Quantitative Methods for Counterparty Risk

Artur Sepp

Joint work with Alex Lipton

Bank of America Merrill Lynch

Quantitative Finance WorkshopTechnical University of Helsinki

September 2, 2009

1

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

References for technical details

1) Lipton, A., Sepp, A. (2009) Credit Value Adjustment for CreditDefault Swaps via the Structural Default Model, The Journal of CreditRisk, 5(2), 127-150http://ssrn.com/abstract=2150669

2) Lipton, A. and Sepp, A. (2011). Credit value adjustment in theextended structural default model. Forthcoming in The Oxford Hand-book of Credit Derivatives, Oxford University Working paperhttp://ukcatalogue.oup.com/product/9780199669486.do

3) Inglis, S., Lipton, A., Savescu, I., Sepp, A. (2008) Dynamic creditmodels, Statistics and its Interface, 1(2), 211-227http://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 stochasticvolatility and jumps, Wilmott Magazine September, 50-62http://ssrn.com/abstract=1412327

3

Simple Example of a CDS contract, I

The reference name defaults at random time τ1

Contract maturity is T , the spread is c

The pay-off for the protection buyer :

V Buy =

{−c, τ1 > T1, τ1 ≤ T

(1)

The pay-off for the protection seller:

V Sell =

{c, τ1 > T−1, τ1 ≤ T

(2)

4

Simple Example of a CDS contract, II

Fair value of the contract for protection buyer:

PV (1) = DF (0, T ) (P(τ1 ≤ T )− cP(τ1 > T )) , (3)

P(τ1 ≤ T ) is the default probabilityDF (0, T ) the risk-free discount factor

Coupon c is set so that PV (1) = 0:

c =P(τ1 ≤ T )

1− P(τ1 ≤ T )(4)

Note that the probability of default is typically small,P(τ1 ≤ 1) ∈ [0.01, .05] for investment grade companies

Thus, the protection seller obliges to pay $1 in return for a muchsmaller fee c (c ∈ [0.01, .05] for investment grade names)!

5

Counterparty Risk

What if the protection seller, the counterparty, is unable to honourits obligations given that the reference name defaults?

Let the default time of the counterparty be τ2

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-off for the protection buyer :

V =

−c, τ1 > T,1, τ1 ≤ T, τ2 > τ10, τ1 ≤ T, τ2 ≤ τ1

(5)

6

Credit Value Adjustment

Now, the fair value of the contract for protection buyer:

PV (1,2) = DF (0, T ) (P(τ1 ≤ T, τ2 > τ1)− cP(τ1 > T )) (6)

We define 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 counterpartyand the correlation between the reference name and the counterparty

We note that in case of perfect correlation P(τ1 ≤ T, τ2 > τ1) = 0 sothat the protection buyer loses the most if there is strong correlationbetween τ1 and τ2

7

CDS basics

Credit default swap (CDS) - provides to the buyer a protection againsta 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 ofCDS is zero

As the time goes by, the mark-to-market (MtM) value of the CDScontract fluctuates

In particular, when the credit quality of the reference name worsens,the MtM increases

8

Counterparty Risk

When the CDS protection is sold by a defaultable counterparty, theprotection buyer faces the risk of losing a part of the mark-to-marketvalue of the CDS, if it is positive for the buyer, due to the counterpartydefault

The loss is profound if the credit quality of both the reference creditand the counterparty worsen simultaneously but the counterparty de-faults first

Because big banks are intermediaries between each other and otherinstitutions (hedge funds, insurers), failure of one of them poses riskfor more failures (”domino effect”)

9

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 SwapDealers 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 outstandingis $403.1 trillion, while the notional amount of the equity derivativesis $8.7 trillion

10

Credit Value Adjustment

Let τ1 and τ2 be default times of the reference name and the coun-terparty, respectively

The Credit Value Adjustment, C(t), is the expected maximal potentialloss due to counterparty default up to CDS maturity T :

C(t) = (1−R2)Et

[∫ TtD(t, t′) max

{Et′

[C(t′) |E(t′)

],0}1{E(t′)}dt

′](8)

C(t) is cash flow of CDS contract (long protection) without counter-party risk discounted to time t

E(t) = {τ1 > t, τ2 = t}

R2 is recovery rate of counterparty obligations

D(t, T ) = e−∫ Tt r(t′)dt′ is the risk-free discount factor

11

Motivation

Model for the counterparty risk evaluation need to:

1) describe realistic dynamics of CDS spreads (jump-diffusions)

2) create profound correlation effects (correlated jump-diffusions withsimultaneous 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 fit term struc-ture effects

12

Credit Modeling

Structural approach (Merton (1974), Black-Cox (1976))

Reduced form models (Jarrow-Turnbull (1995), Duffie-Singleton (1997),Lando (1998))

Hybrid Models

13

Generic 1-d structural model

The value of the firm assets, a(t), is driven by

da(t)

a(t)= (r(t)− ζ(t)− κλ(t))dt+ σ(t)dW (t) + jdN(t), (9)

Jumps j have probability density function $(j)

κ =∫∞−∞ e

j$(j)dj − 1 is the compensator

The firm’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{∫ t

0(r(t′)− ζ(t′))dt′

}(11)

14

Default Time

The default time τ is defined by:

τ = min{t : a(t) ≤ l(t)}

Without the loss of generality we can assume that the default istriggered continuously or over a set of discrete monitoring times (t ∈{td})

Continuous monitoring: convenient choice for analytical develop-ments

Discrete monitoring: probably more realistic as the firm value isobserved over the discrete times (quarterly reports), more suitablefor Monte-Carlo and numerical methods

15

Equity Value

We assume that the model value of equity price per share, s(t), isgiven by:

s(t) =

{a(t)− l(t) = E(t)

(ex(t) − 1

)l(0), if t < τ

0, if t ≥ τ(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 defined by l(0) = RL(0), where R is an average recovery ofthe firm’s liabilities and L(0) is total debt per share.

The volatility of the equity price, σeq(t), is approximately related toσ(t) by:

σeq(t) =

(1 +

l(t)

s(t)

)σ(t) (13)

16

Jump Size Distribution

We assume that jumps have either a discrete negative amplitude ofsize −ν, ν > 0, with

$(j) = δ(j + ν), κ = e−ν − 1 (14)

or jumps have negative exponential distribution with mean size 1ν ,

ν > 0, with:

$(j) = νeνj, j < 0, κ =ν

ν + 1− 1 = −

1

ν + 1(15)

17

Generic 1-d structural model, II

Introduce

x(t) = ln a(t)l(0) - the log of the normalized asset value

dx(t) = µ(t)dt+ σ(t)dW (t) + jdN(t), x(0) ≡ ξ = lna(0)

l(0)(16)

Note that y(t) = x(t) − ξ is an additive process with independenttime-dependent increments (Sato (1999))

µ(t) = −12σ

2(t)− κλ(t)

The default time τ is defined by:

τ = min{t : x(t) ≤ 0}

The default is triggered either continuously or discretely

18

Generic 2-d structural modelWe consider two firms and assume that their asset values are drivenby the following SDEs:

dai (t)

ai (t)= (r(t)−ζi(t)−κiλi (t))dt+σi (t) dWi (t)+

(eji − 1

)dNi (t) (17)

where i = 1,2

Standard Brownian motions W1(t) and W2(t) are correlated with cor-relation ρ.

Jumps in the joint dynamics occur according to the Poisson processN{1,2}(t) with the intensity rate:

λ{1,2}(t) = min{ρ,0}min{λ1(t), λ2(t)}

Idiosyncratic jumps occur according to Poisson processes N1(t) andN2(t) with jump intensities λ{1}(t) and λ{2}(t), respectively, specifiedas follows:

λ{1}(t) = λ1(t)− λ{1,2}(t), λ{2}(t) = λ2(t)− λ{1,2}(t)

19

Jump Size PDF and Instantaneous CorrelationConsider the instantaneous correlations between x1(t) and x2(t) underthe assumption of discrete jumps, ρdis

12 , and that under exponentialjumps, ρexp

12 :

ρdis12 =

ρσ1σ2 + λ{1,2}ν1ν2√σ2

1 + λ1ν21

√σ2

2 + λ2ν22

,

ρexp12 =

ρσ1σ2 +λ{1,2}ν1ν2√

σ21 + 2λ1

ν21

√σ2

2 + 2λ2ν2

2

(18)

If the systematic intensity λ{1,2} is large, ρdis12 ∼ 1 and ρexp

12 ∼12

From experiments: the maximal implied Gaussian correlation that canbe achieved (using ρ = 0.99) is about 90% for the model with discretejumps and about 50% for the model with exponential jumps

The assumption about exponential jumps is not realistic by modellingthe joint dynamics of strongly correlated firms belonging to one in-dustry group (such as financial companies)

20

Multi-dimensional CaseWe consider N firms and assume that their asset values are drivenby the same equations in the two-dimensional case with the index irunning from 1 to N , i = 1, ..., N

We correlate diffusions in the usual way and assume that:

dWi (t) dWj (t) = ρij (t) dt (19)

We correlate jumps following the Marshall-Olkin (1967) idea. LetΠ(N) be the set of all subsets of N names except for the emptysubset {∅}, and π its typical member. With every π we associate aPoisson process Nπ (t) with intensity λπ (t), and represent Ni (t) as:

Ni (t) =∑

π∈Π(N)

1{i∈π}Nπ (t) (20)

λi (t) =∑

π∈Π(N)

1{i∈π}λπ (t) (21)

Thus, we assume that there are both collective and idiosyncratic jumpsources

21

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), {x > 0}V (T, x) = v(x), {x > 0}V (t, x) = g(t, x), {x ≤ 0}V (t, x) →

x→∞ υ(t, x)

(22)

where L(x) is the infinitesimal operator of process x(t):

L(x) = D(x) + λ(t)J (x) (23)

D(x) is a differential operator:

D(x)f(x) =1

2σ2(t)fxx (x) + µ(t)fx (x)− λ (t) f (x) (24)

and J (x) is a jump operator:

J (x)f(x) =∫ 0

−∞f(x+ j)$(j)dj (25)

22

For discrete negative jumps

J (x)f(x) = f(x− ν) (26)

for exponential jumps

J (x)f(x) = ν∫ 0

−∞f(x+ j)eνjdj (27)

One-Dimensional Problem. Discrete Monitoring

When monitoring is discrete, the pricing problem is formulated asfollows:

Vt (t, x) + L(x)V (t, x)− r (t)V (t, x) = c(t, x), {−∞ < x <∞} ,V (T, x) = v(x), {x > 0}V (t, x) = g(t, x), {x ≤ 0} , t ∈ {td1, ..., t

dm}

V (t, x) →x→∞ υp(t, x)

V (t, x) →x→−∞

υm(t, x), t /∈ {td1, ..., tdm}

(28)

23

One-Dimensional Problem. Localization

In case of both the discrete and continuous default monitoring, thecomputational domain is (−∞,∞)

However, for the continuous monitoring, we can switch to the semi-bounded domain [0,∞)

Representing the integral term in problem Eq.(22) as follows:

J (x)f(x) =∫ 0

−∞f(x+ j)$(j)dj

=∫ 0

−xf(x+ j)$(j)dj +

∫ −x−∞

g(x+ j)$(j)dj

≡ J (x)f(x) + Z(x)(x)

(29)

where J (x) is defined by:

J (x)f(x) =∫ 0

−xf(x+ j)$(j)dj (30)

24

and Z(x)(x) is the deterministic function depending on the contractboundary condition g(x).

As a result, we can formulate the pricing problem in the semi-boundeddomain [0,∞) 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→∞ υ(t, x)

(31)

d (t, x) = λ (t)Z(x) (t, x) (32)

One-Dimensional Problem. Green’s FunctionWe 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)†G (T,X) = 0, {X > 0}G(t,X) = δ(X − x)

G(T,X) = 0, {X ≤ 0}G(T, x) →

x→∞ 0

(33)

with L(x)† being the infinitesimal operator adjoint to J (x):

L(x)† = D(x)†+ λ(t)J (x)† (34)

where D(x)† is the differential operator:

D(x)†g(x) =1

2σ2(t)gxx (x)− µ(t)gx (x)− λ (t) g (x) (35)

25

and J (x)† is the jump operator:

J (x)†g(x) =∫ 0

−∞g(x− j)$(j)dj (36)

Two-Dimensional Problem

We denote the value function of the contract by V (t, x1, x2) whichsolves the backward equation:

Vt (t, x1, x2) + L(x1,x2)V (t, x1, x2)− r (t)V (t, x1, x2) = c(t, x1, x2), {x1 > 0, x2 > 0}V (T, x1, x2) = v(x1, x2), {x1 > 0, x2 > 0}V (t, x1, x2) = g1(t, x1, x2), {x1 ≤ 0, x2 > 0}V (t, x1, x2) = g2(t, x1, x2), {x1 > 0, x2 ≤ 0}V (t, x1, x2) = g3(t, x1, x2), {x1 ≤ 0, x2 ≤ 0}V (t, x1, x2) →

x1→∞υ1(t, x1,x2), V (t, x1, x2) →

x2→∞υ2(t, x1, x2)

(37)

where L(x1,x2) is the infinitesimal backward operator corresponding tothe bivariate dynamics:

L(x1,x2) = D(x1) +D(x2) + C(x1,x2) + λ{1}(t)J (x1) + λ{2}(t)J (x2) + λ{1,2}(t)J (x1,x2)

(38)

26

C(x1,x2) is the correlation operator:

C(x1,x2)f(x1, x2) ≡ ρσ1(t)σ2(t)fx1x2(x1, x2)− λ{1,2} (t) f (x1, x2) (39)

and J (x1,x2) is the cross integral operator defined as follows:

J (x1,x2)f(x1, x2) ≡∫ 0

−∞

∫ 0

−∞f(x1 + j1, x2 + j2)$(j1)$(j2)dj1dj2 (40)

Two-Dimensional Problem. Localization

In case of the discrete default monitoring, the PDE is defined on(−∞,∞) × (−∞,∞) and the boundary condition is applied when t ∈{td1, ..., t

dm}.

For the case of continuous monitoring, the integral term in Eq.(37),can be represented as follows:

J (x1,x2)f(x1, x2) =∫ 0

−∞

∫ 0

−∞f(x1 + j1, x2 + j2)$(j1)$(j2)dj1dj2

=∫ 0

−x1

∫ 0

−x2

f(x1 + j1, x2 + j2)$(j1)$(j2)dj1dj2

+

{∫ −x1

−∞

∫ 0

−x2

+∫ 0

−x1

∫ −x2

∞+∫ −x1

−∞

∫ −x2

−∞

}g(x1 + j1, x2 + j2)$(j1)$(j2)dj1dj2

≡ J (x1,x2)f(x1, x2) + Z(x1,x2)1 (x1, x2) + Z

(x1,x2)2 (x1, x2) + Z

(x1,x2)1,2 (x1, x2)

Therefore, we only need to consider the integral term J (x1,x2) definedon the bounded domain and augment the source term by determin-

27

istic functions Z1, Z2, and Z3 defined by integrating of g1, g2, g3,respectively.

Thus, we can localize the problem in the positive quadrant [0,∞) ×[0,∞)

Similar considerations apply for multi-dimensional case.

Multi-Dimensional Problem

For brevity, we assume the continuous monitoring

We can formulate a typical pricing equation in the positive cone R(N)+

as follows:

∂tV (t, ~x) + L(~x)V (t, ~x)− r (t)V (t, ~x) = χ (t, ~x) (41)

V(t, ~x0,k

)= φ0,k (t, ~y) , V (t, ~x) →

xk→∞φ∞,k (t, ~y) (42)

V (T, ~x) = ψ (~x) (43)

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

~x = (x1, ..., xk, ...xN)

~x0,k =(x1, ...,0

k, ...xN

)~yk =

(x1, ...xk−1, xk+1, ...xN

) (44)

28

The corresponding integro-differential operator L(N) can be writtenin the form

L(~x)f (~x) = 12∑iσ2i ∂

2i f (~x) +

∑i,j,j>i

ρijσiσj∂i∂jf (~x)

+∑iµi∂if (~x) +

∑π∈Π(N)

λπ

( ∏i∈πJ (xi)f (~x)− f (~x)

)(45)

For discrete negative jumps

J (xi)f (~x) = H (xi − νi) f (x1, ..., xi − νi, ...xN) (46)

For negative exponential jumps,

J (xi)f (~x) = νi

∫ 0

−xif (x1, ..., xi + ji, ...xN) eνijidji (47)

The corresponding adjoint operator is

L(~x)†g (~x) = 12∑iσ2i ∂

2i g (~x) +

∑i,j,j>i

ρijσiσj∂i∂jg (~x)

−∑iµi∂ig (~x) +

∑π∈Π(N)

λπ

( ∏i∈πJ (xi)†g (~x)− g (~x)

)(48)

where

J (xi)†g (~x) = g (x1, ..., xi + νi, ...xN) (49)

or

J (xi)†g (~x) = νi

∫ 0

−∞g (x1, ..., xi − ji, ...xN) eνijidji (50)

It is easy to check that in both cases∫R

(N)+

[J (xi)f (~x) g (~x)− f (~x) J (xi)†g (~x)

]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)†G(T, ~X

)= 0 (52)

G(T, ~X0k

)= 0, G

(T, ~X

)→

Xk→∞0 (53)

G(t, ~X

)= δ

(~X − ~x

)(54)

By integrating by parts∫ T0

∫R

(N)+

[L(~x)V (t, ~x)G (t, ~x)− V (t, ~x) L(~x)†G (t, ~x)

+∂tV (t, ~x)G (t, ~x)− V (t, ~x) ∂tG (t, ~x)] d~xdt = 0

(55)

we obtain

V (t, ~x) = −∫ Tt

∫R

(N)+

χ(t′, ~x′

)D(t, t′

)G(t, ~x, t′, ~x′

)d~x′dt′ (56)

+∑k

∫ Tt

∫R

(N−1)+

φ0,k

(t′, ~y′

)D(t, t′

)gk(t, ~x, t′, ~y′

)d~y′dt′

+D (t, T )∫R

(N)+

ψ(~x′)G(t, ~x, T, ~x′

)d~x′

where

gk(t, ~x, T, ~Y

)= 1

2σ2k∂kG

(t, ~x, T, ~X

)∣∣∣Xk=0

~Y =(Y1, ..., Yk−1, Yk+1, ..., YN

) (57)

represents the hitting time density for the corresponding piece of theboundary.

This extremely useful formula shows that instead of solving the back-ward pricing problem with non-homogeneous right hand side andboundary conditions, we can solve the forward propagation problemfor Green’s function with homogeneous right hand side and boundaryconditions.

Pricing Credit Products. Survival Probability

The single name survival probability function, Q(x)(t, x, T ), is definedby:

Q(x)(t, x, T ) ≡ EQt [1{τ>T}] (58)

given that τ > t.

Q(x)(t, x, T ) solves the following backward equation:

Q(x)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

(59)

29

Pricing Credit Products. Joint Survival Probability

We define the joint survival probability, Q(x1,x2)(t, x1, x2, T ), as follows:

Q(x1,x2)(t, x1, x2, T ) ≡ EQt [1{τ1>T,τ2>T}]

given that τ1 > t and τ2 > 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→∞

Q(x)(t, x2, T ),

Q(x1,x2)(t, x1, x2, T ) →x2→∞

Q(x)(t, x1, T )

30

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 CDSt (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→∞ −c

∫ TtD(t, t′)dt′

d (t, x) = λ (t)Z(x) (x)

(60)

Assuming that the effective CDS recovery rate, Rex, is floating andrepresents the residual value of the firm assets given the default, weobtain:

Z(x)(x) =

H(ν − x)(1−Rex−ν

), DNJ(

1−R ν1+ν

)e−νx, ENJ

(61)

for discrete and exponential jumps, respectively.

31

Pricing Credit Products. Equity Put Option

Assuming that τ > t, we represent pay-off of put option, vPut(T, s),with strike price K and maturity T , as follows:

vPut (T, s) = (K − s(T ))+ 1{τ>T}+K1{τ≤T} (62)

The value function of V Put(t, x) as function of x solves the followingproblem:

V Putt (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→∞ 0

d (t, x) = λ (t)D (t, T )Z(x) (x)

(63)

Z(x) (x) =

{KH (ν − x) , DNJ

Ke−νx ENJ(64)

32

Pricing Credit Products. Credit Value Adjustment

We denote by x1 the value of driver associated with the firm valueof CDS reference name and by x2 the value of the driver of thecounterparty firm value

Under the bivariate dynamics, the value of the countrparty chargeV CV A(t, x1, x2, T ) defined as the solution to (8) solves the followingproblem:

V CV At (t, x1, x2, T ) + L(x1,x2)V CV A (t, x1, x2, T )− r (t)V CV A (t, x1, x2, T ) = −d (t, x1, x2)

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→∞

0, V (t, x1, x2, T ) →x2→∞

0

d (t, x1, x2) = λ{2} (t)Z(x2) (t, x1, x2)

+ λ{1,2} (t)(Z

(x1,x2)2 (t, x1, x2) + Z

(x1,x2)3 (t, x1, x2)

)33

where

Z(x2) (t, x1, x2) =

H(ν2 − x2)

(V CDS(t, x1, T )

)+

(1−R2e

x2−ν2)

DNJ(V CDS(t, x1, T )

)+

(1−R2

ν21+ν2

)e−ν2x2 ENJ

Z(x1,x2)2 (t, x1, x2) =

H(x1 − ν1)H(ν2 − x2)

(V CDS(t, x1 − ν1, T )

)+

(1−R2e

x2−ν2)

DNJ

ν1∫ 0−x1

(V CDS(t, x1 + j1, T )

)+eνj1dj1

(1−R2

ν21+ν2

)e−ν2x2 ENJ

Z(x1,x2)3 (t, x1, x2) =

H(ν1 − x1)H(ν2 − x2)(1−R1e

x1−ν1) (

1−R2ex2−ν2

)DNJ(

1−R1ν1

1+ν1

) (1−R2

ν21+ν2

)e−ν1x1−ν2x2 ENJ

(65)

Computational Challenges for Above Mentioned Problems

Analytical methods are useful for benchmarking but too restrictivefor practical purposes (mostly are applicable for continuous monitoringin one-dimensional case)

Numerical methods are more robust

Few challenges remain:

1) Drift-dominated problem

For strong credit names, the asset volatility σ (the equity volatility isthe asset volatility times the leverage) is small but the mean of thejump amplitude is large, thus compensator κ is large

For weak credit names, the asset volatility σ is even smaller but thejump frequency is high, thus λ is large

34

Typically, the drift term −κλ dominates the diffusion term σ

2) Non-local integral part

Extra complexity to handle the integral term

Analytical Methods for 1-d problem with Exponential Jumps

For the current setting, we assume constant model parameters, thecontinuous default monitoring, and that the jumps are exponentiallydistributed

Due to the time-homogenuity of the problem under consideration,Green’s function G(t, x, T,X) depends on τ = T − t rather than on t, Tseparately:

G (t, x, T,X) = Γ (τ, x,X)

where Γ (τ, x,X) solves the following problem:

Γτ (τ, x,X)− L(X)†Γ(τ, x,X) = 0,

Γ(0, x,X) = δ(X − x)

Γ(τ, x,0) = 0, Γ(τ, x,X) →X→∞

0(66)

The Laplace transform of Γ(τ, x,X) with respect to τ

Γ(τ, x,X) −→ G (p, x,X) (67)

35

solves the following problem:

− pG (p, x,X) + L(X)†G (p, x,X) = −δ (X − x)

G (p, x,0) = 0, G (p, x,X) →X→∞

0(68)

The corresponding forward characteristic equation is given by:

1

2σ2ψ2 − µψ − (λ+ p) +

λν

ν − ψ= 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

−ψ3(X−x), X ≥ xD1e

−ψ1(X−x) +D2e−ψ2(X−x) +D3e

−ψ3(X−x), 0 < X ≤ x(70)

where

D1 = −2

σ2

(ν + ψ1)

(ψ1 − ψ2) (ψ1 − ψ3), D2 = −

2

σ2

(ν + ψ2)

(ψ2 − ψ1) (ψ2 − ψ3)

D3 = −e(ψ1−ψ3)xD1 − e(ψ2−ψ3)xD2, C3 = D1 +D2 +D3

(71)

The inverse Laplace transform yields Γ (τ, x,X)

We compute the Laplace-transformed survival probability

Q (τ, x) −→ Q (p, x) (72)

as follows:

Q (p, x) =∫ ∞

0G (p, x,X) dX

=∫ ∞x

C3e−ψ3(X−x)dX +

3∑j=1

∫ x0Dje

−ψj(X−x)dX

= E0 + E1eψ1x + E2e

ψ2x

(73)

where

E0 =1

p, E1 =

(ψ1 + ν)ψ2

(ψ1 − ψ2) νp, E2 =

(ψ2 + ν)ψ1

(ψ2 − ψ1) νp(74)

The default time density satisfies the following equation:

q(τ, x) = −∂Q(x)(τ, x)

∂τ(75)

Using Eq.(33) we obtain:

q(τ, x, T ) = −∫ ∞

0

∂Γ(τ, x,X)

∂τdX = g(τ, x) + f(τ, x) (76)

where g(τ, x) is the barrier hitting density:

g(τ, x) =σ2

2

∂Γ(τ, x,X)

∂X

∣∣∣∣∣X=0

(77)

and f(τ, x) is the probability of the overshoot:

f(τ, x) = λ∫ ∞

0

(∫ −X−∞

$(j)dj

)Γ(τ, x,X)dX (78)

Formula (76) is a general result for jump-diffusion with arbitrary jumpsize distributions

When jumps are exponential:

f(τ, x) = λ∫ ∞

0e−νXΓ(τ, x,X)dX (79)

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) =(ν + ψ2)eψ2x − (ν + ψ1)eψ1x

ψ2 − ψ1(81)

and

f(p, x) =2λ

(eψ2x − eψ1x

)σ2(ψ2 − ψ1)(ν + ψ3)

(82)

Alternatively, taking the Laplace transform of (75) and using (73) weobtain:

q(p, x) =(ψ1 + ν)ψ2e

ψ1x

(ψ2 − ψ1) ν+

(ψ2 + ν)ψ1eψ2x

(ψ1 − ψ2) ν(83)

Straightforward but tedious algebra shows that (80)-(82) are equiva-lent with (83)

We express the present value of CDS contract V CDS(τ, x) with coupon

c as:

V CDS(τ, x) = −c∫ τ

0e−rτ

′Q(τ ′, x)dτ ′

+ (1−R)∫ τ

0e−rτ

′g(τ ′, x)dτ ′+

(1−R

ν

1 + ν

) ∫ τ0e−rτ

′f(τ ′, x)dτ ′

(84)

We use (73), (81), and (82) to compute the value of the CDS byLaplace inversion.

Illustration

In Figure we illustrate the jump-diffusion model with exponentialjumps using the following market: s(0) = 40, a(0) = 200, l(0) = 160,r = ζ = 0. We use the following model parameters: ξ = 0.22,σ = 0.05, λ = 0.03, ν = 1/ξ

We also compare outputs from jump-diffusion model with those fromthe diffusion model obtained by taking λ ≡ 0

For the latter model, we use the equivalent diffusion volatility σnr

specified by σnr =√σ2 + 2λ/ν2, so that σnr = 0.074 for the given

model parameters

36

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

20% 40% 60% 80% 100% 120% 140%

K/S

Imp

lied

Vo

lati

lity

Implied Vol, Jump-diffusion

Implied Vol, Diffusion

0.00

0.01

0.02

0.03

0.10 0.60 1.10 1.60 2.10 2.60 3.10 3.60

T

Fai

r sp

read

, s(T

)

s(T), Jump-diffusion

s(T), Diffusion

Left side: the model implied volatility skew for put options withmaturity six months; right side: the model implied CDS spread

The jump-diffusion model generates the implied volatility skew thatis steeper that the diffusion model

Unlike the diffusion model, the jump-diffusion model implies a non-zero probability of defaulting in short term so that its implied spreadis consistent with the spread observed in the market

Asymptotic Solution

We derive an asymptotic solution for the Green’s function solving(66) assuming that the jump intensity parameter λ is small

We introduce new function:

Γ(τ, x,X) = exp

{−(µ2

2σ2+ λ

)τ +

µ

σ2(X − x)

}Γ(τ, x,X) (85)

It solves the following propagation problem:

Γτ (τ, x,X)−1

2σ2ΓXX (τ, x,X)− λν

∫ 0

−∞Γ(τ, x,X − j)eνjdj = 0, {X > 0}

Γ(0, x,X) = δ(X − x)

Γ(0, X) = 0, Γ(T,X) →X→∞

0

(86)

where ν = ν − µ/σ2

37

We assume that λ << 1 and represent Γ(τ, x,X) as follows:

Γ(τ, x,X) = Γ(0)(τ, x,X) + λΓ(1)(τ, x,X) + ... (87)

The zero order solution Γ(0)(τ, x,X) solves the following problem:

Γ(0)τ (τ, x,X)−

1

2σ2Γ(0)

XX (τ, x,X) = 0

Γ(0)(0, x,X) = δ(X − x)

Γ(0)(τ, x,0) = 0, Γ(0)(τ, x,X) →X→∞

0

(88)

The solution to the above problem is

Γ(0)(τ, x,X) =1√ϑ

(n

(X − x√

ϑ

)− n

(X + x√

ϑ

))(89)

where ϑ = σ2τ and n(x) is standard normal PDF.

The first order solution Γ(1)(τ, x,X) solves the following problem:

Γ(1)τ (τ, x,X)−

1

2σ2Γ(1)

XX (τ, x,X) = H(τ, x,X)

Γ(1)(0, x,X) = 0

Γ(1)(τ, x,0) = 0, Γ(1)(τ, x,X) →X→∞

0

(90)

where

H(τ, x,X) = ν∫ 0

−∞Γ(0)(τ, x,X − j)eνjdj

= νP

(−X − x√

ϑ,−ν√ϑ

)− νP

(−X + x

ϑ,−ν√ϑ

) (91)

P(a, b) = exp{ab+ b2/2

}N(a+ b) (92)

and N(x) is standard normal CPDF. We use Duhamel’s principle andrepresent Γ(1)(τ,X) as follows:

Γ(1)(τ, x,X) =∫ τ

0

∫ ∞0

Γ(0)(τ ′, x,X)H(τ ′, X)dXdτ ′ (93)

Fairly involved algebra yields:

Γ(1)(τ, x,X) =ν

νσ2

(νϑP

(−X − x√

ϑ,−ν√ϑ

)

+XP

(−X + x√

ϑ,−ν√ϑ

)− (X − νϑ)P

(−X + x√

ϑ, ν√ϑ

)

−(X + νϑ)e−νxP

(−X√ϑ,−ν√ϑ

)+ (X − νϑ)e−νxP

(−X√ϑ, ν√ϑ

))

Illustration

Model parameters as in the previous figure, T = 10

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X

AnalyticalExpansionLambda=Zero

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X

AnalyticalExpansionLambda=Zero

Left side: λ = 0.03; right side: λ = 0.10

38

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 relativelyeasy and it can be applied for relatively wide of jump-size distributions

39

FFT based method. Characteristic Function

Define the characteristic function of x(t) by:

G(t, T, φ) =∫ ∞−∞

eiφXG(t,0;T,X)dX,

where G(t, x;T,X) is the TPDF of x(t), X ≡ x(T ), φ ∈ R is thetransform variable, and i =

√−1

From the theory of additive processes:

G(t, T, φ) = e−∫ Tt ψ(t′,φ)dt′,

where ψ(t, φ) is the characteristic exponent:

ψ(t, φ) =1

2(σ(t)φ)2 − iµ(t)φ− λ(t)($(φ)− 1), $(φ) =

∫ ∞−∞

eiφj$(j)dj

Accordingly, we can compute TPDF of x(T ) by:

G(t,0;T,X) =1

2π

∫ ∞−∞<[e−iφXG(t, T, φ)

]dφ (94)

40

FFT based method. Backward Problem

Because the increments in x(t) are independent conditional on thecurrent state values:

G(t, x;T,X) ≡ G(t, T,X − x)

The value function U(t, x) can be represented as (ignoring couponand rebate functions):

U(t, x) =∫ ∞−∞

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

2π

∫ ∞−∞

u(X)∫ ∞−∞<[e−iφ(X−x)G(t, T, φ)

]dφdX

=1

2π

∫ ∞−∞<[eiφx

(∫ ∞−∞

e−iφXu(X)dX)G(t, T, φ)

]dφ

(95)

41

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) = ifft(fft(u(x))� G(t, T, φ)

)

By discretisation of the state space of x and φ, the relationship∆x∆φ = 2π

N is required for standard DFT algorithm

42

FFT based method. Forward Equation for function U∗(T,X):

U∗T (T,X) + L†U∗(T,X) = 0,

U∗(t,X) = u∗(x)

where L† is the operator adjoint to L

U∗(T,X) can be represented as the solution to:

U∗(T,X) =∫ ∞−∞

u∗(x)G(t, T,X − x)dx

Applying the Fourier transformed density function (94):

U∗(T,X) =1

2π

∫ ∞−∞

u∗(x)∫ ∞−∞<[e−iφ(X−x)G(t, T, φ)

]dφdx

=1

2π

∫ ∞−∞<[e−iφX

(∫ ∞−∞

eiφxu∗(x)dx)G(t, T, φ)

]dφ

This can be computed by:

U∗(T, x) = fft(ifft(u∗(x))� G(t, T, φ)

)(96)

43

FFT based method. Time Stepping

In case of discrete monitoring, the value function depends on thestate variables observed at discrete times {tm}m=1,...,m

Compute the value function applying the DFT algorithm at each timestep:

1) Apply the terminal condition by U(tm, x) = u(x)

2) Given U(tm, x), compute U(tm−1, x) by:

Um−1(x) = e−∫ tmtm−1,

r(t′)dt′ifft

(fft(U(tm, x))� G(tm−1, tm, φ)

)

3) Set m → m − 1 and, if m > 0, apply the boundary and couponconditions and go to 2)otherwise, if m = 0, the recursion is stopped and the present value iscomputed

44

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 forwardinduction and calibrate the term structure of jump intensity λ(t) bybootstrapping

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 defined on the same grid for pricingand counterparty charge evaluation

45

FFT based method. Illustration

JPM

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

-0.20 -0.16 -0.12 -0.08 -0.04 -0.01 0.03 0.07 0.11 0.15 0.19 0.22y'

1y2y3y4y5y6y7y8y9y10y

Density function for y(t) = x(t)− ξ truncated at y = −ξ at differ-ent maturities

46

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-diffusions 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 dimensionsis O(2N1N2 log(N1N2)))

5) Relatively fast and easy to implement in one and two dimensions

47

FFT based method. Disadvantages

1) DFT assumes that the function is periodic (extend the function)

2) Computational domain is required to be uniform (use fractionalFFT)

3) Convergence is controlled by choosing the grid for transform vari-able φ (look at the decay of G(tm−1, tm, φ) - can be slow if the volatilityparameter is small)

4) The scheme is only first 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

PDE based methods. Considerations

+ PDE methods are not restricted to uniform grids

+ Convection-dominated problems when drift is large and volatilityis small are easier to handle

- Non-local jump term is difficult to handle, especially, in two dimen-sions

- A direct computation of the integral part by, say, trapezoidal ruleleads to O(N2) complexity

- Using the DFT to compute the convolution part (Andersen-Andreasen(2000)) leads to O(N logN) complexity but suffers from unpleasantfeatures of the DFT (uniform grids, periodicity, convergence)

However, explicit algorithms of O(N) complexity can be employed ifjumps are exponential or discrete (Lipton (2003), Carr-Mayo (2007),Toivanen (2008))

49

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 (atintermediate time an artificial boundary condition must be enforced)

50

PDE based method. Time stepping

To compute the value U l−1 at time t = tl given its value U l at timet = tl, l = 1, ..., L, we use the following splitting:

U∗n − U lnδtl

= J lnU l + cl−1n , n = 2, ..., N − 1,

U l−1n − U∗nδtl

= DlnU l−1, n = 2, ..., N − 1

(97)

At the first step, we use the explicit scheme to approximate the inte-gral part and compute the auxiliary function U∗ given U ln

At the second step, we use the implicit scheme to approximate thediffusive step and compute U l−1

n given U∗

51

PDE based method. Integral part for discrete jumps

Given $(j) = δ(j − ν), ν > 0:

J ln = U(tl, xn − ν), n = 2, ..., N − 1

We approximate J ln by linear interpolation with the second order ac-curacy:

J ln = ωnjUlnj−1 + (1− ωnj)U

lnj

where

ωnj =xnj − (xn − ν)

xnj − xnj−1

nj = min{j : xj−1 ≤ xn − ν < xj}

52

PDE based method. Integral part for exponential jumps I

Given $(j) = νeνj, ν > 0:

J(x) = ν∫ 0

−∞eνjU(x+ j)dj

For a small number h, h > 0:

J(x+ h) = ν∫ 0

−∞eνjU(x+ h+ j)dj

z=h+j= νe−νh

∫ h−∞

eνzU(x+ z)dz

= νe−νh(∫ 0

−∞eνzU(x+ z)dz +

∫ h0eνzU(x+ z)dz

)= e−νhJ(x) + J1(x)

53

PDE based method. Integral part for exponential jumps II

Expanding U(x+ z) in Taylor series around z = 0 yields:

J1(x) = νe−νh∫ h

0eνzU(x+ z)dz = a0U(x) + a1U

′+O(h3),

where

a0 = νe−νh∫ h

0eνzdz = 1− e−νh, a1 = νe−νh

∫ h0zeνzdz =

νh−(1− e−νh

)ν

Accordingly, with the second order accuracy

J(x+ h) = e−νhJ(x) + w0(ν, h)U(x) + w1(ν, h)U(x+ h)

where

w0(ν, h) =1− (1 + νh) e−νh

νh, w1(ν, h) =

νh−(1− e−νh

)νh

54

PDE based method. Improving the convergence

Apply the fixed point iterations (d’Halluin et al(2005)):

1) Set V 0 = U l + δtlcl−1n ;

2) For p = 1,2, ..., p apply the scheme (97):

V ∗n − V 0n

δtl= J lnV p−1, n = 2, ..., N − 1,

Vpn − V ∗nδtl

= DlnV p, n = 2, ..., N − 1

3) if norm ||V p − V p−1|| in becomes small, stop and set U l−1 = V p

Typically, p = 2 is enough

55

PDE based method. Summary

1) The scheme has O(N) complexity per each time step

2) Although first order in time, the implicit scheme tends to be morestable than the Crank-Nicolson based scheme (especially for forwardequation and two-dimensional problems)

3) The scheme is second order accurate in the spacial variable if thedrift term is not dominant, otherwise it is first order accurate (D isdiscretisized appropriately)

4) A similar scheme is applied for the forward equation

5) As before, using the same grid, the forward scheme is applied formodel calibration and the backward scheme is applied for pricing

56

Numerical Methods for Two Dimensional ProblemWe 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 diffusion-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) ≡ ρσ1(t)σ2(t)Ux1x2(t, x1, x2)

J12 is the cross integral operator:

J12U(t, x1, x2) ≡ λ{1,2}(t)∫ 0

−∞

∫ 0

−∞U(t, x1+j1, x2+j2)$(j1)$(j2)dj1dj2

57

Counterparty Charge Using Structural Model

Let x1(t) and x2(t) be the stochastic drivers for the reference nameand the counterparty, respectively

The value of the countrparty charge U(t, x1, x2) defined as the solutionto (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 (τ1 < τ2);

U(t, x1, x2) = (1−R2) max{C(t, x1),0}, x1 > b1, x2 ≤ b2, (τ1 > τ2);

U(t, x1, x2) = (1−R2)(1−R1), x1 ≤ b1, x2 ≤ b2, (τ1 = τ2);

limx1→∞

U(t, x1, x2) = 0, limx2→∞

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

Discretisation

We develop a modified Craig-Sneyd (1988) discretization scheme tocompute the solution at time tl−1, U l−1

n,m, given the solution at time

tl, Uln,m, l = 1, ..., L, as follows:

(1− δtlD1)U∗n,m = U ln,m + δtl((D2 +D12 + J1 + J2 + J12)U ln,m + cl−1

n,m

),

(1− δtlD2)U l−1n,m = U∗n,m − δtlD2U

ln,m

In the first line, for each fixed index m we apply the jump operators anddiffusion operator in x1 direction, the correlation operator, and couponpayments (if any); and solve the tridiagonal system of equations toget the auxiliary solution U∗·,m

In the second line, keeping n fixed, we apply the implicit step in x2direction and solve the system of tri-diagonal equations to get thesolution

59

Discretisation of Cross Jump Part

Direct methods are infeasible because of O(N2M2) complexity

DFT method (Clift-Forsyth (2008)) has O(NM logNM) complexitybut suffers from problems associated with the DFT

Explicit methods with O(NM) complexity are available for discreteand exponential jumps (Lipton-Sepp (2009))

The simplest case is if jumps are discrete:

J12U = U(x1 − ν1, x2 − ν2)

This term is approximated by bi-linear interpolation with the secondorder accuracy leading to the O(NM) complexity

60

Discretisation of the Jump Part. Negative exponential jumps

Consider the integral:

J(x1, x2) = ν1ν2

∫ 0

−∞

∫ 0

−∞eν1j1+ν2j2U(x1 + j1, x2 + j2)dj1dj2

Take small numbers hx and hy, hx > 0, hy > 0:

J(x1 + h1, x2 + h2) = ν1ν2

∫ 0

−∞

∫ 0

−∞eν1j1+ν2j2U(x1 + h1 + j1, x2 + h2 + j2)dj1dj2

= ν1ν2e−ν1h1−ν2h2

∫ h1

−∞

∫ h2

−∞eν1z1+ν2z2U(x1 + z1, x2 + z2)dz1dz2

= ν1ν2e−ν1h1−ν2h2

(∫ 0

−∞

∫ 0

−∞+∫ h1

0

∫ 0

−∞+∫ 0

−∞

∫ h2

0+∫ h1

0

∫ h2

0

[eν1z1+ν2z2U(x1 + z1, x2 + z2)dz1dz2

])= e−ν1h1−ν2h2J(x1, x2) + e−ν2h2J10(x1, x2) + e−ν1h1J01(x1, x2) + J11 (x1, x2)

Integrals J10(x, y), J01(x, y), and J11(x, y) can be computed by recur-sion with second order accuracy and O(NM) complexity

61

Discretisation of the Jump Part. Improving the convergence

At each time step, we apply the fixed point iterations as follows (p = 2is enough):

1) Set V 0n,m = U ln,m + δtlCn,mU ln,m + δtlc

l−1n,m;

2) For p = 1,2, ..., p apply the above scheme:

V jn,m = V 0n,m + δtl(J12 + J1 + J2)V p−1,

(1− δtlD1)V ∗n,m = δtlD2Vp−1n,m + V jn,m,

(1− δtlD2)V pn,m = V ∗n,m − δtlD2Vp−1n,m

(99)

3) if norm ||V p − V p−1|| becomes small, stop and set U l−1 = V p

62

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 toO(NM log(NM)) complexity)

2) The scheme is first order accurate in time

3) The scheme is second order accurate in spacial variables (if thedrift is not dominant)

4) The modified scheme is applied for the forward problem, so that,it needed, the calibration problem in two dimensions can be solvedefficiently

63

Example. Input data for model calibration

JPM Cs(0) 36.49 8.47L(0) 604.11 353.07

s(0)/L(0) 16.56 41.68R 40% 40%

l(0) 241.644 141.228v(0) 278.134 149.698

b -0.1406 -0.0582ν(1) 0.1406 0.0582ν(2) 0.0703 0.0291

σ 0.0262 0.0113

Use two choices for the jump size:1) ν ≡ ν1 = −b in the model with discrete jumps and ν ≡ 1

ν1= 1

b in

the model with exponential jumps;2) ν ≡ ν2 = −1

2b in the model with discrete jumps and ν ≡ 1ν2

= 12b in

the model with exponential jumps

64

Example. Input data for model calibration

Spread data for model calibration and the survival probability, defaultleg, and annuity leg implied using the hazard rate model

CDS Spread Survival Prob Default Leg Annuity LegT JPM C JPM C JPM C JPM C

1y 0.0105 0.0286 0.9826 0.9535 0.0174 0.0465 0.9913 0.97662y 0.0118 0.0271 0.9614 0.9137 0.0386 0.0863 1.9633 1.90993y 0.0134 0.0257 0.9348 0.8798 0.0652 0.1202 2.9114 2.80654y 0.0147 0.0249 0.9063 0.8475 0.0937 0.1525 3.8320 3.67015y 0.0160 0.0248 0.8743 0.8138 0.1257 0.1862 4.7223 4.50076y 0.0161 0.0243 0.8498 0.7857 0.1502 0.2143 5.5841 5.30027y 0.0162 0.0238 0.8268 0.7590 0.1732 0.2410 6.4223 6.07258y 0.0163 0.0236 0.8034 0.7319 0.1966 0.2681 7.2374 6.81799y 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

Example. Calibrated Intensity Rates

For both choice of jumps size distributions and the jump sizes, themodel is calibrated to the term structure of CDS spreads given inTable 2 using the forward induction

λdis1 (lambda{dis} {1}) and λdis2 (lambda{dis} {2}) stand for modelwith discrete jumps with sizes ν1 and ν2, respectively

λexp1 (lambda{exp} {1}) and λ

exp2 (lambda{exp} {2}) stand for model

with exponential jumps with sizes 1ν1

and 1ν2

, respectively

66

Example. Input Data

Implied density of the driver x(t) for JMP and C in the model withexponential jump, ν = 1/b, at maturities 1, 5, and 10 years

67

Example. CDS option volatilityThe log-normal CDS option volatility implied from model values ofone year option on five year CDS contract as a function of the money-ness Kα,β/Sα,β

The model implied log-normal volatility σα,β exhibits a positive skew

This effect is in line with the market because the CDS spread volatilityis expected to increase when the CDS spread increases, so that optionsellers charge an extra premium for out-of-the-money CDS options

68

Example. Log-normal equity volatility

Log-normal equity volatility implied from model values of put optionswith maturity 6 months using the Black-Scholes formula

The model implies a remarkable skew in line with that observed inthe market

The smaller the jump size the higher is the implied model volatilitybecause the firm value is expected to have more jumps before thebarrier crossing so that the realized volatility is expected to be higher

69

Example. Implied Gaussian correlationWe compute the model implied Gaussian correlation by equating thefair spread of the first to default swap referencing JPM and C to thatcomputed using the Gaussian copula with implied correlation

We use the two choices for the model correlation parameter: ρ = 0.50and ρ = 0.99

The model with exponential jumps produces lower implied correlations

The model with smaller jump amplitudes implies smaller correlations

70

Illustration. Counterparty charge

We compute the counterparty charge for par CDS on JPM sold by Cand that for par CDS on C sold by JPM as functions of CDS maturityusing two model correlation parameters: ρ = 0.5 and ρ = 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 onJPM corresponding to CDS maturity

For a moderate correlation assumption with ρ = 0.50, the modelwith discrete large jump implies the countrepaty charge in amountof 10% − 15% of the present value of the CDS protection leg onthe underlying name, while, for a high correlation assumption withρ = 0.99, this proportion grows to 30%− 40%

71

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

72

CDS spread with counterparty risk, recent data

30

35

40

45

50

55

60

1 2 3 4 5 6 7 8 9 10

T, years

Sp

read

, bp

InputSpreadRisky Seller, rho=0.75Risky Buyer, rho=0.75Risky Seller, rho=0.0Risky Buyer, rho=0.0Risky Seller, rho=-0.75Risky Buyer, rho=-0.75

96

106

116

126

136

146

156

1 2 3 4 5 6 7 8 9 10

T, years

Sp

read

, 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

Equilibrium spread for protection buyer and protection seller forCDS on JPM with MS as the counterparty, left, and for CDSon MS with JPM as the counterparty, right

73

Counterparty charge. Conclusions

1) The larger is the correlation, the larger is the counterparty chargebecause, given the counterparty default, the protection lost is largerin case of high correlation

2) The larger the jump size, the larger is the counterparty chargebecause the model with higher jumps implies a larger correlation

3) The model with discrete jumps implies a larger counterparty chargethan the model with exponential jumps because the former implieslarger correlation and CDS spread volatility

4) The counterparty charge is not symmetric. It is expected that amore risky counterparty implies a higher counterparty charge

74

Conclusions

1) We have proposed an extended structural model capable of fittingarbitrary term structures of CDS spreads

2) Applying this model, we have obtained a novel method to analysethe counterparty risk

3) We have developed a number of semi-analytical and numericalmethods to solve calibration and pricing problems in an efficient way

75

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