5. Simple Example of a CDS contract, II
Fair value of the contract for protection buyer:
PV (1) = DF(0; T) (P(1 T) cP(1 T)) ; (3)
P(1 T) is the default probability
DF(0; T) the risk-free discount factor
Coupon c is set so that PV (1) = 0:
c = P(1 T)
1 P(1 T)
(4)
Note that the probability of default is typically small,
P(1 1) 2 [0:01; :05] for investment grade companies
Thus, the protection seller obliges to pay $1 in return for a much
smaller fee c (c 2 [0:01; :05] for investment grade names)!
5

6. Counterparty Risk
What if the protection seller, the counterparty, is unable to honour
its obligations given that the reference name defaults?
Let the default time of the counterparty be 2
If the protection seller defaults before the reference name, the pro-tection
buyer has to honour its obligations to pay c
However, the buyer loses the CDS protection
The pay-o for the protection buyer :
V =
8
:
c; 1 T;
1; 1 T; 2 1
0; 1 T; 2 1
(5)
6

7. Credit Value Adjustment
Now, the fair value of the contract for protection buyer:
PV (1;2) = DF(0; T) (P(1 T; 2 1) cP(1 T)) (6)
We de

8. ne the counterparty value adjustment, CV A, by:
CV A = PV (1) PV (1;2) (7)
Note that CV A 0
CVA magnitude depends on the default probability of the counterparty
and the correlation between the reference name and the counterparty
We note that in case of perfect correlation P(1 T; 2 1) = 0 so
that the protection buyer loses the most if there is strong correlation
between 1 and 2
7

9. CDS basics
Credit default swap (CDS) - provides to the buyer a protection against
a reference name in return for coupon payments up to contract ma-turity
or the default event
At contract inception, the coupon is set so that the present value of
CDS is zero
As the time goes by, the mark-to-market (MtM) value of the CDS
contract
uctuates
In particular, when the credit quality of the reference name worsens,
the MtM increases
8

10. Counterparty Risk
When the CDS protection is sold by a defaultable counterparty, the
protection buyer faces the risk of losing a part of the mark-to-market
value of the CDS, if it is positive for the buyer, due to the counterparty
default
The loss is profound if the credit quality of both the reference credit
and the counterparty worsen simultaneously but the counterparty de-faults

11. rst
Because big banks are intermediaries between each other and other
institutions (hedge funds, insurers), failure of one of them poses risk
for more failures (domino eect)
9

12. Counterparty Risk
The volume of CDSs grew by a factor of 100 between 2001 and 2007
According to the most recent survey conveyed by International Swap
Dealers Association, the notional amount outstanding of credit de-fault
swaps decreased to $38:6 trillion as of December 31, 2008, from
$62:2 trillion as of December 31, 2007
Currently, the notional amount of interest rate derivatives outstanding
is $403:1 trillion, while the notional amount of the equity derivatives
is $8:7 trillion
10

13. Credit Value Adjustment
Let 1 and 2 be default times of the reference name and the coun-terparty,
respectively
The Credit Value Adjustment, C(t), is the expected maximal potential
loss due to counterparty default up to CDS maturity T:
C(t) = (1 R2)Et
Z T
t
D(t; t0) max
n
Et0
h
~ C(t0) jE(t0)
i
; 0
o
1fE(t0)gdt0
#
(8)
~ C(t) is cash
ow of CDS contract (long protection) without counter-party
risk discounted to time t
E(t) = f1 t; 2 = tg
R2 is recovery rate of counterparty obligations
D(t; T) = e
R T
t r(t0)dt0
is the risk-free discount factor
11

14. Motivation
Model for the counterparty risk evaluation need to:
1) describe realistic dynamics of CDS spreads (jump-diusions)
2) create profound correlation eects (correlated jump-diusions with
simultaneous jumps)
Model should match observable market data closely:
1) the term structure of CDS spreads
2) the term structure of discount factors
3) equity and CDS options volatilities
4) correlations
Some of model parameters are made time-dependent to

15. t term struc-ture
eects
12

16. Credit Modeling
Structural approach (Merton (1974), Black-Cox (1976))
Reduced form models (Jarrow-Turnbull (1995), Due-Singleton (1997),
Lando (1998))
Hybrid Models
13

17. Generic 1-d structural model
The value of the

18. rm assets, a(t), is driven by
da(t)
a(t)
= (r(t) (t) (t))dt+(t)dW(t)+jdN(t); (9)
Jumps j have probability density function $(j)
=
R1
1 ej$(j)dj 1 is the compensator
The

19. rm's liability per share l(t) is deterministic:
l(t) = E(t)l(0) (10)
where E(t) is the deterministic growth factor:
E(t) = exp
Z t
0
(r(t0) (t0))dt0
(11)
14

20. Default Time
The default time is de

21. ned by:
= minft : a(t) l(t)g
Without the loss of generality we can assume that the default is
triggered continuously or over a set of discrete monitoring times (t 2
ftdg)
Continuous monitoring: convenient choice for analytical develop-ments
Discrete monitoring: probably more realistic as the

22. rm value is
observed over the discrete times (quarterly reports), more suitable
for Monte-Carlo and numerical methods
15

23. Equity Value
We assume that the model value of equity price per share, s(t), is
given by:
s(t) =
(
a(t) l(t) = E(t)
ex(t) 1
l(0); if t
0; if t
(12)
The initial value is set by a(0) = s(0)+l(0)
s(0) is the today's price of the stock
l(0) is de

24. ned by l(0) = RL(0), where R is an average recovery of
the

25. rm's liabilities and L(0) is total debt per share.
The volatility of the equity price, eq(t), is approximately related to
(t) by:
eq(t) =
1+
l(t)
s(t)
!
(t) (13)
16

26. Jump Size Distribution
We assume that jumps have either a discrete negative amplitude of
size , 0, with
$(j) = (j +); = e 1 (14)
or jumps have negative exponential distribution with mean size 1
,
0, with:
$(j) = ej; j 0; =
+1
1 =
1
+1
(15)
17

27. Generic 1-d structural model, II
Introduce
x(t) = ln a(t)
l(0) - the log of the normalized asset value
dx(t) = (t)dt+(t)dW(t)+jdN(t); x(0) = ln
a(0)
l(0)
(16)
Note that y(t) = x(t) is an additive process with independent
time-dependent increments (Sato (1999))
(t) = 12
2(t) (t)
The default time is de

28. ned by:
= minft : x(t) 0g
The default is triggered either continuously or discretely
18

29. Generic 2-d structural model
We consider two

30. rms and assume that their asset values are driven
by the following SDEs:
dai (t)
= (r(t)i(t)ii (t))dt+i (t)dWi (t)+
eji 1
dNi (t) (17)
ai (t)
where i = 1; 2
Standard Brownian motions W1(t) and W2(t) are correlated with cor-relation
.
Jumps in the joint dynamics occur according to the Poisson process
Nf1;2g(t) with the intensity rate:
f1;2g(t) = minf; 0g minf1(t); 2(t)g
Idiosyncratic jumps occur according to Poisson processes N1(t) and
N2(t) with jump intensities f1g(t) and f2g(t), respectively, speci

31. ed
as follows:
f1g(t) = 1(t) f1;2g(t); f2g(t) = 2(t) f1;2g(t)
19

32. Jump Size PDF and Instantaneous Correlation
Consider the instantaneous correlations between x1(t) and x2(t) under
the assumption of discrete jumps, dis
12 , and that under exponential
jumps, exp
12 :
dis
12 =
12 +f1;2g12
q
2
1 +12
1
q
2
2 +22
2
;
exp
12 =
12 +
f1;2g
r 12
2
1 +21
2
1
r
2
2 +22
2
2
(18)
12 1 and exp
If the systematic intensity f1;2g is large, dis
12 1
2
From experiments: the maximal implied Gaussian correlation that can
be achieved (using = 0:99) is about 90% for the model with discrete
jumps and about 50% for the model with exponential jumps
The assumption about exponential jumps is not realistic by modelling
the joint dynamics of strongly correlated

33. rms belonging to one in-dustry
group (such as

34. nancial companies)
20

35. Multi-dimensional Case
We consider N

36. rms and assume that their asset values are driven
by the same equations in the two-dimensional case with the index i
running from 1 to N, i = 1; :::;N
We correlate diusions in the usual way and assume that:
dWi (t)dWj (t) = ij (t) dt (19)
We correlate jumps following the Marshall-Olkin (1967) idea. Let
(N) be the set of all subsets of N names except for the empty
subset f?g, and its typical member. With every we associate a
Poisson process N (t) with intensity (t), and represent Ni (t) as:
Ni (t) =
X
2(N)
1fi2gN (t) (20)
i (t) =
X
2(N)
1fi2g (t) (21)
Thus, we assume that there are both collective and idiosyncratic jump
sources
21

37. One-Dimensional Problem. Continuous Monitoring
The backward problem for the value function V (t; x):
Vt (t; x)+L(x)V (t; x) r (t) V (t; x) = c(t; x); fx 0g
V (T; x) = v(x); fx 0g
V (t; x) = g(t; x); fx 0g
V (t; x) !
x!1
(t; x)
(22)
where L(x) is the in

38. nitesimal operator of process x(t):
L(x) = D(x) +(t)J (x) (23)
D(x) is a dierential operator:
D(x)f(x) =
1
2
2(t)fxx (x)+(t)fx (x) (t) f (x) (24)
and J (x) is a jump operator:
J (x)f(x) =
Z 0
1
f(x+j)$(j)dj (25)
22

39. For discrete negative jumps
J (x)f(x) = f(x ) (26)
for exponential jumps
J (x)f(x) =
Z 0
1
f(x+j)ejdj (27)

40. One-Dimensional Problem. Discrete Monitoring
When monitoring is discrete, the pricing problem is formulated as
follows:
Vt (t; x)+L(x)V (t; x) r (t) V (t; x) = c(t; x); f1 x 1g;
V (T; x) = v(x); fx 0g
V (t; x) = g(t; x); fx 0g ; t 2 ftd
1; :::; td
mg
V (t; x) !
x!1
p(t; x)
V (t; x) !
x!1
m(t; x); t =2 ftd
1; :::; td
mg
(28)
23

41. One-Dimensional Problem. Localization
In case of both the discrete and continuous default monitoring, the
computational domain is (1;1)
However, for the continuous monitoring, we can switch to the semi-bounded
domain [0;1)
Representing the integral term in problem Eq.(22) as follows:
J (x)f(x) =
Z 0
1
f(x+j)$(j)dj
=
Z 0
x
f(x+j)$(j)dj +
Z x
1
g(x+j)$(j)dj
cJ (x)f(x)+Z(x)(x)
(29)
where cJ (x) is de

42. ned by:
cJ (x)f(x) =
Z 0
x
f(x+j)$(j)dj (30)
24

43. and Z(x)(x) is the deterministic function depending on the contract
boundary condition g(x).
As a result, we can formulate the pricing problem in the semi-bounded
domain [0;1) as follows:
Vt (t; x)+ ^ L(x)V (t; x) r (t) V (t; x) = c(t; x) d (t; x)
V (T; x) = v(x)
V (t; 0) = g(t; 0); V (t; x) !
x!1
(t; x)
(31)
d (t; x) = (t)Z(x) (t; x) (32)

44. One-Dimensional Problem. Green's Function
We formulate the problem for Green's function denoted by G(t; x; T;X),
representing the probability of x(T) = X conditional on x(t) = x
We denote G(T;X) G(t; x; T;X) and write:
GT (T; x) L(X)yG(T;X) = 0; fX 0g
G(t;X) = (X x)
G(T;X) = 0; fX 0g
G(T; x) !
x!1
0
(33)
with L(x)y being the in

45. nitesimal operator adjoint to J (x):
L(x)y = D(x)y +(t)J (x)y (34)
where D(x)y is the dierential operator:
D(x)yg(x) =
1
2
2(t)gxx (x) (t)gx (x) (t) g (x) (35)
25

46. and J (x)y is the jump operator:
J (x)yg(x) =
Z 0
1
g(x j)$(j)dj (36)

47. Two-Dimensional Problem
We denote the value function of the contract by V (t; x1; x2) which
solves the backward equation:
Vt (t; x1; x2)+L(x1;x2)V (t; x1; x2) r (t) V (t; x1; x2) = c(t; x1; x2); fx1 0; x2 V (T; x1; x2) = v(x1; x2); fx1 0; x2 0g
V (t; x1; x2) = g1(t; x1; x2); fx1 0; x2 0g
V (t; x1; x2) = g2(t; x1; x2); fx1 0; x2 0g
V (t; x1; x2) = g3(t; x1; x2); fx1 0; x2 0g
V (t; x1; x2) !
x1!1
1(t; x1;x2); V (t; x1; x2) !
x2!1
2(t; x1; x2)
(37)
where L(x1;x2) is the in

48. nitesimal backward operator corresponding to
the bivariate dynamics:
L(x1;x2) = D(x1) +D(x2) +C(x1;x2) +f1g(t)J (x1) +f2g(t)J (x2) +f1;2g(t)J (38)
26

49. C(x1;x2) is the correlation operator:
C(x1;x2)f(x1; x2) 1(t)2(t)fx1x2(x1; x2) f1;2g (t) f (x1; x2) (39)
and J (x1;x2) is the cross integral operator de

50. ned as follows:
J (x1;x2)f(x1; x2)
Z 0
1
Z 0
1
f(x1 +j1; x2 +j2)$(j1)$(j2)dj1dj2 (40)

51. Two-Dimensional Problem. Localization
In case of the discrete default monitoring, the PDE is de

52. ned on
(1;1) (1;1) and the boundary condition is applied when t 2
ftd
1; :::; td
mg.
For the case of continuous monitoring, the integral term in Eq.(37),
can be represented as follows:
J (x1;x2)f(x1; x2) =
Z 0
1
Z 0
1
f(x1 +j1; x2 +j2)$(j1)$(j2)dj1dj2
=
Z 0
x1
Z 0
x2
f(x1 +j1; x2 +j2)$(j1)$(j2)dj1dj2
+
(Z x1
1
Z 0
x2
+
Z 0
x1
Z x2
1
+
Z x1
1
Z x2
1
)
g(x1 +j1; x2 +j2)$(j1)$(j2)dj1dj2
(x1;x2)
1 (x1; x2)+Z
cJ (x1;x2)f(x1; x2)+Z
(x1;x2)
2 (x1; x2)+Z
(x1;x2)
1;2 (x1; x2)
Therefore, we only need to consider the integral term cJ (x1;x2) de

53. ned
on the bounded domain and augment the source term by determin-
27

54. istic functions Z1, Z2, and Z3 de

55. ned by integrating of g1, g2, g3,
respectively.
Thus, we can localize the problem in the positive quadrant [0;1)
[0;1)
Similar considerations apply for multi-dimensional case.

56. Multi-Dimensional Problem
For brevity, we assume the continuous monitoring
(N)
+
We can formulate a typical pricing equation in the positive cone R
as follows:
@tV (t; ~x)+ ^ L(~x)V (t; ~x) r (t) V (t; ~x) = (t; ~x) (41)
V
t; ~x0;k
= 0;k (t; ~y) ; V (t; ~x) !
xk!1
1;k (t; ~y) (42)
V (T; ~x) = (~x) (43)
where ~x, ~x0;k, ~yk are N and N 1 dimensional vectors, respectively,
~x = (x1; :::; xk; :::xN)
~x0;k =
x1; :::;0k
; :::xN
~yk =
x1; :::xk1; xk+1; :::xN
(44)
28

57. The corresponding integro-dierential operator ^ L(N) can be written
in the form
^ L(~x)f (~x) = 1
2
P
i
2
i @2
i f (~x)+
P
i;j;ji
ijij@i@jf (~x)
+
P
i
i@if (~x)+
P
2(N)
Q
i2
cJ (xi)f (~x) f (~x)
!
(45)
For discrete negative jumps
cJ (xi)f (~x) = H(xi i) f (x1; :::; xi i; :::xN) (46)
For negative exponential jumps,
cJ (xi)f (~x) = i
Z 0
xi
f (x1; :::; xi +ji; :::xN) eijidji (47)
The corresponding adjoint operator is
L(~x)yg (~x) = 1
2
P
i
2
i @2
i g (~x)+
P
i;j;ji
ijij@i@jg (~x)
P
i
i@ig (~x)+
P
2(N)
Q
i2
cJ (xi)yg (~x) g (~x)
!
(48)

58. where
cJ (xi)yg (~x) = g (x1; :::; xi +i; :::xN) (49)
or
cJ (xi)yg (~x) = i
Z 0
1
g (x1; :::; xi ji; :::xN) eijidji (50)
It is easy to check that in both cases
Z
R
(N)
+
h
cJ (xi)f (~x) g (~x) f (~x) cJ (xi)yg (~x)
i
d~x = 0 (51)
We introduce Green's function G
T; ~X
, or, more explicitly, G
t; ~x; T; ~X
,
such that
@TG
T; ~X
L
~X
y
G
T; ~X
= 0 (52)
G
T; ~X
0k
= 0; G
T; ~X
!
Xk!1
0 (53)
G
t; ~X
=
~X
~x
(54)

59. By integrating by parts
Z T
0
Z
R
(N)
+
h
^ L(~x)V (t; ~x)G(t; ~x) V (t; ~x) ^ L(~x)yG(t; ~x)
+@tV (t; ~x)G(t; ~x) V (t; ~x) @tG(t; ~x)] d~xdt = 0
(55)
we obtain
V (t; ~x) =
Z T
t
Z
R
(N)
+
t0; ~x0
D
t; t0
G
t; ~x; t0; ~x0
d~x0dt0 (56)
+
X
k
Z T
t
Z
R
(N1)
+
0;k
t0; ~y0
D
t; t0
gk
t; ~x; t0; ~y0
d~y0dt0
+D(t; T)
Z
R
(N)
+
~x0
G
t; ~x; T; ~x0
d~x0
where
gk
t; ~x; T; ~Y
= 1
22
k@kG
t; ~x; T; ~X

62. Xk=0
~Y
=
Y1; :::; Yk1; Yk+1; :::; YN
(57)

63. represents the hitting time density for the corresponding piece of the
boundary.
This extremely useful formula shows that instead of solving the back-ward
pricing problem with non-homogeneous right hand side and
boundary conditions, we can solve the forward propagation problem
for Green's function with homogeneous right hand side and boundary
conditions.

64. Pricing Credit Products. Survival Probability
The single name survival probability function, Q(x)(t; x; T), is de

65. ned
by:
Q(x)(t; x; T) EQt
[1fTg] (58)
given that t.
Q(x)(t; x; T) solves the following backward equation:
(x)
Q
t (t; x; T)+ L^ (x)Q(x) (t; x; T) = 0
Q(x)(T; x; T) = 1
Q(x)(t; 0; T) = 0; Q(x)(t; x; T) !
x!1
1
(59)
29

66. Pricing Credit Products. Joint Survival Probability
We de

67. ne the joint survival probability, Q(x1;x2)(t; x1; x2; T), as follows:
Q(x1;x2)(t; x1; x2; T) EQt
[1f1T;2Tg]
given that 1 t and 2 t.
Q(x1;x2)(t; x1; x2) solves the following equation:
(Q
x1;x2)
t (t; x1; x2; T)+ L^ (x1;x2)Q(x1;x2) (t; x1; x2; T) = 0
Q(x1;x2)(T; x1; x2; T) = 1
Q(x1;x2)(t; x1; 0; T) = 0; Q(x1;x2) (t; 0; x2; T) = 0
Q(x1;x2)(t; x1; x2; T) !
x1!1
Q(x)(t; x2; T);
Q(x1;x2)(t; x1; x2; T) !
x2!1
Q(x)(t; x1; T)
30

68. Pricing Credit Products. Credit Default Swap
The value function of the CDS contract long the protection, V CDS(t; x; T),
solves the following problem:
V CDS
t (t; x; T)+ ^ L(x)V CDS (t; x; T) r (t) V (t; x; T) = c d (t; x)
V CDS(T; x; T) = 0
V CDS(t; 0; T) = (1 R); V CDS(t; x; T) !
x!1
c
Z T
t
D(t; t0)dt0
d (t; x) = (t)Z(x) (x)
(60)
Assuming that the eective CDS recovery rate, Rex, is
oating and
represents the residual value of the

69. rm assets given the default, we
obtain:
Z(x)(x) =
8
:
H( x)
1 Rex
; DNJ
1 R
1+
ex; ENJ
(61)
for discrete and exponential jumps, respectively.
31

70. Pricing Credit Products. Equity Put Option
Assuming that t, we represent pay-o of put option, vPut(T; s),
with strike price K and maturity T, as follows:
vPut (T; s) = (K s(T))+1fTg +K1fTg (62)
The value function of V Put(t; x) as function of x solves the following
problem:
V Put
t (t; x)+ ^ L(x)V Put (t; x) r (t) V Put (t; x) = d (t; x)
V Put(T; x) = (K +l(T) (1 ex))+
V Put(t; 0) = D(t; T)K; V Put(t; x) !
x!1
0
d (t; x) = (t)D(t; T)Z(x) (x)
(63)
Z(x) (x) =
(
KH( x) ; DNJ
Kex ENJ
(64)
32

71. Pricing Credit Products. Credit Value Adjustment
We denote by x1 the value of driver associated with the

72. rm value
of CDS reference name and by x2 the value of the driver of the
counterparty

73. rm value
Under the bivariate dynamics, the value of the countrparty charge
V CV A(t; x1; x2; T) de

74. ned as the solution to (8) solves the following
problem:
V CV A
t (t; x1; x2; T)+ ^ L(x1;x2)V CV A (t; x1; x2; T) r (t) V CV A (t; x1; x2; T) = d (V (T; x1; x2; T) = 0
V (t; x1; 0; T) = (1 R2)
V CDS(t; x1; T)
+
; V (t; 0; x2; T) = 0
V (t; x1; x2; T) !
x1!1
0; V (t; x1; x2; T) !
x2!1
0
d (t; x1; x2) = f2g (t)Z(x2) (t; x1; x2)
+f1;2g (t)
Z
(x1;x2)
3 (t; x1; x2)
(x1;x2)
2 (t; x1; x2)+Z
33

75. where
Z(x2) (t; x1; x2) =
8
:
H(2 x2)
V CDS(t; x1; T)
+
1 R2ex22
DNJ
V CDS(t; x1; T)
+
1 R2
2
1+2
e2x2 ENJ
(x1;x2)
2 (t; x1; x2) =
Z
e2x2 Z
8
:
1 R2ex21
H(x1 1)H(2 x2)
V CDS(t; x1 1; T)
+
R 0
x1
V CDS(t; x1 +j1; T)
+
ej1dj1
1 R2
2
1+2
(x1;x2)
3 (t; x1; x2) =
8
:
H(1 x1)H(2 x2)
1 R1ex11
1 R2ex22
DNJ
1 R1
1
1+1
1 R2
2
1+2
e1x12x2 ENJ
(65)

76. Computational Challenges for Above Mentioned Problems
Analytical methods are useful for benchmarking but too restrictive
for practical purposes (mostly are applicable for continuous monitoring
in one-dimensional case)
Numerical methods are more robust
Few challenges remain:
1) Drift-dominated problem
For strong credit names, the asset volatility (the equity volatility is
the asset volatility times the leverage) is small but the mean of the
jump amplitude is large, thus compensator is large
For weak credit names, the asset volatility is even smaller but the
jump frequency is high, thus is large
34

77. Typically, the drift term dominates the diusion term
2) Non-local integral part
Extra complexity to handle the integral term

78. Analytical Methods for 1-d problem with Exponential Jumps
For the current setting, we assume constant model parameters, the
continuous default monitoring, and that the jumps are exponentially
distributed
Due to the time-homogenuity of the problem under consideration,
Green's function G(t; x; T;X) depends on = T t rather than on t; T
separately:
G(t; x; T;X) = (; x;X)
where (; x;X) solves the following problem:
(; x;X) L(X)y(; x;X) = 0;
(0; x;X) = (X x)
(; x; 0) = 0; (; x;X) !
X!1
0
(66)
The Laplace transform of (; x;X) with respect to
(; x;X) ! ^G
(p; x;X) (67)
35

79. solves the following problem:
p^G
(p; x;X)+L(X)y^G
(p; x;X) = (X x)
^G
(p; x; 0) = 0; ^G
(p; x;X) !
X!1
0
(68)
The corresponding forward characteristic equation is given by:
1
2
2 2 (+p)+
= 0 (69)
This equation has three real-valued roots two of which are negative
Hence, the overall solution has the form:
^G
(p; x;X) =
(
C3e 3(Xx); X x
D1e 1(Xx) +D2e 2(Xx) +D3e 3(Xx); 0 X x
(70)
where
D1 =
2
2
( + 1)
( 1 2) ( 1 3)
; D2 =
2
2
( + 2)
( 2 1) ( 2 3)
D3 = e( 1 3)xD1 e( 2 3)xD2; C3 = D1 +D2 +D3
(71)

80. The inverse Laplace transform yields (; x;X)
We compute the Laplace-transformed survival probability
Q(; x) ! ^Q(p; x) (72)
as follows:
^Q
(p; x) =
Z 1
0
^G
(p; x;X)dX
=
Z 1
x
C3e 3(Xx)dX +
3X
j=1
Z x
0
Dje j(Xx)dX
= E0 +E1e 1x +E2e 2x
(73)
where
E0 =
1
p
; E1 =
( 1 +) 2
( 1 2) p
; E2 =
( 2 +) 1
( 2 1) p
(74)
The default time density satis

81. es the following equation:
q(; x) =
@Q(x)(; x)
@
(75)

82. Using Eq.(33) we obtain:
q(; x; T) =
Z 1
0
@(; x;X)
@
dX = g(; x)+f(; x) (76)
where g(; x) is the barrier hitting density:
g(; x) =
2
2
@(; x;X)
@X

87. X=0
(77)
and f(; x) is the probability of the overshoot:
f(; x) =
Z 1
0
Z X
1
$(j)dj
!
(; x;X)dX (78)
Formula (76) is a general result for jump-diusion with arbitrary jump
size distributions
When jumps are exponential:
f(; x) =
Z 1
0
eX(; x;X)dX (79)

88. Using (70), the Laplace-transformed hitting time density is given by:
^q(p; x) = ^g(p; x)+ ^ f(p; x) (80)
where
^g(p; x) =
( + 2)e 2x ( + 1)e 1x
2 1
(81)
and
^ f(p; x) =
2
e 2x e 1x
2( 2 1)( + 3)
(82)
Alternatively, taking the Laplace transform of (75) and using (73) we
obtain:
^q(p; x) =
( 1 +) 2e 1x
( 2 1)
+
( 2 +) 1e 2x
( 1 2)
(83)
Straightforward but tedious algebra shows that (80)-(82) are equiva-lent
with (83)
We express the present value of CDS contract V CDS(; x) with coupon

89. c as:
V CDS(; x) = c
Z
0
er0
Q(0; x)d0
+(1 R)
Z
0
er0
g(0; x)d0 +
1 R
1+
Z
0
er0
f(0; x)d0
(84)
We use (73), (81), and (82) to compute the value of the CDS by
Laplace inversion.

90. Illustration
In Figure we illustrate the jump-diusion model with exponential
jumps using the following market: s(0) = 40, a(0) = 200, l(0) = 160,
r = = 0. We use the following model parameters: = 0:22,
= 0:05, = 0:03, = 1=
We also compare outputs from jump-diusion model with those from
the diusion model obtained by taking 0
For the latter model, we use the equivalent diusion volatility nr
speci

91. ed by nr =
q
2 +2=2, so that nr = 0:074 for the given
model parameters
36

92. 100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
K/S
20% 40% 60% 80% 100% 120% 140%
Implied Volatility
Implied Vol, Jump-diffusion
Implied Vol, Diffusion
0.03
0.02
0.01
0.00
T
0.10 0.60 1.10 1.60 2.10 2.60 3.10 3.60
Fair spread, s(T)
s(T), Jump-diffusion
s(T), Diffusion
Left side: the model implied volatility skew for put options with
maturity six months; right side: the model implied CDS spread
The jump-diusion model generates the implied volatility skew that
is steeper that the diusion model
Unlike the diusion model, the jump-diusion model implies a non-zero
probability of defaulting in short term so that its implied spread
is consistent with the spread observed in the market

93. Asymptotic Solution
We derive an asymptotic solution for the Green's function solving
(66) assuming that the jump intensity parameter is small
We introduce new function:
(; x;X) = exp
(
!
+
2
22 +
2(X x)
)
~(; x;X) (85)
It solves the following propagation problem:
~
(; x;X)
1
2
2~
XX (; x;X)
Z 0
1
~(; x;X j)ejdj = 0; fX 0g
~(0; x;X) = (X x)
~(0;X) = 0; ~(T;X) !
X!1
0
(86)
where = =2
37

94. We assume that 1 and represent ~(; x;X) as follows:
~(; x;X) = ~
(0)(; x;X)+~
(1)(; x;X)+::: (87)
The zero order solution ~
(0)(; x;X) solves the following problem:
~
(0)
(; x;X)
1
2
2~
(0)
XX (; x;X) = 0
~
(0)(0; x;X) = (X x)
~
(0)(; x; 0) = 0; ~
(0)(; x;X) !
X!1
0
(88)
The solution to the above problem is
~
(0)(; x;X) =
1
p
#
n
X x
p
#
!
n
X +x
p
#
!!
(89)
where # = 2 and n(x) is standard normal PDF.

95. The

96. rst order solution ~
(1)(; x;X) solves the following problem:
~
(1)
(; x;X)
1
2
2~
(1)
XX (; x;X) = H(; x;X)
~
(1)(0; x;X) = 0
~
(1)(; x; 0) = 0; ~
(1)(; x;X) !
X!1
0
(90)
where
H(; x;X) =
Z 0
1
~(0)(; x;X j)ejdj
= P
X x
p
#
;
!
P
p
#
X +x
#
;
(91)
p
#
P(a; b) = exp
n
ab+b2=2
o
N(a+b) (92)
and N(x) is standard normal CPDF. We use Duhamel's principle and
represent (1)(;X) as follows:
~
(1)(; x;X) =
Z
0
Z 1
0
(0)(0; x;X)H(0;X)dXd0 (93)

97. Fairly involved algebra yields:
~
(1)(; x;X) =
2
#P
X x
p
#
;
!
p
#
+XP
X +x
p
#
;
!
(X #)P
p
#
X +x
p
#
;
!
p
#
(X +#)exP
X
p
#
;
!
+(X #)exP
p
#
X
p
#
;
!!
p
#

98. Illustration
Model parameters as in the previous

99. gure, T = 10
3.00
2.50
2.00
1.50
1.00
0.50
0.00
X
Analytical
Expansion
Lambda=Zero
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
3.00
2.50
2.00
1.50
1.00
0.50
0.00
X
Analytical
Expansion
Lambda=Zero
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Left side: = 0:03; right side: = 0:10
38

100. FFT Methods
FFT based method is applicable to case of the discrete default mon-itoring
Employs the characteristic function of process x(t)
The advantage of this method is that its implementation is relatively
easy and it can be applied for relatively wide of jump-size distributions
39

101. FFT based method. Characteristic Function
De

102. ne the characteristic function of x(t) by:
bG
(t; T; ) =
Z 1
1
eiXG(t; 0; T;X)dX;
where G(t; x; T;X) is the TPDF p
of x(t), X x(T), 2 R is the
transform variable, and i =
1
From the theory of additive processes:
bG
(t; T; ) = e
R T
t (t0;)dt0
;
where (t; ) is the characteristic exponent:
(t; ) =
1
2
((t))2 i(t) (t)(c$() 1); c$() =
Z 1
1
eij$(j)dj
Accordingly, we can compute TPDF of x(T) by:
G(t; 0; T;X) =
1
2
Z 1
1
h
eiX bG
(t; T; )
i
d (94)
40

103. FFT based method. Backward Problem
Because the increments in x(t) are independent conditional on the
current state values:
G(t; x; T;X) G(t; T;X x)
The value function U(t; x) can be represented as (ignoring coupon
and rebate functions):
U(t; x) =
Z 1
1
u(X)G(t; T;X x)dX
Applying the Fourier transformed density function (94) and exchang-ing
the integration order we obtain (Carr-Madan (1999), Lewis (2001),
Lipton (2001)):
U(t; x) =
1
2
Z 1
1
u(X)
Z 1
1
h
ei(Xx) bG
(t; T; )
i
ddX
=
1
2
Z 1
1
eix
Z 1
1
eiXu(X)dX
bG
(t; T; )
d
(95)
41

104. FFT based method. DFT Algorithm
Observe that Eq.(95) can be computed by applying the two opera-tions
of the DFT algorithm:
U(t; x) = it
t(u(x))

105. eG
(t; T; )
By discretisation of the state space of x and , the relationship
x = 2
N is required for standard DFT algorithm
42

106. FFT based method. Forward Equation for function U(T;X):
U
T (T;X)+LyU(T;X) = 0;
U(t;X) = u(x)
where Ly is the operator adjoint to L
U(T;X) can be represented as the solution to:
U(T;X) =
Z 1
1
u(x)G(t; T;X x)dx
Applying the Fourier transformed density function (94):
U(T;X) =
1
2
Z 1
1
u(x)
Z 1
1
h
ei(Xx) bG
(t; T; )
i
ddx
=
1
2
Z 1
1
eiX
Z 1
1
eixu(x)dx
bG
(t; T; )
d
This can be computed by:
U(T; x) = t
it(u(x))

107. bG
(t; T; )
(96)
43

108. FFT based method. Time Stepping
In case of discrete monitoring, the value function depends on the
state variables observed at discrete times ftmgm=1;:::;m
Compute the value function applying the DFT algorithm at each time
step:
1) Apply the terminal condition by U(tm; x) = u(x)
2) Given U(tm; x), compute U(tm1; x) by:
Um1(x) = e
R tm
tm1; r(t0)dt0
it
t(U(tm; x))

109. bG
(tm1; tm; )
3) Set m ! m 1 and, if m 0, apply the boundary and coupon
conditions and go to 2)
otherwise, if m = 0, the recursion is stopped and the present value is
computed
44

110. FFT based method. Implication
Solutions to forward and backward problems are consistent
We use the forward induction to compute:
TPDF G(0; T;X) ) survival probability at T ) CDS spread at T
Given volatility and jump amplitude parameters, we use the forward
induction and calibrate the term structure of jump intensity (t) by
bootstrapping
European equity and CDS options can also by computed by the for-ward
induction to calibrate volatility and jump amplitude parameters
We use the backward algorithm de

111. ned on the same grid for pricing
and counterparty charge evaluation
45

112. FFT based method. Illustration
JPM
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
-0.20 -0.16 -0.12 -0.08 -0.04 -0.01 0.03 0.07 0.11 0.15 0.19 0.22
y'
1y
2y
3y
4y
5y
6y
7y
8y
9y
10y
Density function for y(t) = x(t) truncated at y = at dier-
ent maturities
46

113. FFT based method. Advantages
1) Can be applied for problems with discrete monitoring and piece-wise
constant model parameters
2) Can be applied for jump-diusions with known characteristic func-tions
(the jump size PDF $(j) can be arbitrary)
3) Can be extended to two-dimensional problems with discrete mon-itoring
4) Complexity of the method using the standard DFT in one dimen-sion
is O(2N logN) per time step (the complexity in two dimensions
is O(2N1N2 log(N1N2)))
5) Relatively fast and easy to implement in one and two dimensions
47

114. FFT based method. Disadvantages
1) DFT assumes that the function is periodic (extend the function)
2) Computational domain is required to be uniform (use fractional
FFT)
3) Convergence is controlled by choosing the grid for transform vari-able
(look at the decay of bG
(tm1; tm; ) - can be slow if the volatility
parameter is small)
4) The scheme is only

115. rst order accurate in the space variable be-cause
of the discontinuity at the barrier (use Hilbert transform (Feng-
Linetsky (2008)))
5) Becomes slow if the number of discrete monitoring times is large
48

116. PDE based methods. Considerations
+ PDE methods are not restricted to uniform grids
+ Convection-dominated problems when drift is large and volatility
is small are easier to handle
- Non-local jump term is dicult to handle, especially, in two dimen-sions
- A direct computation of the integral part by, say, trapezoidal rule
leads to O(N2) complexity
- Using the DFT to compute the convolution part (Andersen-Andreasen
(2000)) leads to O(N logN) complexity but suers from unpleasant
features of the DFT (uniform grids, periodicity, convergence)
However, explicit algorithms of O(N) complexity can be employed if
jumps are exponential or discrete (Lipton (2003), Carr-Mayo (2007),
Toivanen (2008))
49

117. PDE based method. Discretisation
For continuous monitoring:
The computational domain is [0; xmax]
The default boundary is enforced continuously
For discrete monitoring:
The computational domain is [xmin; xmax]
The default boundary is enforced only at default monitoring times (at
intermediate time an arti

118. cial boundary condition must be enforced)
50

119. PDE based method. Time stepping
To compute the value Ul1 at time t = tl given its value Ul at time
t = tl, l = 1; :::;L, we use the following splitting:
U
n Uln
tl
= JlnUl +cl1
n ; n = 2; :::;N 1;
Ul1
n U
n
tl
= Dln
Ul1; n = 2; :::;N 1
(97)
At the

120. rst step, we use the explicit scheme to approximate the inte-gral
part and compute the auxiliary function U given Uln
At the second step, we use the implicit scheme to approximate the
diusive step and compute Ul1
n given U
51

121. PDE based method. Integral part for discrete jumps
Given $(j) = (j ), 0:
Jln
= U(tl; xn ); n = 2; :::;N 1
We approximate Jln
by linear interpolation with the second order ac-curacy:
Jln
j1 +(1 !nj)Uln
= !njUln
j
where
!nj =
xnj (xn )
xnj xnj1
nj = minfj : xj1 xn xjg
52

122. PDE based method. Integral part for exponential jumps I
Given $(j) = ej, 0:
J(x) =
Z 0
1
ejU(x+j)dj
For a small number h, h 0:
J(x+h) =
Z 0
1
ejU(x+h+j)dj
z=h+j
= eh
Z h
1
ezU(x+z)dz
= eh
Z 0
1
ezU(x+z)dz +
Z h
0
ezU(x+z)dz
!
= ehJ(x)+ ~ J1(x)
53

123. PDE based method. Integral part for exponential jumps II
Expanding U(x+z) in Taylor series around z = 0 yields:
~ J1(x) = eh
Z h
0
ezU(x+z)dz = ~a0U(x)+~a1U0 +O(h3);
where
~a0 = eh
Z h
0
ezdz = 1 eh; ~a1 = eh
Z h
0
zezdz =
h
1 eh
Accordingly, with the second order accuracy
J(x+h) = ehJ(x)+w0(; h)U(x)+w1(; h)U(x+h)
where
w0(; h) =
1 (1+h) eh
h
; w1(; h) =
h
1 eh
h
54

124. PDE based method. Improving the convergence
Apply the

125. xed point iterations (d'Halluin et al(2005)):
1) Set V 0 = Ul +tlcl1
n ;
2) For p = 1; 2; :::; p apply the scheme (97):
V
n V 0
n
tl
= Jln
V p1; n = 2; :::;N 1;
V p
n V
n
tl
= Dln
V p; n = 2; :::;N 1
3) if norm jjV p V p1jj in becomes small, stop and set Ul1 = V p
Typically, p = 2 is enough
55

126. PDE based method. Summary
1) The scheme has O(N) complexity per each time step
2) Although

127. rst order in time, the implicit scheme tends to be more
stable than the Crank-Nicolson based scheme (especially for forward
equation and two-dimensional problems)
3) The scheme is second order accurate in the spacial variable if the
drift term is not dominant, otherwise it is

128. rst order accurate (D is
discretisized appropriately)
4) A similar scheme is applied for the forward equation
5) As before, using the same grid, the forward scheme is applied for
model calibration and the backward scheme is applied for pricing
56

129. Numerical Methods for Two Dimensional Problem
We consider the backward problem for the value function U(t; x1; x2):
Ut +MU = c(t; x1; x2)
U(T; x1; x2) = u(x1; x2)
(98)
M= D1 +D2 +D12 +J1 +J2 +J12
D1 and D2 are 1-d diusion-convection operators in x1 and x2 direc-tions,
respectively
J1 and J2 are 1-d orthogonal integral operators in x1 and x2 direc-tions,
respectively
D12 is the correlation operator, D12U(t; x1; x2) 1(t)2(t)Ux1x2(t; x1; x2)
J12 is the cross integral operator:
J12U(t; x1; x2) f1;2g(t)
Z 0
1
Z 0
1
U(t; x1+j1; x2+j2)$(j1)$(j2)dj1dj2
57

130. Counterparty Charge Using Structural Model
Let x1(t) and x2(t) be the stochastic drivers for the reference name
and the counterparty, respectively
The value of the countrparty charge U(t; x1; x2) de

131. ned as the solution
to (8) solves the following problem:
Ut +MU(t; x1; x2) = 0;
U(T; x1; x2) = 0;
U(t; x1; x2) = 0; x1 b1; x2 b2 (1 2);
U(t; x1; x2) = (1 R2) maxfC(t; x1); 0g; x1 b1; x2 b2; (1 2);
U(t; x1; x2) = (1 R2)(1 R1); x1 b1; x2 b2; (1 = 2);
lim
x1!1
U(t; x1; x2) = 0; lim
x2!1
U(t; x1; x2) = 0
C(t; x) is the value of CDS contract without counterparty risk
Joint defaults are possible under the discrete monitoring
58

132. Discretisation
We develop a modi

133. ed Craig-Sneyd (1988) discretization scheme to
compute the solution at time tl1, Ul1
n;m, given the solution at time
tl, Ul
n;m, l = 1; :::;L, as follows:
n;m = Ul
(1 tlD1)U
n;m +tl
(D2 +D12 +J1 +J2 +J12)Ul
n;m +cl1
n;m
;
(1 tlD2)Ul1
n;m tlD2Ul
n;m = U
n;m
In the

134. rst line, for each

135. xed index m we apply the jump operators and
diusion operator in x1 direction, the correlation operator, and coupon
payments (if any); and solve the tridiagonal system of equations to
get the auxiliary solution U
;m
In the second line, keeping n

136. xed, we apply the implicit step in x2
direction and solve the system of tri-diagonal equations to get the
solution
59

137. Discretisation of Cross Jump Part
Direct methods are infeasible because of O(N2M2) complexity
DFT method (Clift-Forsyth (2008)) has O(NM logNM) complexity
but suers from problems associated with the DFT
Explicit methods with O(NM) complexity are available for discrete
and exponential jumps (Lipton-Sepp (2009))
The simplest case is if jumps are discrete:
J12U = U(x1 1; x2 2)
This term is approximated by bi-linear interpolation with the second
order accuracy leading to the O(NM) complexity
60

138. Discretisation of the Jump Part. Negative exponential jumps
Consider the integral:
J(x1; x2) = 12
Z 0
1
Z 0
1
e1j1+2j2U(x1 +j1; x2 +j2)dj1dj2
Take small numbers hx and hy, hx 0, hy 0:
J(x1 +h1; x2 +h2) = 12
Z 0
1
Z 0
1
e1j1+2j2U(x1 +h1 +j1; x2 +h2 +j2)dj1dj2
= 12e1h12h2
Z h1
1
Z h2
1
e1z1+2z2U(x1 +z1; x2 +z2)dz1dz2
= 12e1h12h2
Z 0
1
Z 0
1
+
Z h1
0
Z 0
1
+
Z 0
1
Z h2
0
+
Z h1
0
Z h2
0
h
e1z1+2z2U(x1 += e1h12h2J(x1; x2)+e2h2Je10(x1; x2)+e1h1Je01(x1; x2)+Je11 (x1; x2)
Integrals Je10(x; y), Je01(x; y), and Je11(x; y) can be computed by recur-sion
with second order accuracy and O(NM) complexity
61

139. Discretisation of the Jump Part. Improving the convergence
At each time step, we apply the

140. xed point iterations as follows (p = 2
is enough):
1) Set V 0
n;m = Ul
n;m +tlCn;mUl
n;m +tlcl1
n;m;
2) For p = 1; 2; :::; p apply the above scheme:
V j
n;m = V 0
n;m +tl(J12 +J1 +J2)V p1;
(1 tlD1)V
n;m = tlD2V p1
n;m +V j
n;m;
(1 tlD2)V p
n;m tlD2V p1
n;m
n;m = V
(99)
3) if norm jjV p V p1jj becomes small, stop and set Ul1 = V p
62

141. Discretisation. Final Remarks
1) The overall complexity of this method per time step is O(NM)
operations (using DFT method to compute the convolution leads to
O(NM log(NM)) complexity)
2) The scheme is

142. rst order accurate in time
3) The scheme is second order accurate in spacial variables (if the
drift is not dominant)
4) The modi

143. ed scheme is applied for the forward problem, so that,
it needed, the calibration problem in two dimensions can be solved
eciently
63

144. Example. Input data for model calibration
JPM C
s(0) 36.49 8.47
L(0) 604.11 353.07
s(0)=L(0) 16.56 41.68
R 40% 40%
l(0) 241.644 141.228
v(0) 278.134 149.698
b -0.1406 -0.0582
(1) 0.1406 0.0582
(2) 0.0703 0.0291
0.0262 0.0113
Use two choices for the jump size:
1) 1 = b in the model with discrete jumps and 1
1
= 1
b in
the model with exponential jumps;
2) 2 = 1
2b in the model with discrete jumps and 1
2
= 1
2b in
the model with exponential jumps
64

145. Example. Input data for model calibration
Spread data for model calibration and the survival probability, default
leg, and annuity leg implied using the hazard rate model
CDS Spread Survival Prob Default Leg Annuity Leg
T JPM C JPM C JPM C JPM C
1y 0.0105 0.0286 0.9826 0.9535 0.0174 0.0465 0.9913 0.9766
2y 0.0118 0.0271 0.9614 0.9137 0.0386 0.0863 1.9633 1.9099
3y 0.0134 0.0257 0.9348 0.8798 0.0652 0.1202 2.9114 2.8065
4y 0.0147 0.0249 0.9063 0.8475 0.0937 0.1525 3.8320 3.6701
5y 0.0160 0.0248 0.8743 0.8138 0.1257 0.1862 4.7223 4.5007
6y 0.0161 0.0243 0.8498 0.7857 0.1502 0.2143 5.5841 5.3002
7y 0.0162 0.0238 0.8268 0.7590 0.1732 0.2410 6.4223 6.0725
8y 0.0163 0.0236 0.8034 0.7319 0.1966 0.2681 7.2374 6.8179
9y 0.0164 0.0234 0.7804 0.7056 0.2196 0.2944 8.0292 7.5366
10y 0.0165 0.0233 0.7582 0.6801 0.2418 0.3199 8.7985 8.2294
65

146. Example. Calibrated Intensity Rates
For both choice of jumps size distributions and the jump sizes, the
model is calibrated to the term structure of CDS spreads given in
Table 2 using the forward induction
dis
1 (lambda^fdisg f1g) and dis
2 (lambda^fdisg f2g) stand for model
with discrete jumps with sizes 1 and 2, respectively
exp
1 (lambda^fexpg f1g) and exp
2 (lambda^fexpg f2g) stand for model
with exponential jumps with sizes 1
1
and 1
2
, respectively
66

147. Example. Input Data
Implied density of the driver x(t) for JMP and C in the model with
exponential jump, = 1=b, at maturities 1, 5, and 10 years
67

148. Example. CDS option volatility
The log-normal CDS option volatility implied from model values of
one year option on

149. ve year CDS contract as a function of the money-ness
K;

150. =S;

151. The model implied log-normal volatility ;

152. exhibits a positive skew
This eect is in line with the market because the CDS spread volatility
is expected to increase when the CDS spread increases, so that option
sellers charge an extra premium for out-of-the-money CDS options
68

153. Example. Log-normal equity volatility
Log-normal equity volatility implied from model values of put options
with maturity 6 months using the Black-Scholes formula
The model implies a remarkable skew in line with that observed in
the market
The smaller the jump size the higher is the implied model volatility
because the

154. rm value is expected to have more jumps before the
barrier crossing so that the realized volatility is expected to be higher
69

155. Example. Implied Gaussian correlation
We compute the model implied Gaussian correlation by equating the
fair spread of the

156. rst to default swap referencing JPM and C to that
computed using the Gaussian copula with implied correlation
We use the two choices for the model correlation parameter: = 0:50
and = 0:99
The model with exponential jumps produces lower implied correlations
The model with smaller jump amplitudes implies smaller correlations
70

157. Illustration. Counterparty charge
We compute the counterparty charge for par CDS on JPM sold by C
and that for par CDS on C sold by JPM as functions of CDS maturity
using two model correlation parameters: = 0:5 and = 0:99
We use R = 0 for the counterparty recovery and normalize the coun-terparty
charge by the present value of the default leg of CDS on
JPM corresponding to CDS maturity
For a moderate correlation assumption with = 0:50, the model
with discrete large jump implies the countrepaty charge in amount
of 10% 15% of the present value of the CDS protection leg on
the underlying name, while, for a high correlation assumption with
= 0:99, this proportion grows to 30% 40%
71

158. Counterparty charge for CDS on JPM sold by C (JMP-C, top)
and for CDS on C sold by JPM (C-JPM, bottom)
72

159. CDS spread with counterparty risk, recent data
60
55
50
45
40
35
30
T, years
Spread, bp
InputSpread
Risky Seller, rho=0.75
Risky Buyer, rho=0.75
Risky Seller, rho=0.0
Risky Buyer, rho=0.0
Risky Seller, rho=-0.75
Risky Buyer, rho=-0.75
1 2 3 4 5 6 7 8 9 10
156
146
136
126
116
106
96
T, years
Spread, bp
InputSpread
Risky Seller, rho=0.75
Risky Buyer, rho=0.75
Risky Seller, rho=0.0
Risky Buyer, rho=0.0
Risky Seller, rho=-0.75
Risky Buyer, rho=-0.75
1 2 3 4 5 6 7 8 9 10
Equilibrium spread for protection buyer and protection seller for
CDS on JPM with MS as the counterparty, left, and for CDS
on MS with JPM as the counterparty, right
73

160. Counterparty charge. Conclusions
1) The larger is the correlation, the larger is the counterparty charge
because, given the counterparty default, the protection lost is larger
in case of high correlation
2) The larger the jump size, the larger is the counterparty charge
because the model with higher jumps implies a larger correlation
3) The model with discrete jumps implies a larger counterparty charge
than the model with exponential jumps because the former implies
larger correlation and CDS spread volatility
4) The counterparty charge is not symmetric. It is expected that a
more risky counterparty implies a higher counterparty charge
74

161. Conclusions
1) We have proposed an extended structural model capable of

162. tting
arbitrary term structures of CDS spreads
2) Applying this model, we have obtained a novel method to analyse
the counterparty risk
3) We have developed a number of semi-analytical and numerical
methods to solve calibration and pricing problems in an ecient way
75

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