Estimating Joint Default Probabilities by Efficient Importance - - PowerPoint PPT Presentation

Estimating Joint Default Probabilities by Efficient Importance Sampling

Chuan-Hsiang Han

• Dept. of Quantitative Finance
• Natl. Tsing-Hua University, Taiwan

WSAF, Hong Kong July 3, 2009

Outline

• Credit Risk Modeling: Classical Models
• Monte Carlo Simulations, Importance Sam-

pling, and Large Deviations

• Homogenization by Singular Perturbation

and Effect of (Stochastic) Correlation

1

Motivation I: Loss Density Function A loss random variable is defined by L(T) =

N

• i=1

ci I(τi ≤ T), so that, say the VaR, can be calculated from solving

P (L(T) > VaRα) = α.

2

Motivation II: Evaluation of Credit Swaps premium = I E {(1 − R) × B(0, τ) × I(τ < T)} I E

N

j=1 △j−1, j × B(0, tj) × I(τ > tj)

• Notations: τ: default time, R: recovery rate,

B(0, t): discount factor, △j−1, j: time incre- ment. CDS: τ is the time to default of an asset. CDO: τ is often an order statistics of τ1, τ2, · · · , τn.

3

Modeling Default Times

• Intensity-Based (Reduced Form)

View firm’s default as extraneous and model the hazard rate of the firm. I P(τ ≤ t) = F(t) = 1 − exp

• −

t

0 h(s)ds

• .
• Asset Value-Based (Structural Form)

Firm assets follow correlated geometric Brow- nian motions: dSi t = µi Si t dt+σi Si tdWi t, d < Wi, Wj >= ρijdt. Joint default event: Πn

i=1I(Si T ≤ Bi) (Mer-

ton’s Model).

4

Then we ask a basic question JDP = E

• Πn

i=1I(τi ≤ T)

• ?

And hope this leads to the estimation of (1) P(L(T) = i) = pi, (2) P(τ ≤ T), where τ is the k-th oder statis- tics of {τi}n

i=1 . JDP = joint default probability

5

Correlation under Reduced Form Model: Copula Method∗ 1. Default Time:

• τi = F −1

i

(Ui)

n

i=1 , U’s

are [0,1]-uniform random variables.

• 2. Copula is a distribution function on [0, 1]n

with uniform marginal distributions. 3. Through a copula function, one can build up correlations between default times.

∗Cherubini,

Luciano, Vecchiato (2004), Nel- son(2006).

6

Characterization of Default Events Gaussian Copula Factor Model (Laurent and Gregory (2003)) {τi = F −1

i

(Φ(Wi)) ≤ T} =

• Wi = ρiZ0 +
• 1 − ρ2

i Zi ≤ Φ−1(Fi(T))

• =

    Z0 ≤

Φ−1(Fi(T)) −

• 1 − ρ2

i zi

ρi

     whenZi = zi

=

    Zi ≤ Φ−1(Fi(T)) − ρiz0

• 1 − ρ2

i

     whenZ0 = z0.

7

(Conditional) Importance Sampling Estimate the JDP I E

n

i=1 I(τi ≤ T)

• by

(1) Condition on common factor I E

• ˜

I E(u1,···,un)

• Πn

i=1I(Zi ≤ ci − ρiZ0

• 1 − ρ2

i

)Πn

i=1L(Zi; ui) | Z0

• (2) Condition on marginal factors

I E

• ˜

I E

• Πn

i=1I

• Z0 ≤ ci −
• 1 − ρ2

i Zi

ρi

• L(Z0; u)|Z1, · · · , Zn
• (3) Direct Change of Measure

˜ I E

n

i=1 I(Wi ≤ ci)Πn i=1L(Wi; wi)

• ,

where ci = Φ−1(Fi(T)) and L(·, ·) the Likeli- hood ratio.

8

Comparison of Variance Reduction

9

Asymptotic Optimality of Direct Change of Measure Let W be a centered multivariate normal JDP = E{I(W < c)} = Eµ {I(W < c)Πn

i=1L(Wi; µi)} ,

where L(w; µ) = eµ2/2e−µw is the likelihood function. Theorem The variance of I{W<c}Πn

i=1L(Wi; µi)

is optimally minimized at µ = c when each component in the vector −c is sufficiently large.

10

More Applications

• Computation of tail probability for multi-

variate normals. (versus a Matlab program mvncdf.m, based on Genz and Bretz (’99))

• Connection to Merton’s structural-form model

model in high dimension to estimate I E {Πn

i=1I(SiT ≤ Bi)}

• Extend to Student-t copula (Conditional

IS) (see also mvtcdf.m)

11

Tail Probability Estimation: N=5, ρ=0.5 Multivariate Normal D

• log (P_MC)
• log (P_IS)
• log(P_MATLAB)
• 0.5
• 1
• 1.5
• 2
• 2.5
• 3
• 3.5
• 4
• 4.5
• 5
• 5.5
• 6

1.05 1.05 1.04 1.64 1.67 1.68 2.55 2.51 2.52 3.70 3.59 3.57 4.87 4.85 6.34 6.37 8.11 8.13 10.14 10.14 12.43 12.40 14.88 14.92 17.65 17.68 20.49 20.71 23.99 23.98

7.5 15.0 22.5 30.0 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5 -6

IS versus QMC

• log(P,10)

D: Default Level

• log (P_MC)
• log (P_IS)
• log(P_MATLAB)

12

D

• log (P_MC)
• log (P_IS)
• log (P_MATLAB)

0.0

• 0.5
• 1.0
• 1.5
• 2.0
• 2.5
• 3.0
• 3.5
• 4.0
• 4.5
• 5.0
• 5.5
• 6.0

1.046 1.041 1.04 1.622 1.644 1.636 2.330 2.335 2.317 3.018 3.082 3.021 3.553 3.676 3.717 4.398 5.102 4.358 4.398 4.866 4.914 5.41 5.43 5.712 5.836 6.693 6.236 7.650 6.907 7.076 8.038 8.166 7.077

2.5 5.0 7.5 10.0 0.0 -0.5

• 1.5
• 2.5
• 3.5
• 4.5
• 5.5

IS versus QMC

• log(P,10)

D: Default Level

• log (P_MC)
• log (P_IS)
• log (P_MATLAB)

文字

Tail Probability Estimation: N=5, ρ=0.5 d.f.=10 (Multivariate Student-T)

13

Credit Risk Modeling: Structural Form Approach Multi-Names Dynamics: for 1 ≤ i ≤ n dSit = µiSitdt + σi SitdWit, d

• Wit, Wjt
• = ρijdt.

Each default time τi for the ith name is de- fined as τi = inf{t ≥ 0 : Sit ≤ Bi}, where Bi denotes the ith debt level. The ith default event is defined as {τi ≤ T}.

14

Joint Default Probability: First Passage Time Problem in High Dim. Q: How to compute JDP = I E

• Πn

i=1I(τi ≤ T)

• under structural-form models?

Explicit Formulas exist for 1-name case (Black and Cox ’76) and 2-name case (Zhou ’01). Note that Carmona, Fouque, and Vestal (’08) dealed with a similar problem by means of Interacting Particle Systems.

15

Multi-Dimensional Girsanov Theorem Given the Radon-Nikodym derivative dI P d ˜ I P = Qh

T = e

T

0 h(s,Ss)·d ˜

Ws−1

2

T

0 ||h(s,Ss)||2ds

• ,

˜ Wt = Wt +

t

0 h(s, Ss)ds is a vector of Brown-

ian motions under ˜ I

• P. Thus

DP = ˜ I E

• Πn

i=1I(τi ≤ T)Qh T

• .

16

Monte Carlo Simulations: Importance Sampling An importance sampling method is devel-

• ped to satisfy

˜ I E {SiT|F0} = Bi, i = 1, · · · , n. The new measure is characterized by solving the linear system Σi

j=1ρijhj = µi σi − ln Bi/Si0 σi T

so that by Girsanov Theorem JDP = ˜ I E {Πn

i=1I(τi ≤ T) QT} .

17

Trajectories under different measures Single Name Case

18

Single Name Default Probability

B BMC Exact Sol Importance Sampling 50 0.0886 (0.0028) 0.0945 0.0890 (0.0016) 20 0 (0) 7.7 ∗ 10−5 7.2 ∗ 10−5(2.3 ∗ 10−6) 1 0 (0) 1.3 ∗ 10−30 1.8 ∗ 10−30(3.4 ∗ 10−31)

The number of simulations is 104 and the Euler discretization takes time step size T/400, where T is one year. Other parameters are S0 = 100, µ = 0.05 and σ = 0.4. Standard errors are shown in parenthesis. 19

Asymptotic Optimality Efficient Importance Sampling Assume ln (Bi/Si0) = −1

ε

for each 1 ≤ i ≤ N. Let the JDP, Pε, and the second moment, M2ε under a new measure, are defined by Pε = I E

• ΠN

i=1I

• inf

0≤t≤T Sit ≤ Bi

• M2ε

= ˜ I E

• ΠN

i=1I

• inf

0≤t≤T Sit ≤ Bi

• QT
• Theorem: By M2ε ≈ (Pε)2 for small ε (spa-

tial scale) we observe the optimality of cho- sen measure.

20

Sketch of Proof: Single-Name Default

• Prob. Approximation

I P

  inf

0≤t≤T St = S0 e

• µ−σ2

2

• t+σWt ≤ B

 

(scaling in space by ln (B/S0) = −1 ε ) = I E

• I
• inf

0≤t≤T ε

• µ − σ2

2

• t + εσWt ≤ −1
• :=

Pε ≈ exp

• −1

ε2 2 σ2 T

• .( by Freidlin-WentzellThm )

21

Importance Sampling: 2nd Moment Approximation ˜ I E

• I
• inf

0≤t≤T St ≤ B

• e2 h ˜

WT −h2 T

• St = S0 e
• µ−σ2

2 −σh

• t+σ ˜

Wt, h = µ

σ − ln B/S0 σ T = ˆ I E

 I   inf

0≤t≤T S0 e

• µ−σ2

2 +σh

• t+σ ˆ

Wt ≤ B

    eh2 T

22

2nd Moment Approximation (Cont.) = ˆ I E

• I
• inf

0≤t≤T

• ε
• 2µ − σ2

2

• + 1

T

• t + εσ ˆ

Wt ≤ −1

• × e
• r

σ+ 1 εσT

2

T

(scaling by ln (B/S0) = −1 ε ) := M2ε ≈ exp

• −1

ε2 σ2 T

• .

( by F-W Thm ) Observe that M2ε ≈ (Pε)2 when ε is small.

23

Three-Names Joint Default Probability

ρ BMC Importance Sampling 0.3 0.0049(6.98 ∗ 10−4) 0.0057(1.95 ∗ 10−4) 3.00 ∗ 10−4(1.73 ∗ 10−4) 6.40 ∗ 10−4(6.99 ∗ 10−5)

• 0.3

0(0) 2.25 ∗ 10−5(1.13 ∗ 10−5)

Parameters are S10 = S20 = S30 = 100, µ1 = µ2 = µ3 = 0.05, σ1 = σ2 = 0.4, σ3 = 0.3 and B1 = B2 = 50, B3 = 60. Effect of Correlations! Debt to Asset-Value Ratios (Bi/Si0) are not small.

24

Loss Density Function: 25 Names Diffusion Model Note: Consider both survival and default prob- abilities. Applications: Pricing CDOs, Risk Manage- ment of credit portfolios, etc.

25

Loss Density Function: 25 Names Jump Diffusion Model

5 10 15 20 25 10

10

10

10

10

10 Loss density function of N=25 Number of defaults P(L=n)

Use the compound poisson jump as a com- mon factor. But optimal efficiency can not be obtained.

26

A Modification: Stochastic Correlation∗

        

dS1

t = rS1 t dt + σ1S1 t dW 1 t

dS2

t = rS2 t dt + σ2S2 t (ρ(Yt)dW 1 t +

• 1 − ρ2(Yt)dW 2

t )

dYt = 1

ε(m − Yt)dt + √ 2β √ε dZt (Scaling in Time)

Joint default probability P ε(t, x1, x2, y) := I Ex1,x2,y

• Π2

i=1 I( min t≤u≤T Si u ≤ Bi)

• ∗Hull, Presescu, White (2005)

27

Full Expansion of P ε Theorem P ε(t, x1, x2, y) =

∞

• i=0

εiPi(t, x1, x2, y), where P ′

is can be obtained recursively by solv-

ing a seq. of Poisson eqns. Proof: by means of Singular Perturbation Techniques. Accuracy results are ensured given smooth- ness of terminal condition.

28

Leading Order Term P0(t, x1, x2) solves the homogenized PDE (y-independent).

• L1,0 + ¯

ρ L1,1

• P0(t, x1, x2) = 0

¯ ρ =< ρ(y) >, average taken wrt the invar- tiant measure of Y. Differential operators are L1,0 = ∂ ∂t +

2

• i=1

σ2

i x2 i

2 ∂2 ∂x2

i

+

2

• i=1

µixi ∂ ∂xi L1,1 = σ1σ2x1x2 ∂2 ∂x1∂x2 .

29

Other Terms Pn+1(t, x1, x2, y) =

i+j=n+1

• i≥0,j≥1

ϕ(n+1)

i,j

(y) Li

1,0 Lj 1,1 Pn

where a seq. of Poisson eqns to be solved: L0 ϕ(n+1)

i+1,j (y) =

• ϕ(n)

i,j (y)− < ϕ(n) i,j (y) >

• L0 ϕ(n+1)

i,j+1 (y) =

• ρ(y) ϕ(n)

i,j (y)− < ρ ϕ(n) i,j >

• ,

where L0 = β2 ∂2

∂y2 + (m − y) ∂ ∂y.

30

Stochastic Correlation I

α = 1

ε

BMC Importance Sampling 0.1 0.0037(6 ∗ 10−4) 0.0032(1 ∗ 10−4) 1 0.0074(9 ∗ 10−4) 0.0065(2 ∗ 10−4) 10 0.0112(1 ∗ 10−3) 0.0116(4 ∗ 10−4) 50 0.0163(1 ∗ 10−3) 0.0137(5 ∗ 10−4) 100 0.016(1 ∗ 10−3) 0.0132(4 ∗ 10−4)

Parameters are S10 = S20 = 100, B1 = 50, B2 = 40, m = π/4, ν = 0.5, ρ(y) = |sin(y)|. Using the homogenized term in IS, note the effect of correlation.

31

Stochastic Correlation II

α = 1

ε

BMC Importance Sampling 0.1 0(0) 9.1 ∗ 10−7(7 ∗ 10−8) 1 0(0) 7.5 ∗ 10−6(6 ∗ 10−7) 10 0(0) 2.4 ∗ 10−5(2 ∗ 10−6) 50 1 ∗ 10−4(1 ∗ 10−4) 2.9 ∗ 10−5(3 ∗ 10−6) 100 1 ∗ 10−4(1 ∗ 10−4) 2.7 ∗ 10−5(2 ∗ 10−6)

Parameters are S10 = S20 = 100, B1 = 30, B2 = 20, m = π/4, ν = 0.5. Note the effect of correlation.

32

Conclusions

• Simple yet efficient importance sampling

methods are proposed, justified by large deviation theory.

• Full expansion of joint default probability

under stochastic correlation by singular perturbation.

• Of course all these ideas can be applied

to option pricing...

33

European Vanilla Call Option Prices S0 Martingale C.V. Efficient I.S. 10 2.27E − 07 3.29E − 10 40 2.63E − 03 1.722E − 03 80 0.0156 0.1163 120 0.0223 0.0728 160 0.0234 0.0215 200 0.0252 0.0069 Model Parameters: K = 100, σ = 0.4, r = 0.05, T = 1.

34

Acknowledgments

• S.-J. Sheu(Academia Sinica), N.-R. Shieh(NTU),

Kan Lee (NCTU), Doug Vestal(Julius, NY).

Thank You

35