Simulation of an SPDE Model for a Credit Basket...The buyer of a kth-to-default credit default swap...

Simulation ofan SPDE Model

for a Credit Basket

Christoph ReisingerJoint work with N. Bush, B. Hambly, H. Haworth

[email protected].

MCFG, Mathematical Institute, Oxford University

C. Reisinger – p.1

Outline

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Introduction, structural models of credit

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Example: Credit default swap spreads

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A general multi-factor model

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Large basket limit and CDO pricing

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Numerical simulation of the SPDE

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Calibration and pricing examples

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Improvements, extensions


Framework

Limit

Results

Extensions

Structural setup

As in Merton (1974), and Black and Cox (1976),

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At the company’s asset value, governed by

dAt

At= µdt + σ dWt

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µ the mean return, σ the volatility, W a standard Brownianmotion

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denoting the default threshold barrier by b, say constant, definethe distance to default, Xt, as

Xt =1

σ( logAt − logb ) .

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Merton: company defaults if XT < 0

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Black/Cox: default time τ is given by the first time Xt hits 0


Framework

Limit

Results

Extensions

Default probabilities

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By first exit time theory, the probability of survival to T is

Q(T ≤ τ |Xt) = Q(Xs ≥ 0, t ≤ s ≤ T |Xt)

= H(Xt, T − t),

where

H(x, s) = Φ

(

x + ms√s

)

− e−2mxΦ

(

−x + ms√s

)

.

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Φ the standard Gaussian CDF, m the risk-neutral drift of Xs


Framework

Limit

Results

Extensions

More companies

Consider firm values, for i = 1, . . . , N (risk-neutral measure),

dAit = (rf − qi)A

it dt + σiA

it dWi(t)

where

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rf risk-free rate

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qi dividend yields

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σi volatilities

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Wi(t) Brownian motions and

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cov(Wi(t), Wj(t)) = ρijt.


Framework

Limit

Results

Extensions

More companies

Consider firm values, for i = 1, . . . , N (risk-neutral measure),

dAit = (rf − qi)A

it dt + σiA

it dWi(t)

where

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rf risk-free rate

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qi dividend yields

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σi volatilities

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Wi(t) Brownian motions and

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cov(Wi(t), Wj(t)) = ρijt.

We assume that each company has an exponential default barrier,

bi(t) = Kie−γi(T−t)

for constants Ki, γi. T represents the maturity of the product.


Framework

Limit

Results

Extensions

Transformation

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Setting

Xit = ln

(

Ait

Ai0

e−γit

)

= αit + σiWi(t)

with αi = rf − qi − γi − 12σ2

i , leads to a Brownian motion with

drift and constant barrier Bi = ln(

bi(0)Ai

0

)

≤ 0, Xi0 = 0.

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Framework

Limit

Results

Extensions

Transformation

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Setting

Xit = ln

(

Ait

Ai0

e−γit

)

= αit + σiWi(t)

with αi = rf − qi − γi − 12σ2

i , leads to a Brownian motion with

drift and constant barrier Bi = ln(

bi(0)Ai

0

)

≤ 0, Xi0 = 0.

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Defining the running minimum

Xit = min

0≤s≤tXi

s,

and default time, τi, as the first hitting time of the defaultbarrier,

τ i = inf{t : Xit = Bi},

the survival probability is then

Q(τ i > s) = Q(Xis ≥ Bi).


Framework

Limit

Results

Extensions

2D case

For two firms, this problem is ‘analytically’ tractable, e. g.

Q(t) = Q(X1t ≥ B1, X

2t ≥ B2)

=2

βtea1B1+a2B2+bt

∞∑

n=1

e−r2

0/2t sin

(

nπθ0

β

)∫ β

0

sin

(

nπθ

β

)

gn(θ) dθ

where

gn(θ) =

∫ ∞

0

re−r2/2teA(θ)rI( nπβ

)

(rr0

t

)

dr

a1 =α1σ2 − ρα2σ1

(1 − ρ2)σ21σ2

, a2 =α2σ1 − ρα1σ2

(1 − ρ2)σ1σ22

b = −α1a1 − α2a2 +1

2σ2

1a21 + ρσ1σ2a1a2 +

1

2σ2

2a22

tanβ = −√

1 − ρ2

ρ, β ∈ [0, π]

etc, and I( nπβ

)

(

rr0

t

)

is a modified Bessel’s function.C. Reisinger – p.7

Framework

Limit

Results

Extensions

CDS spread calculation

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The buyer of a kth-to-default credit default swap (CDS) on abasket of n companies pays a premium, the CDS spread, forthe life of the CDS – until maturity or the kth default.

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In the event of default by the kth underlying referencecompany, the buyer receives a default payment and thecontract terminates.


Framework

Limit

Results

Extensions

CDS spread calculation

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The buyer of a kth-to-default credit default swap (CDS) on abasket of n companies pays a premium, the CDS spread, forthe life of the CDS – until maturity or the kth default.

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In the event of default by the kth underlying referencecompany, the buyer receives a default payment and thecontract terminates.

Equating discounted spread payment and discounted defaultpayment,

DSP = cK

∫ T

0

e−rf sQ(τk > s) ds

DDP = (1 − R)K

∫ T

0

e−rf sQ(s ≤ τk ≤ s + ds)

gives the market kth-to-default CDS spread, ck, as

ck =(1 − R)

{

1 − e−rf T Q(τk > T ) −∫ T

0rfe−rf sQ(τk > s) ds

}

∫ T

0e−rf sQ(τk > s) ds

.


Framework

Limit

Results

Extensions

Varying T

First-to-default CDS, varying T

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Correlation

Spr

ead

1 year2 years3 years4 years5 years

Second-to-default CDS, varying T

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Correlation

Spr

ead

1 year2 years3 years4 years5 years

σ1 = σ2 = 0.2, K1 = 100, rf = 0.05, q1 = q2 = 0,

γ1 = γ2 = 0.03, initial distance-to-default = 2, R = 0.5


Framework

Limit

Results

Extensions

Varying σ

First-to-default CDS, varying σ

−1 −0.5 0 0.5 10

1

2

3

4

5

6

7

8

9

10

Correlation

Spr

ead

σi=0.1

σi=0.15

σi=0.2

σi=0.25

σi=0.3

Second-to-default CDS, varying σ

−1 −0.5 0 0.5 1−0.5

0

0.5

1

1.5

2

2.5

3

Correlation

Spr

ead

σi=0.1

σi=0.15

σi=0.2

σi=0.25

σi=0.3

K1 = 100, rf = 0.05, q1 = q2 = 0, R = 0.5

γ1 = γ2 = 0.03, initial distance to default = 2, T = 5


Framework

Limit

Results

Extensions

Portfolio model

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Consider a portfolio of N different companies.

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Denoting the companies’ asset values at time t by Ait, we

assume that under the risk neutral measure they follow adiffusion process given by

dAit = µ(t, Ai

t) dt + σ(t, Ait) dW i

t +

m∑

j=1

σij(t, Ait) dM j

t ,

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W it and M j

t are Brownian motions satisfying

d⟨

W it , M

jt

⟩

= 0 ∀i, j

and

d⟨

W it , W

jt

⟩

= d⟨

M it , M j

t

⟩

= δij dt.


Framework

Limit

Results

Extensions

Numerical PDE solution

Solve Kolmogorov equation numerically?

Isotropic grid for N = 2 directions (firms):

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Framework

Limit

Results

Extensions




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Framework

Limit

Results

Extensions




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Framework

Limit

Results

Extensions




‘Curse of dimensionality’:

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After n refinements, unknowns ∼ h−N ∼ 2nN

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Accuracy ǫ has complexity C(ǫ) ∼ ǫ−N/2 (for an order 2method)

→ exponential growth of required computational resourcesC. Reisinger – p.12

Framework

Limit

Results

Extensions

What is high dimensional?

Example, 30 firms: If each asset is represented by onlytwo states, the total number of variables is already

230

= 1 073 741 824.


Framework

Limit

Results

Extensions

What is high dimensional?

Example, 30 firms: If each asset is represented by onlytwo states, the total number of variables is already

230

= 1 073 741 824.

If we choose a reasonable number of points in each di-rection, say 32 = 25, the same total number is alreadyobtained for

dim = 6.


Framework

Limit

Results

Extensions

Sparse grids

Full grid, h−N points Sparse, h−1| log h|N−1

But: require too much smoothness and have ‘large constants’.


Framework

Limit

Results

Extensions

Correlation data (typical)1.00 0.62 0.65 0.59 0.33 0.61 0.57 0.66 0.01 0.28 0.61 0.38 0.60

0.62 1.00 0.77 0.74 0.45 0.74 0.59 0.76 0.21 0.34 0.64 0.27 0.72

0.65 0.77 1.00 0.73 0.54 0.77 0.57 0.83 0.10 0.44 0.69 0.33 0.68

0.59 0.74 0.73 1.00 0.43 0.68 0.51 0.70 0.25 0.40 0.56 0.30 0.66

0.33 0.45 0.54 0.43 1.00 0.51 0.42 0.54 0.37 0.27 0.59 0.46 0.44

0.61 0.74 0.77 0.68 0.51 1.00 0.62 0.75 0.31 0.38 0.66 0.28 0.83

0.57 0.59 0.57 0.51 0.42 0.62 1.00 0.46 0.11 0.26 0.51 0.14 0.69

0.66 0.76 0.83 0.70 0.54 0.75 0.46 1.00 0.20 0.33 0.68 0.46 0.65

0.01 0.21 0.10 0.25 0.37 0.31 0.11 0.20 1.00 0.17 0.31 0.23 0.21

0.28 0.34 0.44 0.40 0.27 0.38 0.26 0.33 0.17 1.00 0.02 0.09 0.23

0.61 0.64 0.69 0.56 0.59 0.66 0.51 0.68 0.31 0.02 1.00 0.42 0.64

0.38 0.27 0.33 0.30 0.46 0.28 0.14 0.46 0.23 0.09 0.42 1.00 0.19

0.60 0.72 0.68 0.66 0.44 0.83 0.69 0.65 0.21 0.23 0.64 0.19 1.00

0.55 0.52 0.63 0.63 0.51 0.50 0.35 0.63 0.34 0.23 0.59 0.35 0.44

0.39 0.67 0.66 0.58 0.55 0.57 0.37 0.64 0.21 0.36 0.51 0.27 0.59

0.51 0.57 0.55 0.62 0.30 0.40 0.64 0.49 0.03 0.12 0.52 0.16 0.52

0.59 0.69 0.75 0.61 0.53 0.75 0.62 0.73 0.15 0.21 0.71 0.41 0.70

0.65 0.82 0.8 0.89 0.49 0.73 0.51 0.76 0.25 0.47 0.64 0.31 0.72

0.34 0.63 0.52 0.56 0.42 0.6 0.33 0.62 0.19 0.1 0.47 0.18 0.71

0.3 0.41 0.4 0.42 0.23 0.42 0.27 0.41 0.03 0.41 0.12 0.15 0.3


Framework

Limit

Results

Extensions

Correlation data1.00 0.62 0.65 0.59 0.33 0.61 0.57 0.66 0.01 0.28 0.61 0.38 0.60

0.62 1.00 0.77 0.74 0.45 0.74 0.59 0.76 0.21 0.34 0.64 0.27 0.72

0.65 0.77 1.00 0.73 0.54 0.77 0.57 0.83 0.10 0.44 0.69 0.33 0.68

0.59 0.74 0.73 1.00 0.43 0.68 0.51 0.70 0.25 0.40 0.56 0.30 0.66

0.33 0.45 0.54 0.43 1.00 0.51 0.42 0.54 0.37 0.27 0.59 0.46 0.44

0.61 0.74 0.77 0.68 0.51 1.00 0.62 0.75 0.31 0.38 0.66 0.28 0.83

0.57 0.59 0.57 0.51 0.42 0.62 1.00 0.46 0.11 0.26 0.51 0.14 0.69

0.66 0.76 0.83 0.70 0.54 0.75 0.46 1.00 0.20 0.33 0.68 0.46 0.65

0.01 0.21 0.10 0.25 0.37 0.31 0.11 0.20 1.00 0.17 0.31 0.23 0.21

0.28 0.34 0.44 0.40 0.27 0.38 0.26 0.33 0.17 1.00 0.02 0.09 0.23

0.61 0.64 0.69 0.56 0.59 0.66 0.51 0.68 0.31 0.02 1.00 0.42 0.64

0.38 0.27 0.33 0.30 0.46 0.28 0.14 0.46 0.23 0.09 0.42 1.00 0.19

0.60 0.72 0.68 0.66 0.44 0.83 0.69 0.65 0.21 0.23 0.64 0.19 1.00

0.55 0.52 0.63 0.63 0.51 0.50 0.35 0.63 0.34 0.23 0.59 0.35 0.44

0.39 0.67 0.66 0.58 0.55 0.57 0.37 0.64 0.21 0.36 0.51 0.27 0.59

0.51 0.57 0.55 0.62 0.30 0.40 0.64 0.49 0.03 0.12 0.52 0.16 0.52

0.59 0.69 0.75 0.61 0.53 0.75 0.62 0.73 0.15 0.21 0.71 0.41 0.70

0.65 0.82 0.8 0.89 0.49 0.73 0.51 0.76 0.25 0.47 0.64 0.31 0.72

0.34 0.63 0.52 0.56 0.42 0.6 0.33 0.62 0.19 0.1 0.47 0.18 0.71

0.3 0.41 0.4 0.42 0.23 0.42 0.27 0.41 0.03 0.41 0.12 0.15 0.3

5 10 15 20 25 30

0.005

0.01

0.05

0.1

0.5

1

l

k

σ(Σ)λ1

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Spectral gap after λ1

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Exponential decay from λ2

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Asymptotic expansion? Not enough regularity.C. Reisinger – p.16

Framework

Limit

Results

Extensions

A different viewpoint

It is irrelevant which of the firms default. Assume interchangeable.

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Let νN,t the empirical measure for the entire portfolio,

νN,t =1

N

N∑

i=1

δAit

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Let φ ∈ C∞c (R) and for measure νt write

〈φ, νt〉 =

∫

φ(x)νt(dx).

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Define a family of processes FN,φt by

FN,φt = 〈φ, νN,t〉 =

1

N

N∑

i=1

φ(Ait).


Framework

Limit

Results

Extensions

Evolution of measure

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Applying Ito’s lemma to FN,φt we have

FN,φt =

1

N

N∑

i=1

∫ t

0

φ′(Ais)

µ(s, Ais) dt + σ(s, Ai

s) dW is +

m∑

j=1

σij(s, Ais) dM j

s

+1

N

N∑

i=1

∫ t

0

1

2φ′′(Ai

s)

m∑

j=1

σ2ij + σ2

ds.

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We now assume that σij = σj for all i = 1, 2 . . . , then

FN,φt = FN,φ

0 +

∫ t

0

〈Aφ, νN,s〉 ds +

∫ t

0

m∑

j=1

〈σjφ′, νN,s〉 dM j

s +

+

∫ t

0

1

N

N∑

i=1

φ′(Ais) σ(s, Ai

s) dW is .

A = µ∂

∂x+

1

2σ̄2 ∂2

∂x2, σ̄2 =

m∑

j=1

σ2j + σ2,


Framework

Limit

Results

Extensions

Limiting equation

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The idiosyncratic component becomes deterministic in theinfinite dimensional limit, see also [Kurtz & Xiong, 1999].

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As the market factors affect each company to the samedegree, we can write

FN,φt → Fφ

t = 〈φ, νt〉 as N → ∞,

with

dFφt = 〈Aφ, νt〉 dt +

m∑

j=1

〈σjφ′, νt〉 dM j

t .

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Alternatively, we can write this in integrated form as

〈φ, νt〉 = 〈φ, ν0〉 +

∫ t

0

〈Aφ, νs〉 ds +

m∑

j=1

∫ t

0

〈σjφ′, νs〉 dM j

s


Framework

Limit

Results

Extensions

Density

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Assume the measure νt to be absolutely continuous withrespect to the Lebesgue measure, to write νt(dx) = v(t, x)dx

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In differential form,

dv = −µ∂v

∂xdt +

1

2σ̄2 ∂2v

∂x2dt −

m∑

j=1

∂

∂x(σj(t, x)v) dM j

t .

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This is a stochastic PDE that describes the evolution of aninfinite portfolio of assets whose dynamics were given before.

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For solubility see [Krylov, 1994].

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Can now use this to approximate the loss distribution for aportfolio of fixed size N .


Framework

Limit

Results

Extensions

Simplification

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Assume asset processes are correlated via a single factor.

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Specifically, let the SDEs for the asset processes be given by

dAit

Ait

= r dt +√

1 − ρσ dW it +

√ρσ dMt, Ai

0 = ai,

where ρ ∈ [0, 1), d⟨

W it , Mt

⟩

= 0.

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The distance-to-default

Xit =

1

σ

(

log Ait − log Bi

t

)

evolves according to

dXit = µdt +

√

1 − ρdW it +

√ρdMt, Xi

0 = xi

with µ = 1σ

(

r − 12σ2)

and xi = 1σ

(

log ai − log Bi0

)

.


Framework

Limit

Results

Extensions

Portfoilio loss

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Constant coefficient SPDE for v.

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The portfolio loss variable

LNt = (1 − R)

1

N

N∑

i=1

1{τi≤t}

measures the losses at time t.

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Assume that a fraction R of losses, 0 ≤ R ≤ 1, the recoveryrate, is recovered after default.

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Absorbing barrier condition:

v(t, 0) = 0 for t ≥ 0.

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The loss distribution is then

Lt = (1 − R)

(

1 −∫ ∞

0

v(t, x) dx

)

.


Framework

Limit

Results

Extensions

Properties

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Matching the initial conditions for both measures, we need

v(0, x) =1

N

N∑

i=1

δAi0

.

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Given this initial condition,

L0 = N

(

1 − 1

N

N∑

i=1

∫ ∞

Bt

δAi0

)

= 0,

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Also,0 ≤ Lt ≤ 1, for t ≥ 0

Q(Ls ≥ K) ≤ Q(Lt ≥ K), for s ≤ t,

which ensure that there is no arbitrage in the loss distribution.


Framework

Limit

Results

Extensions

CDOs

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CDOs slice up the losses into tranches, defined by theirattachment and detachment points.

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Typical numbers are 0-3%, 3%-6%, 6%-9%, 9%-12%,12%-22%, 22%-100%.

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For an attachment point a and detachment point d > a, theoutstanding tranche notional

Xt = max(d − Lt, 0) − max(a − Lt, 0)

and tranche loss

Yt = (d − a) − Xt = max(Lt − a, 0) − max(Lt − d, 0)

determine the spread and default payments for that tranche.

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Spreads are quoted as an annual payment, as a ratio of thenotional, but assumed to be paid quarterly. a

athere is a variation for the equity tranche, but we do not go into details.


Framework

Limit

Results

Extensions

Spread calculation

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Assume the notional to be 1, c the spread payment;

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n the maximum number of payments up to expiry T ;

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Ti the payment dates for 1 ≤ i ≤ n, δ = 0.25 the intervalbetween payments.

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The expected sum of discounted fee payments, referred to asthe fee leg, is given by

cV fee = cn∑

i=1

δe−rTiEQ[XTi],

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The protection leg is

V prot =

n∑

i=1

e−rTiEQ[XTi−1− XTi

].

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The fair spread payment is then given by s = V prot

V fee .C. Reisinger – p.25

Framework

Limit

Results

Extensions

Computation

Recall the SPDE

dv = − 1

σ

(

r − 1

2σ2

)

vx dt +1

2vxx dt −√

ρvx dMt, t > 0, x > 0

v(0, x) = v0(x), x > 0

v(t, 0) = 0 t ≥ 0

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Monte Carlo on top of a PDE solver computationally costly.

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For each simulated path of the market factor, a PDE needs tobe solved, e. g. by a finite difference method.

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From the PDE solution, quantities like tranche losses can becomputed,

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then averaged over the simulated paths.


Framework

Limit

Results

Extensions

Discrete defaults

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First make the assumption that defaults are only observed at adiscrete set of times,

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taken quarterly to coincide with the payment dates,

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i.e. if a firm’s value is below the default barrier on one of theobservation dates Ti, removed from the basket.

We therefore solve the modified SPDE problem

dv = − 1

σ

(

r − 1

2σ2

)

vx dt +1

2vxx dt −√

ρvx dMt, t ∈ (Tk, Tk+1),

v(0, x) = v0(x),

v(Tk, x) = 0 ∀x ≤ 0, 0 < k ≤ n


Framework

Limit

Results

Extensions

Lagrangian coordinate

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The boundary condition is not active in intervals (Tk, Tk+1).

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The Brownian driver only introduces a random shift.

The solution can therefore be written as

v(t, x) =

8

<

:

0 x ≤ 0 ∧ t ∈ {Tk, 1 ≤ i ≤ n}v(i)(t − Tk, x −√

ρ(Mt − MTk)) else if t ∈ (Tk, Tk+1], 0 ≤ k < n

where v(k) is the solution to the (deterministic) problem

v(k)t =

1

2(1 − ρ)v(k)

xx − 1

σ

(

r − 1

2σ2

)

v(k)x , t ∈ (0, τ) = (0, Tk+1 − Tk)

v(k)(0, x) = v(Tk, x)

assuming payment dates are equally spaced, τ = Tk+1 − Tk.


Framework

Limit

Results

Extensions

Algorithm

This suggests the following inductive strategy for k = 0, . . . , n − 1:

1. Start with v(0)(0, x) = v0(x).

2. Solve the PDE numerically in the interval (0, T1), to obtainv(0)(T1, x).

3. Simulate MT1, evaluate v(T1, x).

4. For k > 0, having computed v(Tk, x) in the previous step, usethis as initial condition for v(k), and repeat until k = n.


Framework

Limit

Results

Extensions

Finite differences

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The initial distribution is assumed localised.

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Approximate the distribution by one with support [xmin, xmax]with xmin < 0 and xmax > 0.

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Can ensure that the expected error of this approximation ismuch smaller than the standard error of the Monte Carloestimates.

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Grid x0 = xmin, x1 = xmin + ∆x, . . . , xmin + j∆x, . . . , xJ =xmin + J∆x = xmax, where ∆x = (xmax − xmin)/J ,

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timesteps t0 = 0, t1 = ∆t, . . . , tI = I∆t = τ , where ∆t = τ/I.

Define an approximation vij to v(ti, xj) as solution to a FD/FE

scheme with θ timestepping.


Framework

Limit

Results

Extensions

Stability

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Standard central differencing is second order accurate in ∆x.

��

��

The backward Euler scheme θ = 1 is of first order accurate (in∆t) and strongly stable.

��

��

The Crank-Nicolson scheme θ = 12 is of second order

accurate, and is unconditionally stable in the l2-norm.

We deal with initial conditions of the form

v(0, x) =1

N

N∑

i=1

δ(x − xi),

��

��

Crank-Nicolson timestepping gives spurious oscillations forDirac initial conditions, and

��

��

reduces the convergence order for discontinuous intitialconditions.


Framework

Limit

Results

Extensions

Rannacher timestepping

Rannacher proposed the following modification:

��

��

Replace first Crank-Nicolson steps with backward Euler steps.

��

��

Need to balance between accuracy and stability.

��

��

Analysis by Giles and Carter of the heat equation suggests toreplace the first two Crank-Nicolson steps by four backwardEuler steps of half the stepsize.

��

��

We do this at t = 0, and also at t = Tk where the interfaceconditions introduce discontinuities at x = 0.

��

��

This restores second order convergence in time.


Framework

Limit

Results

Extensions

Averaging

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The sum of δ-distributions needs approximation on the grid.

��

��

Could collect the firms into symmetric intervals of width ∆xaround grid points,

v0j =

1

N∆x

∫ xj+∆x/2

xj−∆x/2

δ(xi − x) dx,

��

��

however, this reduces the overall order of the finite differencescheme to 1:

��

��

The approximation cannot distinguish between initial positionsxi in an interval of length ∆x.


Framework

Limit

Results

Extensions

Projection

��

��

To achieve higher (i. e. second) order, the δ’s are split betweenadjacent grid points. The correct weighting for a single firmwith distance-to-default xi in the interval [xj , xj+1), is

v0k =

∆x−2(xj+1 − xi) k = j

∆x−2(xi − xj) k = j + 1

0 else

��

��

This can be written more elegantly as L2-projection onto thebasis of ‘hat functions’ 〈Φk〉0≤k≤N where

Φk(x) =1

∆xmin (max(x − xk + ∆x), max(xk − x, 0)) ,

then v0k = 〈Φk, v0〉 =

∫ xmax

xmin

Φk(x)v0(x) dx.

��

��

Note that ∆x∑N

k=0 v0k = 1.


Framework

Limit

Results

Extensions

Interface condition

At t = Tk, we have to evaluate the grid function at shiftedarguments that do not normally coincide with the grid.

��

��

First, define a piecewise linear reconstruction from theapproximation vI

k obtained in the last step over the previousinterval [Tk−1, Tk], as

v∆x(Tk, x) =

N∑

j=0

Φj(x − ∆M)vIj .

��

��

Then, approximate the shift by setting

v0j =

∫ xj+∆x/2

max(xj−∆x/2,0)

v∆x(Tk, x) dx,

with ∆M = MTk− MTk−1

.


Framework

Limit

Results

Extensions

Interface condition

��

��

Use this as initial condition for the next interval (the integral isunderstood to be 0 if the lower limit is larger than the upperlimit).

��

��

This ensures that

∆x

J∑

j=0

v0j =

∫ xmax

0

vh(Tk, x) dx,

so the cumulative density of firms with firm values greater than0 is preserved.

��

��

Also, the solution is smoothened at x = 0


Framework

Limit

Results

Extensions

Simulations

��

��

For a given realisation of the market factor, we canapproximate the loss functional LTk

at time Tk by

L∆xTk

= 1 −∫ xmax

0

v∆x(Tk, x) dx

��

��

Explicitly include the dependency L∆xTk

(Φ), where Φi are drawnindependently from a standard normal distribution,

��

��

Then for Nsims simulations with samples Φl = (Φlk)1≤k≤n,

1 ≤ l ≤ Nsims,

EQ[XTk] ≈ EQ[max(d − L∆x

Tk(Φ), 0) − max(a − L∆x

Tk(Φ), 0)]

≈ 1

Nsims

Nsims∑

l=1

(

max(d − L∆xTk

(Φl), 0) − max(a − L∆xTk

(Φl), 0))


Framework

Limit

Results

Extensions

Data

��

��

The simulation error has two components, the discretisationerror and the variance of the Monte Carlo estimate.

��

��

For the following simulations we have used these data:

��

��

T = 5, r = 0.027, σ = 0.24, R = 0.7, ρ = 0.13.

��

��

Initial positions for individual firms, calibrated to their individualCDS spreads,

��

��

were well within the range [xmin, xmax] = [−10, 20].


Framework

Limit

Results

Extensions

Grid convergence

First consider the discretisation error in ∆t and ∆x.

101

102

103

104

10−6

10−5

10−4

10−3

10−2

10−1

100

J

ǫ J

100

101

102

103

10−8

10−7

10−6

10−5

10−4

10−3

I

ǫ I

��

��

Extrapolation-based estimator for the discretisation error ofLTn

for increasing J (left) and I (right) for a single realisation ofthe path of the market factor.

��

��

We clearly see second order convergence in ∆t and ∆x.


Framework

Limit

Results

Extensions

MC convergence

s [0, 3%] [3%, 6%] [6%, 9%] [9%, 12%] [12%, 22%] [22%, 100%]

1 6.2725e-03 2.0969e-02 2.61110e-02 2.95131e-02 1.000000e-01 7.8000000e-01

2 7.9104e-03 2.2701e-02 2.73422e-02 2.95842e-02 9.985866e-02 7.8000000e-01

3 7.385e-03 2.2685e-02 2.82221e-02 2.97561e-02 9.996077e-02 7.8000000e-01

4 7.6036e-03 2.2834e-02 2.80317e-02 2.95556e-02 9.987046e-02 7.8000000e-01

5 7.5088e-03 2.2772e-02 2.80370e-02 2.95103e-02 9.984416e-02 7.7999902e-01

6 7.4316e-03 2.2754e-02 2.80541e-02 2.94878e-02 9.982312e-02 7.7999919e-01

7 7.3909e-03 2.2683e-02 2.80568e-02 2.94938e-02 9.982567e-02 7.7999904e-01

8 7.4092e-03 2.2681e-02 2.80461e-02 2.94908e-02 9.982834e-02 7.7999870e-01

9 7.4051e-03 2.2676e-02 2.80520e-02 2.94934e-02 9.982809e-02 7.7999860e-01

10 7.4068e-03 2.2678e-02 2.80526e-02 2.94938e-02 9.982806e-02 7.7999866e-01

Monte Carlo estimates for expected outstanding tranche notionals forNsims = 64 ∗ 4s−1 Monte Carlo runs, s = 1, ..., 10. The finite difference parameterswere fixed at J = 256, I = 4.


Framework

Limit

Results

Extensions

MC convergence

2 4 6 8 10 12 140

0.005

0.01

0.015

0.02

0.025

0.03

0.035

log4 Nsims

EQ[X

Tn],

a=

0,b

=0.

03

2 4 6 8 10 12 140

1

2

3x 10

−4

log4 Nsims

EQ[X

Tn],

a=

0.06,b

=0.

09

2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−7

log4 Nsims

EQ[X

Tn],

a=

0.12,b

=0.

22

��

��

Monte Carlo estimates with standard error bars

��

��

for expected losses in tranches [0, 3%], [6%, 9%], [12%, 22%]

��

��

for Nsims = 16 · 4k−1, k = 1, ..., 10, J = 256, I = 4.


Framework

Limit

Results

Extensions

Index spreads

The fixed coupons, traded spreads and model spreads for theiTraxx Main Series 10 index.

Maturity Date Fixed Coupon (bp) Traded Spread (bp) Model Spread (bp)

20/12/2011 30 21 19.6

20/12/2013 40 30 30.7

20/12/2016 50 41 41.0

February 22, 2007. Parameters used for the model spreads arer = 0.042, σ = 0.22, R = 0.4.

Maturity Date Fixed Coupon (bp) Traded Spread (bp) Model Spread (bp)

20/12/2013 120 215 207

20/12/2015 125 195 195

20/12/2018 130 175 176

December 5, 2008. Parameters used for the model spreads arer = 0.033, σ = 0.136, R = 0.4.


Framework

Limit

Results

Extensions

Implied correlation

Implied Correlation Skew for iTraxx Main Series 6 Tranches, Feb 22, 2007. Theimplied correlation for each tranche is the value of correlation that gives a modeltranche spread equal to the market tranche spread.

3% 6% 9% 12% 22% 100%0

10

20

30

40

50

60

70

Tranche Detachment Point

Impl

ied

Cor

rleat

ion

(%)

5 Year7 Year10 Year

Model parameters are r = 0.042, σ = 0.22, R = 0.4.


Framework

Limit

Results

Extensions

Results

5 Year

Tranche Market ρ = 0.3 ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8 ρ

0%-3% 71.5 % 81.88 % 75.9 % 69.56 % 63.02 % 56.25 % 49.16 % 41.65

3%-6% 1576.3 2275.2 1978.5 1743.2 1546.8 1374.6 1222.8 1090.1

6%-9% 811.5 1273.1 1168.2 1079.7 1001.4 931.3 864.6 796.3

9%-12% 506.1 775.7 765.8 748.6 724.7 695.8 663.2 629.1

12%-22% 180.3 307.8 353.3 384.7 405.5 418.1 423.4 420.5

22%-100% 77.9 9.2 16.5 25 34.3 44.5 55.7 68.1

Model tranche spreads (bp) for varying values of the correlation parameter. Theequity tranches are quoted as an upfront assuming a 500bp running spread.

The model is calibrated to the iTraxx Main Series 10 index for Dec 5, 2008. Marketlevels shown are for this date; model parameters are r = 0.033, σ = 0.136, R = 0.4.


Framework

Limit

Results

Extensions

Results

7 Year

Tranche Market ρ = 0.3 ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8 ρ

0%-3% 72.9 % 84.03 % 78.98 % 73.26 % 66.93 % 60 % 52.41 % 44.13

3%-6% 1473.2 2327.3 1985.7 1715.2 1493.4 1308 1147.8 1001.3

6%-9% 804.2 1344.2 1199 1085.2 988.2 900.7 820.9 747.9

9%-12% 512.4 855.4 808.4 765.3 725.3 684.8 643 600.4

12%-22% 182.6 375.4 401.7 417.6 425.6 427.4 423.1 411.8

22%-100% 75.8 14 22 30.6 39.6 49.3 59.7 71.2




Framework

Limit

Results

Extensions

Results

10 Year

Tranche Market ρ = 0.3 ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8

0%-3% 73.8 % 85.13 % 80.57 % 74.99 % 68.51 % 61.31 % 53.31 %

3%-6% 1385.5 2270.8 1895.7 1611.1 1385.8 1195.3 1032

6%-9% 824.7 1332.2 1164.2 1033.7 925.5 833.5 749.8

9%-12% 526.1 870.8 798.8 740.7 689.3 640.5 592.1

12%-22% 174.1 406.1 414.9 417.5 415.6 409.8 400.2

22%-100% 76.3 18.3 26.1 34 42.1 50.6 59.7




Framework

Limit

Results

Extensions

Extensions

��

��

Dynamic properties, forward starting CDOs, options ontranches,...

��

��

Simulation of SPDEs driven by jumps, with Karolina Bujok.

��

��

Calibration, including jumps.

��

��

Analytic work by Hambly/Jin on jumps, contagion, granularity.

��

��

Continuous defaults, with Mike Giles, via Multi-Level MonteCarlo.

��

��

Convergence analysis through mean-square stability.

��

��

Importance sampling for senior tranches, with Tom Dean.


Framework

Limit

Results

Extensions

Importance sampling

Using large deviations theory, see also [Glasserman et al, 1999,2007], preliminary observations:

without importance sampling

0 1 2 3 4 5 6 7 8 9 100.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

with importance sampling

0 1 2 3 4 5 6 7 8 9 100.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

Probability of more than 3% defaults occuring for 1000 × 2n Monte Carlo paths


Framework

Limit

Results

Extensions

Importance sampling

Using large deviations theory, see also [Glasserman et al, 1999,2007], preliminary observations:

without importance sampling

1 2 3 4 5 6 7 8 9 10 110.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8x 10

−3

with importance sampling

0 2 4 6 8 10 120.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8x 10

−3

Probability of more than 10% defaults occuring for 1000 × 2n Monte Carlo paths


Literature

References[Black & Cox(1976)] Black, F. & Cox, J. (1976) Valuing Corporate

Securities: Some Effects of Bond Indenture Provisions,J.Finance, 31, 351–367.

[Cameron & Martin(1947)] Cameron, R. H., & Martin, W. T. (1947)The Orthogonal Development of Non-Linear Functionals inSeries of Fourier-Hermite Functionals, Ann. Math, 48, 385-392.

[Giles & Carter(2006)] Giles, M. & Carter, R. (2006) Convergenceanalysis of Crank-Nicolson and Rannacher time-marching, J.Comp. Fin., 9(4), 89–112.

[Hambly & Jin (2008)] Hambly, B.M. & Jin, L. (2008) SPDEapproximations for large basket portfolio credit modelling, inpreparation.

[Haworth & Reisinger(2007)] Haworth, H., & Reisinger, C. (2007)Modelling Basket Credit Default Swaps with Default Contagion,J. Credit Risk, 3(4), 31–67.

[Hull & White(2001)] Hull, J., & White, A. (2001) Valuing CreditC. Reisinger – p.49

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Simulation of an SPDE Model for a Credit Basket...The buyer of a kth-to-default credit default swap...

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