Simulation of an SPDE Model for a Credit Basket Christoph - - PowerPoint PPT Presentation

Simulation of an SPDE Model for a Credit Basket

Christoph Reisinger Joint work with N. Bush, B. Hambly, H. Haworth

christoph.reisinger@maths.ox.ac.uk. MCFG, Mathematical Institute, Oxford University

• C. Reisinger – p.1

Outline

• Introduction, structural models of credit
• Example: Credit default swap spreads
• A general multi-factor model
• Large basket limit and CDO pricing
• Numerical simulation of the SPDE
• Calibration and pricing examples
• Improvements, extensions
• C. Reisinger – p.2

Framework Limit Results Extensions

Structural setup

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

• At the company’s asset value, governed by

dAt At = µ dt + σ dWt

• µ the mean return, σ the volatility, W a standard Brownian

motion

• denoting the default threshold barrier by b, say constant, define

the distance to default, Xt, as Xt = 1 σ ( logAt − logb ) .

• Merton: company defaults if XT < 0
• Black/Cox: default time τ is given by the first time Xt hits 0
• C. Reisinger – p.3

Framework Limit Results Extensions

Default probabilities

• 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

• .
• Φ the standard Gaussian CDF, m the risk-neutral drift of Xs
• C. Reisinger – p.4

Framework Limit Results Extensions

More companies

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

t = (r f − qi)Ai t dt + σiAi t dWi(t)

where

• r

f risk-free rate

• qi dividend yields
• σi volatilities
• Wi(t) Brownian motions and
• cov(Wi(t), Wj(t)) = ρijt.
• C. Reisinger – p.5

Framework Limit Results Extensions

More companies

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

t = (r f − qi)Ai t dt + σiAi t dWi(t)

where

• r

f risk-free rate

• qi dividend yields
• σi volatilities
• Wi(t) Brownian motions and
• 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.

• C. Reisinger – p.5

Framework Limit Results Extensions

Transformation

• Setting

Xi

t = ln

Ai

t

Ai e−γit

• = αit + σiWi(t)

with αi = r

f − qi − γi − 1 2σ2 i , leads to a Brownian motion with

drift and constant barrier Bi = ln

• bi(0)

Ai

• ≤ 0, Xi

0 = 0.

• C. Reisinger – p.6

Framework Limit Results Extensions

Transformation

• Setting

Xi

t = ln

Ai

t

Ai e−γit

• = αit + σiWi(t)

with αi = r

f − qi − γi − 1 2σ2 i , leads to a Brownian motion with

drift and constant barrier Bi = ln

• bi(0)

Ai

• ≤ 0, Xi

0 = 0.

• Defining the running minimum

Xi

t = min 0≤s≤t Xi s,

and default time, τi, as the first hitting time of the default barrier, τ i = inf{t : Xi

t = Bi},

the survival probability is then Q(τ i > s) = Q(Xi

s ≥ Bi).

• C. Reisinger – p.6

Framework Limit Results Extensions

2D case

For two firms, this problem is ‘analytically’ tractable, e. g. Q(t) = Q(X1

t ≥ B1, X2 t ≥ B2)

= 2 βtea1B1+a2B2+bt

∞

• n=1

e−r2

0/2t sin

nπθ0 β β sin nπθ β

• gn(θ) dθ

where gn(θ) = ∞ re−r2/2teA(θ)rI( nπ

β )

rr0 t

• dr

a1 = α1σ2 − ρα2σ1 (1 − ρ2)σ2

1σ2

, a2 = α2σ1 − ρα1σ2 (1 − ρ2)σ1σ2

2

b = −α1a1 − α2a2 + 1 2σ2

1a2 1 + ρσ1σ2a1a2 + 1

2σ2

2a2 2

tan β = −

• 1 − ρ2

ρ , β ∈ [0, π] etc, and I( nπ

β )

rr0

t

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

Framework Limit Results Extensions

CDS spread calculation

• The buyer of a kth-to-default credit default swap (CDS) on a

basket of n companies pays a premium, the CDS spread, for the life of the CDS – until maturity or the kth default.

• In the event of default by the kth underlying reference

company, the buyer receives a default payment and the contract terminates.

• C. Reisinger – p.8

Framework Limit Results Extensions

CDS spread calculation

• The buyer of a kth-to-default credit default swap (CDS) on a

basket of n companies pays a premium, the CDS spread, for the life of the CDS – until maturity or the kth default.

• In the event of default by the kth underlying reference

company, the buyer receives a default payment and the contract terminates. Equating discounted spread payment and discounted default payment,

DSP

= cK T e−r

f sQ(τk > s) ds

DDP

= (1 − R)K T e−r

f sQ(s ≤ τk ≤ s + ds)

gives the market kth-to-default CDS spread, ck, as ck = (1 − R)

• 1 − e−r

f T Q(τk > T) −

T

0 r fe−r

f sQ(τk > s) ds

• T

0 e−r

f sQ(τk > s) ds

.

• C. Reisinger – p.8

Framework Limit Results Extensions

Varying T

First-to-default CDS, varying T

−1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3 Correlation Spread 1 year 2 years 3 years 4 years 5 years

Second-to-default CDS, varying T

−1 −0.5 0.5 1 −0.2 0.2 0.4 0.6 0.8 1 1.2 Correlation Spread 1 year 2 years 3 years 4 years 5 years

σ1 = σ2 = 0.2, K1 = 100, r

f = 0.05, q1 = q2 = 0,

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

• C. Reisinger – p.9

Framework Limit Results Extensions

Varying σ

First-to-default CDS, varying σ

−1 −0.5 0.5 1 1 2 3 4 5 6 7 8 9 10 Correlation Spread σi=0.1 σi=0.15 σi=0.2 σi=0.25 σi=0.3

Second-to-default CDS, varying σ

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 Correlation Spread σi=0.1 σi=0.15 σi=0.2 σi=0.25 σi=0.3

K1 = 100, r

f = 0.05, q1 = q2 = 0, R = 0.5

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

• C. Reisinger – p.10

Framework Limit Results Extensions

Portfolio model

• Consider a portfolio of N different companies.
• Denoting the companies’ asset values at time t by Ai

t, we

assume that under the risk neutral measure they follow a diffusion process given by dAi

t = µ(t, Ai t) dt + σ(t, Ai t) dW i t + m

• j=1

σij(t, Ai

t) dM j t ,

• W i

t and M j t are Brownian motions satisfying

d

• W i

t , M j t

• = 0

∀i, j and d

• W i

t , W j t

• = d
• M i

t , M j t

• = δij dt.
• C. Reisinger – p.11

Framework Limit Results Extensions

Numerical PDE solution

Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms):

• C. Reisinger – p.12

Framework Limit Results Extensions

Numerical PDE solution

Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms):

• C. Reisinger – p.12

Framework Limit Results Extensions

Numerical PDE solution

Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms):

• C. Reisinger – p.12

Framework Limit Results Extensions

Numerical PDE solution

Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms): ‘Curse of dimensionality’:

• After n refinements, unknowns ∼ h−N ∼ 2nN
• Accuracy ǫ has complexity C(ǫ) ∼ ǫ−N/2 (for an order 2

method) → exponential growth of required computational resources

• C. Reisinger – p.12

Framework Limit Results Extensions

What is high dimensional?

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

230 = 1 073 741 824.

• C. Reisinger – p.13

Framework Limit Results Extensions

What is high dimensional?

Example, 30 firms: If each asset is represented by only two 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 already

• btained for

dim = 6.

• C. Reisinger – p.13

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’.

• C. Reisinger – p.14

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

• C. Reisinger – p.15

Framework Limit Results Extensions

Correlation data

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 0.3 0.41 0.4 0.42 0.23 0.42 0.27 0.41 0.03 0.41 0.12 0.15 5 10 15 20 25 30 0.005 0.01 0.05 0.1 0.5 1 l

k

σ(Σ) λ1

• Spectral gap after λ1
• Exponential decay from λ2
• 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.

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

νN,t = 1 N

N

• i=1

δAi

t

• Let φ ∈ C∞

c (R) and for measure νt write

φ, νt =

• φ(x)νt(dx).
• Define a family of processes F N,φ

t

by F N,φ

t

= φ, νN,t = 1 N

N

• i=1

φ(Ai

t).

• C. Reisinger – p.17

Framework Limit Results Extensions

Evolution of measure

• Applying Ito’s lemma to F N,φ

t

we have F N,φ

t

= 1 N

N

• i=1

t φ′(Ai

s)

 µ(s, Ai

s) dt + σ(s, Ai s) dW i s + m

• j=1

σij(s, Ai

s) dM j s

  + 1 N

N

• i=1

t 1 2φ′′(Ai

s)

 

m

• j=1

σ2

ij + σ2

  ds.

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

F N,φ

t

= F N,φ + t Aφ, νN,s ds + t

m

• j=1

σjφ′, νN,s dM j

s +

+ t 1 N

N

• i=1

φ′(Ai

s) σ(s, Ai s) dW i s.

A = µ ∂ ∂x + 1 2 ¯ σ2 ∂2 ∂x2 , ¯ σ2 =

m

• j=1

σ2

j + σ2,

• C. Reisinger – p.18

Framework Limit Results Extensions

Limiting equation

• The idiosyncratic component becomes deterministic in the

infinite dimensional limit, see also [Kurtz & Xiong, 1999].

• As the market factors affect each company to the same

degree, we can write F N,φ

t

→ F φ

t = φ, νt

as N → ∞,

with dF φ

t = Aφ, νt dt + m

• j=1

σjφ′, νt dM j

t .

• Alternatively, we can write this in integrated form as

φ, νt = φ, ν0 + t Aφ, νs ds +

m

• j=1

t σjφ′, νs dM j

s

• C. Reisinger – p.19

Framework Limit Results Extensions

Density

• Assume the measure νt to be absolutely continuous with

respect to the Lebesgue measure, to write νt(dx) = v(t, x)dx

• In differential form,

dv = −µ∂v ∂xdt + 1 2 ¯ σ2 ∂2v ∂x2 dt −

m

• j=1

∂ ∂x (σj(t, x)v) dM j

t .

• This is a stochastic PDE that describes the evolution of an

infinite portfolio of assets whose dynamics were given before.

• For solubility see [Krylov, 1994].
• Can now use this to approximate the loss distribution for a

portfolio of fixed size N.

• C. Reisinger – p.20

Framework Limit Results Extensions

Simplification

• Assume asset processes are correlated via a single factor.
• Specifically, let the SDEs for the asset processes be given by

dAi

t

Ai

t

= r dt +

• 1 − ρσ dW i

t + √ρσ dMt,

Ai

0 = ai,

where ρ ∈ [0, 1), d

• W i

t , Mt

• = 0.
• The distance-to-default

Xi

t = 1

σ

• log Ai

t − log Bi t

• evolves according to

dXi

t = µdt +

• 1 − ρdW i

t + √ρdMt,

Xi

0 = xi

with µ = 1

σ

• r − 1

2σ2

and xi = 1

σ

• log ai − log Bi
• .
• C. Reisinger – p.21

Framework Limit Results Extensions

Portfoilio loss

• Constant coefficient SPDE for v.
• The portfolio loss variable

LN

t = (1 − R) 1

N

N

• i=1

1{τi≤t} measures the losses at time t.

• Assume that a fraction R of losses, 0 ≤ R ≤ 1, the recovery

rate, is recovered after default.

• Absorbing barrier condition:

v(t, 0) = 0

for t ≥ 0.

• The loss distribution is then

Lt = (1 − R)

• 1 −

∞ v(t, x) dx

• .
• C. Reisinger – p.22

Framework Limit Results Extensions

Properties

• Matching the initial conditions for both measures, we need

v(0, x) = 1 N

N

• i=1

δAi

0.

• Given this initial condition,

L0 = N

• 1 − 1

N

N

• i=1

∞

Bt

δAi

• = 0,
• 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.

• C. Reisinger – p.23

Framework Limit Results Extensions

CDOs

• CDOs slice up the losses into tranches, defined by their

attachment and detachment points.

• Typical numbers are 0-3%, 3%-6%, 6%-9%, 9%-12%,

12%-22%, 22%-100%.

• For an attachment point a and detachment point d > a, the
• utstanding 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.

• Spreads are quoted as an annual payment, as a ratio of the

notional, but assumed to be paid quarterly.

a

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

• C. Reisinger – p.24

Framework Limit Results Extensions

Spread calculation

• Assume the notional to be 1, c the spread payment;
• n the maximum number of payments up to expiry T;
• Ti the payment dates for 1 ≤ i ≤ n, δ = 0.25 the interval

between payments.

• The expected sum of discounted fee payments, referred to as

the fee leg, is given by cV fee = c

n

• i=1

δe−rTiEQ[XTi],

• The protection leg is

V prot =

n

• i=1

e−rTiEQ[XTi−1 − XTi].

• 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

• Monte Carlo on top of a PDE solver computationally costly.
• For each simulated path of the market factor, a PDE needs to

be solved, e. g. by a finite difference method.

• From the PDE solution, quantities like tranche losses can be

computed,

• then averaged over the simulated paths.
• C. Reisinger – p.26

Framework Limit Results Extensions

Discrete defaults

• First make the assumption that defaults are only observed at a

discrete set of times,

• taken quarterly to coincide with the payment dates,
• i.e. if a firm’s value is below the default barrier on one of the
• bservation 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

• C. Reisinger – p.27

Framework Limit Results Extensions

Lagrangian coordinate

• The boundary condition is not active in intervals (Tk, Tk+1).
• The Brownian driver only introduces a random shift.

The solution can therefore be written as

v(t, x) = 8 < : 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.

• C. Reisinger – p.28

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 obtain

v(0)(T1, x).

• 3. Simulate MT1, evaluate v(T1, x).
• 4. For k > 0, having computed v(Tk, x) in the previous step, use

this as initial condition for v(k), and repeat until k = n.

• C. Reisinger – p.29

Framework Limit Results Extensions

Finite differences

• The initial distribution is assumed localised.
• Approximate the distribution by one with support [xmin, xmax]

with xmin < 0 and xmax > 0.

• Can ensure that the expected error of this approximation is

much smaller than the standard error of the Monte Carlo estimates.

• Grid x0 = xmin, x1 = xmin + ∆x, . . . , xmin + j∆x, . . . , xJ =

xmin + J∆x = xmax, where ∆x = (xmax − xmin)/J,

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

Define an approximation vi

j to v(ti, xj) as solution to a FD/FE

scheme with θ timestepping.

• C. Reisinger – p.30

Framework Limit Results Extensions

Stability

• 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 = 1

2 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 for

Dirac initial conditions, and

• reduces the convergence order for discontinuous intitial

conditions.

• C. Reisinger – p.31

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 to

replace the first two Crank-Nicolson steps by four backward Euler steps of half the stepsize.

• We do this at t = 0, and also at t = Tk where the interface

conditions introduce discontinuities at x = 0.

• This restores second order convergence in time.
• C. Reisinger – p.32

Framework Limit Results Extensions

Averaging

• The sum of δ-distributions needs approximation on the grid.
• Could collect the firms into symmetric intervals of width ∆x

around grid points, v0

j =

1 N∆x xj+∆x/2

xj−∆x/2

δ(xi − x) dx,

• however, this reduces the overall order of the finite difference

scheme to 1:

• The approximation cannot distinguish between initial positions

xi in an interval of length ∆x.

• C. Reisinger – p.33

Framework Limit Results Extensions

Projection

• To achieve higher (i. e. second) order, the δ’s are split between

adjacent grid points. The correct weighting for a single firm with distance-to-default xi in the interval [xj, xj+1), is v0

k =

   ∆x−2(xj+1 − xi) k = j ∆x−2(xi − xj) k = j + 1 else

• This can be written more elegantly as L2-projection onto the

basis of ‘hat functions’ Φk0≤k≤N where Φk(x) = 1 ∆x min (max(x − xk + ∆x), max(xk − x, 0)) , then v0

k = Φk, v0 =

xmax

xmin

Φk(x)v0(x) dx.

• Note that ∆x N

k=0 v0 k = 1.

• C. Reisinger – p.34

Framework Limit Results Extensions

Interface condition

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

• First, define a piecewise linear reconstruction from the

approximation vI

k obtained in the last step over the previous

interval [Tk−1, Tk], as v∆x(Tk, x) =

N

• j=0

Φj(x − ∆M)vI

j .

• Then, approximate the shift by setting

v0

j =

xj+∆x/2

max(xj−∆x/2,0)

v∆x(Tk, x) dx, with ∆M = MTk − MTk−1.

• C. Reisinger – p.35

Framework Limit Results Extensions

Interface condition

• Use this as initial condition for the next interval (the integral is

understood to be 0 if the lower limit is larger than the upper limit).

• This ensures that

∆x

J

• j=0

v0

j =

xmax vh(Tk, x) dx, so the cumulative density of firms with firm values greater than 0 is preserved.

• Also, the solution is smoothened at x = 0
• C. Reisinger – p.36

Framework Limit Results Extensions

Simulations

• For a given realisation of the market factor, we can

approximate the loss functional LTk at time Tk by L∆x

Tk = 1 −

xmax v∆x(Tk, x) dx

• Explicitly include the dependency L∆x

Tk (Φ), where Φi are drawn

independently from a standard normal distribution,

• Then for Nsims simulations with samples Φl = (Φl

k)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∆x

Tk (Φl), 0) − max(a − L∆x Tk (Φl), 0)

• C. Reisinger – p.37

Framework Limit Results Extensions

Data

• The simulation error has two components, the discretisation

error 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 individual

CDS spreads,

• were well within the range [xmin, xmax] = [−10, 20].
• C. Reisinger – p.38

Framework Limit Results Extensions

Grid convergence

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

10

10

10

10

10

10

10

10

10

10

10

J ǫJ

10 10

10

10

10

10

10

10

10

10

I ǫI

• Extrapolation-based estimator for the discretisation error of

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

• We clearly see second order convergence in ∆t and ∆x.
• C. Reisinger – p.39

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 7.8000000 2 7.9104e-03 2.2701e-02 2.73422e-02 2.95842e-02 9.985866e-02 7.8000000 3 7.385e-03 2.2685e-02 2.82221e-02 2.97561e-02 9.996077e-02 7.8000000 4 7.6036e-03 2.2834e-02 2.80317e-02 2.95556e-02 9.987046e-02 7.8000000 5 7.5088e-03 2.2772e-02 2.80370e-02 2.95103e-02 9.984416e-02 7.7999902 6 7.4316e-03 2.2754e-02 2.80541e-02 2.94878e-02 9.982312e-02 7.7999919 7 7.3909e-03 2.2683e-02 2.80568e-02 2.94938e-02 9.982567e-02 7.7999904 8 7.4092e-03 2.2681e-02 2.80461e-02 2.94908e-02 9.982834e-02 7.7999870 9 7.4051e-03 2.2676e-02 2.80520e-02 2.94934e-02 9.982809e-02 7.7999860 10 7.4068e-03 2.2678e-02 2.80526e-02 2.94938e-02 9.982806e-02 7.7999866 Monte Carlo estimates for expected outstanding tranche notionals for Nsims = 64 ∗ 4s−1 Monte Carlo runs, s = 1, ..., 10. The finite difference parameters were fixed at J = 256, I = 4.

• C. Reisinger – p.40

Framework Limit Results Extensions

MC convergence

2 4 6 8 10 12 14 0.005 0.01 0.015 0.02 0.025 0.03 0.035

log4 Nsims EQ[XTn], a = 0, b = 0.03

2 4 6 8 10 12 14 1 2 3 x 10

log4 Nsims EQ[XTn], a = 0.06, b = 0.09

2 4 6 8 10 12 14 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

log4 Nsims EQ[XTn], 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.
• C. Reisinger – p.41

Framework Limit Results Extensions

Index spreads

The fixed coupons, traded spreads and model spreads for the iTraxx 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 are r = 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 are r = 0.033, σ = 0.136, R = 0.4.

• C. Reisinger – p.42

Framework Limit Results Extensions

Implied correlation

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

3% 6% 9% 12% 22% 100% 10 20 30 40 50 60 70 Tranche Detachment Point Implied Corrleation (%) 5 Year 7 Year 10 Year

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

• C. Reisinger – p.43

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 % 4 3%-6% 1576.3 2275.2 1978.5 1743.2 1546.8 1374.6 1222.8 1 6%-9% 811.5 1273.1 1168.2 1079.7 1001.4 931.3 864.6 7 9%-12% 506.1 775.7 765.8 748.6 724.7 695.8 663.2 6 12%-22% 180.3 307.8 353.3 384.7 405.5 418.1 423.4 4 22%-100% 77.9 9.2 16.5 25 34.3 44.5 55.7 6 Model tranche spreads (bp) for varying values of the correlation parameter. The equity 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. Market levels shown are for this date; model parameters are r = 0.033, σ = 0.136, R = 0.4.

• C. Reisinger – p.44

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 % 4 3%-6% 1473.2 2327.3 1985.7 1715.2 1493.4 1308 1147.8 1 6%-9% 804.2 1344.2 1199 1085.2 988.2 900.7 820.9 7 9%-12% 512.4 855.4 808.4 765.3 725.3 684.8 643 6 12%-22% 182.6 375.4 401.7 417.6 425.6 427.4 423.1 4 22%-100% 75.8 14 22 30.6 39.6 49.3 59.7 7 Model tranche spreads (bp) for varying values of the correlation parameter. The equity 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. Market levels shown are for this date; model parameters are r = 0.033, σ = 0.136, R = 0.4.

• C. Reisinger – p.45

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 Model tranche spreads (bp) for varying values of the correlation parameter. The equity 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. Market levels shown are for this date; model parameters are r = 0.033, σ = 0.136, R = 0.4.

• C. Reisinger – p.46

Framework Limit Results Extensions

Extensions

• Dynamic properties, forward starting CDOs, options on

tranches,...

• 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 Monte

Carlo.

• Convergence analysis through mean-square stability.
• Importance sampling for senior tranches, with Tom Dean.
• C. Reisinger – p.47

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 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

with importance sampling

1 2 3 4 5 6 7 8 9 10 0.045 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

• C. Reisinger – p.48

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 11 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 x 10

with importance sampling

2 4 6 8 10 12 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 x 10

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

• C. Reisinger – p.48

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 in Series of Fourier-Hermite Functionals, Ann. Math, 48, 385-392. [Giles & Carter(2006)] Giles, M. & Carter, R. (2006) Convergence analysis of Crank-Nicolson and Rannacher time-marching, J.

• Comp. Fin., 9(4), 89–112.

[Hambly & Jin (2008)] Hambly, B.M. & Jin, L. (2008) SPDE approximations for large basket portfolio credit modelling, in preparation. [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 Credit

• C. Reisinger – p.49